U.S. patent application number 11/090999 was filed with the patent office on 2005-09-01 for method for synchronizing symbol timing.
This patent application is currently assigned to Pentomics, Inc.. Invention is credited to Fu, Dengwei, Willson, Alan N. JR..
Application Number | 20050193047 11/090999 |
Document ID | / |
Family ID | 34315934 |
Filed Date | 2005-09-01 |
United States Patent
Application |
20050193047 |
Kind Code |
A1 |
Fu, Dengwei ; et
al. |
September 1, 2005 |
Method for synchronizing symbol timing
Abstract
A method of synchronizing symbol timing in a digital device
includes the steps of receiving complex data samples of one or more
symbols, correlating the data samples with a complex conjugate of a
preamble data set, selecting between real and imaginary samples and
generating a complex number based on the result. The method then
determines an angle in a complex plane that represents symbol
synchronization for the digital device. Disclosed embodiments
include applications to communications devices, and to carrier
recovery in such devices. In such embodiments the resultant angle
may represent a carrier phase offset for the device.
Inventors: |
Fu, Dengwei; (Santa Clara,
CA) ; Willson, Alan N. JR.; (Pasadena, CA) |
Correspondence
Address: |
STERNE, KESSLER, GOLDSTEIN & FOX PLLC
1100 NEW YORK AVENUE, N.W.
WASHINGTON
DC
20005
US
|
Assignee: |
Pentomics, Inc.
Pasadena
CA
|
Family ID: |
34315934 |
Appl. No.: |
11/090999 |
Filed: |
March 28, 2005 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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11090999 |
Mar 28, 2005 |
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09698249 |
Oct 30, 2000 |
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6874006 |
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60162391 |
Oct 29, 1999 |
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Current U.S.
Class: |
708/300 |
Current CPC
Class: |
H04L 27/22 20130101 |
Class at
Publication: |
708/300 |
International
Class: |
G06F 017/10 |
Goverment Interests
[0005] This invention was made with Government support under grant
no. MIP 9632698 awarded by the National Science Foundation. The
U.S. Government has certain rights in this invention.
Claims
We claim:
1. In a digital device, a method of symbol timing synchronization,
the method comprising the steps of: (1) receiving complex data
samples of one or more symbols; (2) correlating said complex data
samples with a complex conjugate of a preamble data set, resulting
in correlated complex data samples, each correlated complex data
sample represented by a real sample and an imaginary sample; (3)
selecting between said real samples and said imaginary samples,
resulting in a set of selected samples; (4) generating a complex
number based on said set of selected samples; and (5) determining
an angle in a complex plane associated with said complex number,
whereby said angle represents symbol synchronization for the
digital device.
2. The method of claim 1, further comprising the step of: (6)
multiplying said angle by .pi./2 to determine an offset .mu. that
indicates symbol synchronization.
3. The method of claim 1, wherein step (2) comprises the step of
multiplying said received complex data samples with said preamble
data set.
4. The method of claim 1, wherein step (3) comprises the step of
selecting the larger of said real samples and said imaginary
samples.
5. The method of claim 1, wherein step (4) comprises the steps of:
(a) determining a Fourier transform based on said set of selected
data samples; and (b) evaluating said Fourier transform at
.pi./2.
6. The method of claim 1, wherein step (4) comprises the steps of:
(a) determining which of said selected data samples has the largest
magnitude; (b) selecting n adjacent samples from the selected data
samples that includes said largest magnitude sample; (c)
determining a Fourier transform of said n adjacent data samples;
and (d) evaluating said Fourier transform at .pi./2, resulting in
said complex number.
7. The method of claim 1, wherein said complex number is in a
rectangular format, and wherein step (5) comprises the step of:
converting said complex number to polar format having a magnitude
and said angle.
8. The method of claim 1, where step (4) comprises the steps of:
(a) determining which of said selected data samples has the largest
magnitude; (b) selecting four adjacent samples from the selected
data samples, represented by r(-1), r(0), r(1), and r(2), wherein
said largest magnitude data sample is one of r(0) and r(1); (c)
determining a Fourier transform of said four adjacent data samples;
and (d) evaluating said Fourier transform at .pi./2, resulting in
said complex number.
9. The method of claim 8, wherein step (c) comprises the steps of:
(i) determining r(0)-r(2), to produce a real part of said complex
number; and (ii) determining r(-1)-r(1), to produce an imaginary
part of said complex number.
10. In a digital device, a method of carrier recovery, the method
comprising the steps of: (1) receiving complex data samples of one
or more symbols; (2) correlating said complex data samples with a
complex conjugate of a preamble data set, resulting in correlated
complex data samples; (3) selecting one of said correlated complex
data samples; (4) determining an angle in said complex plane based
on said selected correlated complex data sample, whereby said angle
represents a carrier phase offset in the digital device.
11. The method of claim 10, wherein step (3) comprises the step of
selecting a largest of said correlated complex data samples.
12. The method of claim 10, wherein said complex number is in a
rectangular format, wherein step (4) comprises the step of
converting said complex number to a polar format having a magnitude
and said angle.
Description
[0001] This application claims the benefit of U.S. Provisional
Application No: 60/162,391, filed on Oct., 29, 1999, which is
incorporated herein by reference.
CROSS-REFERENCE TO OTHER APPLICATIONS
[0002] The following applications of common assignee are related to
the present application, have the same filing date as the present
application, and are herein incorporated by reference in their
entireties:
[0003] "Apparatus and Method for Trigonometric Interpolation,";
and
[0004] Apparatus and Method for Angle Rotation,".
BACKGROUND OF THE INVENTION
[0006] 1. Field of the Invention
[0007] The present invention is related to digital signal
processing and digital communications. More specifically the
present invention is related to interpolation, angle rotation,
rectangular-to-polar conversion, and carrier and symbol timing
recovery for digital processing and digital communications
applications.
[0008] 2. Related Art
[0009] Advances in technology have enabled high-quality, low-cost
communications with global coverage, and provide the possibility
for fast Internet access and multimedia to be added to existing
services. Exemplary emerging technologies include cellular mobile
radio and digital video broadcasting, both of which are described
briefly as follows.
[0010] In recent years, cellular mobile radio has experienced rapid
growth due to the desire for mobility while enjoying the two-way
voice services it provides. GSM, IS-136 and personal digital
cellular (PDC) are among the most successful second-generation
personal communications (PCS) technologies in the world today, and
are responsible for providing cellular and PCS services globally.
As the technology advances, customers will certainly demand more
from their wireless services. For example, with the explosive
growth of the world wide web over the wired networks, it is
desirable to provide Internet services over mobile radio networks.
One effort to specify the future global wireless access system is
known as IMT-2000 (Buchanan, K., et al., IEEE Pers. Comm. 4:8-13
(1997)). The goal of IMT-2000 is to provide not only traditional
mobile voice communications, but also a variety of voice and data
services with a wide range of applications such as multimedia
capabilities, Internet access, imaging and video conferencing. It
is also an aim to unify many existing diverse systems (paging,
cordless, cellular, mobile satellite, etc.) into a seamless radio
structure offering a wide range of services. Another principle is
to integrate mobile and fixed networks in order to provide fixed
network services over the wireless infrastructure. Such systems
might well utilize broadband transport technologies such as
asynchronous transfer mode (ATM).
[0011] For the applications of IMT-2000, a high-bit-rate service is
needed. Moreover, for multimedia applications, the system should
provide a multitude of services each requiring 1) a different rate,
and 2) a different quality-of-service parameter. Thus, a flexible,
variable-rate access with data rates approaching 2 Mb/s is proposed
for IMT-2000.
[0012] The advent of digital television systems has transformed the
classical TV channel into a fast and reliable data transmission
medium. According to the specifications of the DVB project
(Reimers, U., IEEE Comm. Magazine 36:104-110 (1998)), digital TV is
no longer restricted to transmitting sound and images but instead
has become a data broadcasting mechanism which is fully transparent
to all contents. Digital TV broadcasting by satellite, cable and
terrestrial networks is currently under intensive development. A
typical system looks like this: a DVB signal is received from a
satellite dish, from cable, or from an antenna (terrestrial
reception). A modem built into an integrated receiver/decoder (IRD)
will demodulate and decode the signal. The information received
will be displayed on a digital TV or a multimedia PC. In addition
to being used as a digital TV, DVB can receive data streams from
companies who wish to transmit large amounts of data to many
reception sites. These organizations may be banks, chains of retail
stores, or information brokers who wish to offer access to selected
Internet sites at high data rates. One such system is MultiMedia
Mobile (M.sup.3), which has a data rate of 16 Mb/s.
[0013] For proper operation, these third generation systems require
proper synchronization between the transmitter and the receiver.
More specifically, the frequency and phase of the receiver local
oscillator should substantially match that of the transmitter local
oscillator. When there is a mismatch, then an undesirable rotation
of the symbol constellation will occur at the receiver, which will
seriously degrade system performance. When the carrier frequency
offset is much smaller than the symbol rate, the phase and
frequency mismatches can be corrected at baseband by using a phase
rotator. It is also necessary to synchronize the sampling clock
such that it extracts symbols at the correct times. This can be
achieved digitally by preforming appropriate digital resamples.
[0014] The digital resampler and the direct digital frequency
synthesizer (DDS) used by the phase rotator are among the most
complex components in a receiver (Cho, K., "A frequency-agile
single-chip QAM modulator with beamforming diversity," Ph.D.
dissertation, University of California, Los Angeles (1999)). Their
performance is significant in the overall design of a
communications modem. For multimedia communications, the
high-data-rate requirement would impose a demand for high
computational power. However, for mobile personal communication
systems, low cost, small size and long battery life are desirable.
Therefore, it would be desirable to have an efficient
implementation of the phase rotator, re-sampler, and DDS in order
to perform fast signal processing that operates within the
available resources. Furthermore, it would be desirable to have an
efficient synchronization mechanism that uses a unified approach to
timing and carrier phase corrections.
[0015] For Internet services it is important to provide
instantaneous throughput intermittently. Packet data systems allow
the multiplexing of a number of users on a single channel,
providing access to users only when they need it. This way the
service can be made more cost-effective. However, the user data
content of such a transmission is usually very short. Therefore, it
is essential to acquire the synchronization parameters rapidly from
the observation of a short signal-segment.
[0016] For applications where low power and low complexity are the
major requirements, such as in personal communications, it is
desirable to sample the signal at the lowest possible rate, and to
have a synchronizer that is as simple as possible. Therefore, it is
also desirable to have an efficient synchronizer architecture that
achieves these goals.
[0017] For applications utilizing Orthogonal Frequency Division
Multiplexing (OFDM), sampling phase shift error produces a rotation
of the Fast Fourier Transform (FFT) outputs (Pollet T., and Peters,
M., IEEE Comm. Magazine 37:80-86 (1999)). A phase correction can be
achieved at the receiver by rotating the FFT outputs. Therefore, it
is also desirable to have an efficient implementation structure to
perform rotations of complex numbers.
SUMMARY OF THE INVENTION
[0018] The present invention is directed at a
rectangular-to-polar-convert- er that receives a complex input
signal (having X.sub.0 and Y.sub.0 components) and determines an
angle .phi., which represents the position of the complex input
signal in the complex plane. In doing so, the rectangular-to-polar
converter determines a coarse angle .phi..sub.1 and a fine angle
.phi..sub.2, where .phi.=.phi..sub.1+.phi..sub.2.
[0019] The coarse angle .phi..sub.1 is obtained using a small
arctangent table and a reciprocal table. These tables provide just
enough precision such that the remaining fine angle .phi..sub.2 is
small enough to approximately equal its tangent value. Therefore
the fine angle .phi..sub.2 can be obtained without a look-up table,
and the fine angle computations are consolidated into a few small
multipliers, given a precision requirement.
[0020] More specifically, the coarse angle computation is performed
by retrieving a pre-computed Z.sub.0=1/[X.sub.0] value from a
reciprocal lookup table (e.g. memory device), where [X.sub.0] is a
bit truncated approximation of X.sub.0. The Z.sub.0 value is
multiplied by the Y.sub.0 component, resulting in a
[Y.sub.0Z.sub.0] value. The coarse approximation angle .phi..sub.1
is retrieved from a second lookup table that stores pre-computed
arctan values of [Y.sub.0Z.sub.0]. Next, the input complex signal
is multiplied by the [Y.sub.0Z.sub.0] value. This multiplication
effectively rotates the input complex number by the coarse angle
.phi..sub.1 back toward the X-axis of the complex plane, resulting
in an intermediate complex number having an X.sub.1 component and a
Y.sub.1 component. Next the reciprocal lookup table is re-used to
determine an approximation of Z.sub.1=1/[X.sub.1]. Then the tangent
of the fine angle .phi..sub.2 is determined based on
[Z.sub.1Y.sub.1], assuming that tan .phi..sub.2 can be
substantially approximated as [Z.sub.1Y.sub.1]. In embodiments, the
Newton Raphson method is implemented to get a more accurate tan
.phi..sub.2 result. Finally, based on the smallness of tan
.phi..sub.2, the trigonometric function value tan .phi..sub.2 is
used as an approximation to .phi..sub.2, hence requiring no arctan
table.
[0021] Applications of the rectangular-to-polar converter include
symbol and carrier synchronization, including symbol
synchronization for bursty transmissions of packet data systems.
Other applications include any application requiring the
rectangular-to-polar conversion of a complex input signal.
[0022] Further features and advantages of the invention, as well as
the structure and operation of various embodiments of the
invention, are described in detail below with reference to the
accompanying drawings. The drawing in which an element first
appears is typically indicated by the leftmost character(s) and/or
digit(s) in the corresponding reference number.
BRIEF DESCRIPTION OF THE FIGURES
[0023] FIG. 1A illustrates a PSK transmitter.
[0024] FIG. 1B illustrates a PSK receiver.
[0025] FIG. 1C illustrates a block diagram of an OFDM system.
[0026] FIG. 1D illustrates a PSK receiver with carrier and timing
recovery.
[0027] FIG. 2 illustrates an Interpolation environment.
[0028] FIG. 3 illustrates a Lagrange basis polynomials.
[0029] FIG. 4 illustrates a Farrow structure that implements (2.5)
and (2.6).
[0030] FIG. 5 illustrates a flowchart 500 representing
trigonometric interpolation according to embodiments of the present
invention.
[0031] FIG. 6A illustrates an impulse response of a Lagrange
interpolator.
[0032] FIG. 6B illustrates an impulse response of a Trigonometric
interpolator according to embodiments of the present invention.
[0033] FIG. 7A illustrates a frequency response for N=4 according
to embodiments of the present invention.
[0034] FIG. 7B illustrates a frequency response for N=32 according
to embodiments of the present invention.
[0035] FIG. 8A illustrates a signal with two samples/symbol and
100% excess BW according to embodiments of the present
invention.
[0036] FIG. 8B illustrates an NMSE of the interpolated signal
according to embodiments of the present invention.
[0037] FIG. 9 illustrates the critical path of the Lagrange cubic
interpolator.
[0038] FIG. 10 illustrates a trigonometric interpolator with N=4
according to embodiments of the present invention.
[0039] FIG. 11 illustrates a trigonometric interpolator with N=8
according to embodiments of the present invention.
[0040] FIG. 12 illustrates a conceptual modification of input
samples according to embodiments of the present invention.
[0041] FIG. 13 illustrates correcting the offset due to
modification of original samples according to embodiments of the
present invention.
[0042] FIG. 14 illustrates the modified trigonometric interpolator
for N=4 according to embodiments of the present invention.
[0043] FIG. 15 illustrates the modified trigonometric interpolator
for N=8 according to embodiments of the present invention.
[0044] FIG. 16A-D illustrates a comparision of the amount of
interpolation error using (A) Language cubic, (B) the trigonometric
interpolator 1000, (C) the trigonometric interpolator, (D) the
optimal structure (to be discussed in Section 4).
[0045] FIG. 17 illustrates a flowchart 1700 representing
trigonometric interpolation according to embodiments of the present
invention.
[0046] FIG. 18 illustrates trigonometric interpolation using a
table lookup for angle rotation according to embodiments of the
present invention.
[0047] FIG. 19 illustrates trigonometric interpolation using
modified samples according to embodiments of the present
invention.
[0048] FIG. 20 illustrates normalized impulse responses f of the
interpolation filters according to embodiments of the present
invention.
[0049] FIG. 21 illustrates normalized frequency responses F of the
interpolation filters according to embodiments of the present
invention.
[0050] FIG. 22 illustrates analysis of the frequency responses
according to embodiments of the present invention.
[0051] FIG. 23 illustrates the effect of a more gradual transition
at the band edge according to embodiments of the present
invention.
[0052] FIG. 24 illustrates reducing the transition bandwidth by
increasing N according to embodiments of the present invention.
[0053] FIGS. 25A-B illustrate (A) impulse response of the original
filter and the modified filter; (B) The equivalent window,
according to embodiments of the present invention.
[0054] FIG. 26 illustrates forming the frequency response of the
discrete-time fractional-delay filter according to embodiments of
the present invention.
[0055] FIGS. 27A-B illustrate a fractional-delay filter with (A)
.mu.=0.12 and (B) .mu.=0.5, using the preliminary N=8 interpolator
according to embodiments of the present invention.
[0056] FIGS. 28A-D illustrate modification to F(.OMEGA.), and the
corresponding F.sub..mu.(.omega.) according to embodiments of the
present invention.
[0057] FIGS. 29A-B illustrate F.sub..mu.(.omega.), with .mu.=0.5,
N=8, (A) before and (B) after optimization according to embodiments
of the present invention.
[0058] FIGS. 30A-30B illustrate F.sub..mu.(.omega.) for .mu.=0.5,
N=4, (A) before and (B) after modification according to embodiments
of the present invention.
[0059] FIGS. 31A-31B illustrate F.sub..mu.(.omega.), .mu.=0.5,
simplified N=4 structure, (A) before and (B) after modification
according to embodiments of the present invention.
[0060] FIG. 32 illustrates real and imaginary components of the 1 F
^ ( 1 ) j 2
[0061] value according to embodiments of the present invention.
[0062] FIG. 33 illustrates signal with two samples/symbol and 40%
excess bandwidth embodiments of the present invention.
[0063] FIG. 34 illustrates a flowchart 3400 for optimizing
trigonometric interpolation according to embodiments of the present
invention.
[0064] FIG. 35 illustrates a flowchart 3500 for optimizing
trigonometric interpolation according to embodiments of the present
invention.
[0065] FIG. 36 illustrates an optimized interpolator 3600 according
to embodiments of the present invention.
[0066] FIG. 37 illustrates an optimized interpolator 3700 according
to embodiments of the present invention.
[0067] FIG. 38 illustrates an angle rotator 3800 according to
embodiments of the present invention.
[0068] FIG. 39 illustrates an angle rotator 3900 according to
embodiments of the present invention.
[0069] FIG. 40 illustrates an angle rotator 3900 and example
multiplier sizes according to embodiments of the present
invention.
[0070] FIG. 41 illustrates a flowchart 4100 for angle rotation
according to embodiments of the present invention.
[0071] FIG. 42 illustrates an angle rotator 3900 and multiplier
sizes to achieve 90.36 dB SFDR according to embodiments of the
present invention.
[0072] FIG. 43 illustrates an output spectrum showing 90.36 dB SFDR
according to embodiments of the present invention.
[0073] FIG. 44 illustrates a modified angle rotator 4400 when only
one output is needed according to embodiments of the present
invention.
[0074] FIG. 45 illustrates a flowchart 4500 for angle rotation when
only one output is needed according to embodiments of the present
invention.
[0075] FIG. 46 illustrates a phase accumulator 4600.
[0076] FIG. 47 illustrates a Quadrature Direct Digital Frequency
Synthesizer/Mixer (QDDFSM) 4700 according to embodiments of the
present invention.
[0077] FIG. 48 illustrates an angle rotator 4800 according to
embodiments of the present invention.
[0078] FIG. 49 illustrates a Booth multiplier according to
embodiments of the present invention.
[0079] FIG. 50 illustrates an original Booth table 5000.
[0080] FIG. 51 illustrates an negating booth table 5100 according
to embodiments of the present invention.
[0081] FIG. 52 illustrates an negating Booth multiplier 5200.
[0082] FIG. 53 illustrates a conditionally negating Booth decoder
5300.
[0083] FIG. 54 illustrates a conditionally negating multiplier
5400.
[0084] FIG. 55 illustrates as an angle rotator configured as
quadrature direct digital synthesizer (QDDS) 5500 according to
embodiments of the present invention.
[0085] FIG. 56 illustrates an angle rotator as a cosine only direct
digital synthesizer based on angle rotator 3900 according to
embodiments of the present invention.
[0086] FIG. 57 illustrates an angle rotator as a cosine only direct
digital synthesizer based on angle rotator 4400 according to
embodiments of the present invention.
[0087] FIG. 58 illustrates a common packet format for packet based
communications.
[0088] FIG. 59 illustrates a system model for packet based
communications.
[0089] FIG. 60 illustrates mean values of a preamble correlator
output, for .theta.=0, according to embodiments of the present
invention.
[0090] FIG. 61 illustrates a synchronizer 6100 according to
embodiments of the present invention.
[0091] FIG. 62 illustrates a flowchart 6200 associated with the
synchronizer 6100 according to embodiments of the present
invention.
[0092] FIG. 63 illustrates bias due to truncation.
[0093] FIG. 64 illustrates a synchronizer 6400 according to
embodiments of the present invention.
[0094] FIG. 65A-65B illustrate a flowchart 6200 associated with the
synchronizer 6100 according to embodiments of the present
invention.
[0095] FIG. 66 illustrates timing variance, .alpha.=0.4.
[0096] FIG. 67 illustrates timing jitter variance, .alpha.=0.1.
[0097] FIG. 68 illustrates phase jitter variance, .alpha.=0.1.
[0098] FIG. 69 illustrates Cartesian-to-polar conversion.
[0099] FIGS. 70A-70B illustrate using Newton-Raphson iteration to
find 1/X.sub.1.
[0100] FIG. 71 illustrates a rectangular-to-polar converter 7100
according to embodiments of the present invention.
[0101] FIG. 72 illustrates angle rotation associated with the
rectangular-to-polar converter 7100 according to embodiments of the
present invention.
[0102] FIG. 73 illustrates a flowchart 7300 associated with the
synchronizer 7100 according to embodiments of the present
invention.
[0103] FIG. 74 illustrates Interpolation in a non-center interval
according to embodiments of the present invention.
[0104] FIGS.75A-B illustrate impulse responses of the
non-center-interval interpolation filter (A) before and (b) after
optimization, according to embodiments of the present
invention.
[0105] FIGS. 76A-B illustrate frequency responses of the
non-center-interval interpolator (A) before optimization and (B)
after optimization, according to embodiments of the present
invention.
[0106] FIG. 77 illustrates an exemplary computer system 7702,
according to embodiments of the present invention.
[0107] FIG. 78 illustrates a data rate expansion circuit 7800
according to embodiments of the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0108]
1 Table of Contents 1. Introduction 1.1 Exemplary Modulation
Schemes and Synchronization Issues 1.2 Overview 2. Interpolation
Using a Trigonometric Polynomial 2.1 Interpolation Using an
Algebraic Polynomial 2.2 The Trigonometric Polynomial Method 2.3
Performance Comparisons 2.4 Efficient Implementation Structures
2.4.1 Using a Lookup Table 2.4.2 Using an Angle Rotation Processor
2.5 Delays in the Computation 2.6 Simplifications of the
Preliminary Structures 2.6.1 The Simplified Structure for N = 4
2.6.2 The Simplified Structure for N = 8 2.6.3 Performance
Comparisons with Other Structures 2.7 Trigonometric Interpolator
Application 2.8 Trigonometric Interpolator Summary 3. Interpolation
Filters with Arbitrary Frequency Response 3.1 Formulating the
Trigonometric Interpolator as an Interpolation Filter 3.2 Analysis
of the Frequency Response 3.3 Implementing the Modified Algorithm
3.4 Conditions for Zero ISI 3.5 Optimization Algorithm 3.6
Conclusion 4. Design of Optimal Resamplers 4.1 Motivation 4.2
Resampler Optimizations 4.3 Implementations 4.4 Simulation Results
4.5 Conclusion 5. A High-Speed Angle Rotation Processor 5.1 The
angle rotation problem 5.1.1 Single-Stage Angle Rotation 5.1.2
Rotation by a Small Angle 5.1.3 Partitioning into Coarse and Fine
Rotations 5.2 Simplification of the Coarse Stage 5.3 Reduction of
Multiplier Size in the Fine Stage 5.4 Scaling Multiplier
Simplification 5.5 Computational Accuracy and Wordlength 5.6
Comparison with the Single-Stage Mixer 5.7 A Modified Structure
When Only One Output is Needed 5.7.1 Modifications to the Coarse
Stage 5.7.2 Scaling Multiplier Simplification 5.8 Application of
Angle Rotation Processors 5.8.1 Using the Angle Rotation Processor
in a Quadrature Direct Digital Frequency Synthesizer/Mixer 5.8.1.1
A General Angle Rotator for Arbitrary Input Angles 5.8.1.2 Adapting
the General Angle Rotator to Make a QDDFSM 5.8.2 How to Use the
Conditionally Negating Multipliers in the General Angle Rotator
5.8.2.1 Booth Multiplier 5.8.2.2 How to Make a Negating Booth
Multiplier 5.8.2.3 How to Make a Conditionally Negative Booth
Multiplier 5.8.3 Using the Angle Rotation Processor in a Quadrature
Direct Digital Frequency Synthesizer 5.9 Conclusion 6. Symbol
Synchronization for Bursty Transmissions 6.1 Initial Parameter
Estimations for Burst Modems 6.2 Background Information 6.3 Symbol
Timing Estimation Assuming .theta. = 0 6.4 Bias in Symbol Timing
Estimation due to Truncating the Sequence 6.5 Carrier-Independent
Symbol Timing Recovery 6.6 Carrier Phase Computation 6.7 Simulation
Result 6.8 Conclusion 7. A High-Speed Processor for
Rectangular-to-Polar Conversion 7.1 Partitioning the Angle 7.2 The
Two-Stage Algorithm 7.2.1 Simplification in the Coarse Computation
Stage 7.2.1.1 The Reciprocal Table 7106 7.2.1.2 The Arctangent
Table 7110 7.2.2 Hardware Reduction in the Fine Computation Stage
7124 7.3 Magnitude Calculation 7.4 Converting Arbitrary Inputs 7.5
Test Result 7.6 Conclusion 8. Exemplary Computer System 9.
Appendices 9.1 Appendix A: Proof of the Zero ISI Condition. 9.2
Appendix B: Impulse Response of the Simplified Interpolators 9.3
Appendix C: Fourier Transform of g(nT.sub.s-.mu.) 9.4 Appendix D:
Interpolation on Non-Center Intervals 9.5 Appendix E 10.
Conclusion
1. Introduction
[0109] As discussed herein, third generation and other cutting edge
communications systems require proper synchronization between the
transmitter and the receiver. More specifically, the frequency and
phase of the receiver local oscillator should substantially match
that of the transmitter local oscillator and accurate symbol timing
must be achieved. The following section discuss some exemplary
modulation schemes and configurations, and their related
synchronization issues. These example configurations are not meant
to be limiting, and are provided for example purposes only. After
which, an overview of the present invention is provided.
[0110] 1.1 Exemplary Modulation Schemes and Synchronization
Issues
[0111] A key to the evolution of third-generation PCS is the
ability to provide higher data rates via increased spectral
efficiency of the access scheme. The IS-136 community intends to
add a 200-KHz carrier bandwidth and adopt 8PSK modulation. This
allows for data rates up to 384 Kb/s.
[0112] A simplified 8PSK transmitter 102 and receiver 104 are shown
in FIG. 1A and FIG. 1B, respectively. The receiver 104, as shown,
performs baseband sampling. Alternatively, the received signal
could be sampled at an IF frequency, where the down-conversion to
baseband is performed digitally. However, since it does not alter
the main subject in the present invention, the baseband-sampled
system is used as an example.
[0113] Referring to FIG. 1B, PSK receiver 104 down-converts an IF
input signal 106 to baseband by multiplication with a local
oscillator signal 108 using mixers 110. After filtering 111, A/D
converters 112 sample the down-converted signal according to a
sampling clock 114 in preparation for logic examination. After
further filtering 116 and equalization 118, the logic decision
devices 120 examine the sampled signal to determine a logic output
for the two channels.
[0114] During down-conversion, an undesirable rotation of the
symbol constellation will occur if the frequency and phase of the
oscillator signal 108 does not match the oscillator signal of the
transmitter 102. This symbol rotation can seriously degrade system
performance. When the carrier frequency offset is much smaller than
the symbol rate, the phase and frequency mismatches can be
corrected at baseband, using a phase rotator 124, as shown in FIG.
1D.
[0115] The sampling clock 114 is generated locally in the receiver
104. The logic decision devices 120 make more accurate decisions
when the sampling instant is optimal, i.e., synchronous to the
incoming symbols.
[0116] If the timing information can be extracted from the signal
106, it can be used to adjust the phase of the sampling clock 114.
This adjustment would require a voltage controlled oscillator (VCO)
to drive the A/D converters 112 . In this scenario, the digital
portion of the circuit 104 needs to keep in synchronization with
the A/D converters 112, which places strict requirements on the
VCO. Moreover, changing the phase of the sampling clock 114 would
cause jitter. High data-rate receivers are more sensitive to such
jitter when used in multimedia communications.
[0117] Another solution to timing errors is to correct them
entirely in the digital domain, with the equivalent of A/D sampling
adjustment performed by a digital resampler 122, as shown in FIG.
1D. This resampler 122 is controlled by a timing recovery circuit
(not shown) and it attempts to supply the optimal samples (i.e.
synchronous) to the decision circuits 120. Using the digital
resampler 122, the timing recovery loop is closed entirely in the
digital domain. This allows the complete separation of digital
components from analog components.
[0118] The digital resampler 122 and a direct digital frequency
synthesizer (not shown) used by the phase rotator 124 are among the
most complex components in a receiver (Cho, K., "A frequency-agile
single-chip QAM modulator with beamforming diversity," Ph.D.
dissertation, University of California, Los Angeles (1999)). Their
performance is significant in the overall design of the modem. For
multimedia communications, the high-data-rate requirement imposes a
demand for high computational power. However, for mobile personal
communication systems, low cost, small size, and long battery life
are desirable. Therefore, efficient implementation is the key to
implementing fast signal processing within the available resources.
It is also desirable to provide an efficient synchronization
mechanism by using a unified approach to timing and carrier phase
corrections. This can be accomplished by sharing resources between
the resampler 122 and the phase rotator 124.
[0119] As for the digital video broadcasting system (DVB) systems,
the most challenging of all DVB transmissions is the one used in
terrestrial channels (DVB-T) due to the presence of strong echoes
which characterize the propagation medium. A common approach for
DVB-T is based on Coded-OFDM (orthogonal frequency division
multiplexing). The major benefit of OFDM is that the serial
baseband bitstream which needs to be transmitted is distributed
over many individual subcarriers. Such spreading makes the signal
robust against the effects of multipath and narrowband
interference. The simplified block diagram of an OFDM modem 108 is
shown in FIG. 1C.
[0120] FIG. 1C illustrates an orthogonal frequency division
multiplexing system (OFDM) 126 having an OFDM transmitter 128 and
an OFDM receiver 130. For the OFDM system 126, synchronization
errors produce a rotation of the fast Fourier Transform (FFT)
outputs of the OFDM receiver 130. (Pollet T., and Peeters, M., IEEE
Comm. Magazine 37:80-86 (1999)). A sampling phase correction for
the received signals can be achieved by rotating the FFT outputs at
the receiver. For FFT rotation, it is desirable to have an
efficient implementation structure to perform rotations of complex
numbers.
[0121] The example applications and modulation schemes described
above in this section were provided for illustrative purposes only,
and are not meant to be limiting. Other applications and
combinations of such applications will be apparent to persons
skilled in the relevant art(s) based on the teachings contained
herein. These other applications are within the scope and spirit of
the present invention.
[0122] 1.2 Overview
[0123] The following is an overview of the sections that
follow.
[0124] Sections 2, 3 and 4, discussed herein, present a novel
interpolation method for digital resampling using a trigonometric
polynomial. In Section 2, after a brief review of interpolation
methods, particularly those using a conventional polynomial, the
trigonometric interpolation method is introduced. Efficient
implementation structures for trigometric interpolation are given.
The performance, hardware complexity and computational delay are
compared with conventional polynomial interpolators. The
trigonometric-polynomial based resampler can use the same hardware
as is employed in the phase rotator for carrier synchronization,
thus further reducing the total complexity in the synchronization
circuitry.
[0125] In Section 3, a signal processing approach is used to
analyze the interpolation method devised in Section 2. It shows how
an arbitrary frequency response is achieved by applying a simple
modification to the original interpolation algorithm. This enables
the interpolator to also perform matched filtering of the received
signal.
[0126] The approaches in Section 3 can be employed to improve the
interpolator performance by optimizing the frequency response of
the continuous-time interpolation filter. This method is based on
optimizing the performance by conceptually reconstructing the
continuous-time signal from existing samples. From the point of
view of designing digital resamplers, however, what we are actually
interested in are new samples corresponding to the new sampling
instants. In Section 4, we optimize the interpolation filter such
that the error in producing a new sample corresponding to every
resampling instant is minimized, hence further improving the
overall interpolation accuracy.
[0127] Section 5 presents an angle rotation processor that can be
used to efficiently implement the trigonometric resampler and the
carrier phase rotator. This structure can also implement the
resampler for an OFDM receiver, which rotates the FFT outputs. It
has many other practical applications.
[0128] The discussions in the previous Sections have assumed that
the sampling mismatch that is supplied to the resampler is known.
The problem of obtaining the synchronization parameters is studied
in Section 6. For burst mode transmissions in packet data systems,
we present an efficient architecture for feedforward symbol-timing
and carrier-phase estimation.
[0129] Section 7 presents an efficient implementation of a key
component in the feedforward synchronizer of Section 6, as well as
in many other such synchronizers. This involves computing the angle
from the real and imaginary components of a complex number. The
discussion, however, extends to a general problem of
Cartesian-to-polar conversion, which is encountered in many
communication applications. An architecture that efficiently
accomplishes this conversion is presented.
[0130] Section 8 presents an exemplary computer system in which the
invention can be operated.
[0131] Section 9 includes various appendices.
[0132] Further discussions related to materials in Sections 2-7 are
included in Dengwei Fu, "Efficient Synchronization for Multimedia
Communications," Ph.D dissertation, University of California, Los
Angeles, 2000, which is incorporated-by-reference, in its
entirety.
[0133] Additionally, the following articles are herein incorporated
by reference:
[0134] D. Fu and A. N. Willson, Jr., "A high-speed processor for
digital sine/cosine generation and angle rotation," in Conf. Record
32nd Annual Asilomar Conference on Signals, Systems and Computers,
vol. 1, pp. 177-181, November 1998;
[0135] D. Fu and A. N. Willson, Jr., "Interpolation in timing
recovery using a trigonometric polynomial and its implementation,"
in Proc. GLOBECOM 1998, Comm. Theory Mini-Conference Record, pp.
173-178, November 1998;
[0136] D. Fu and A. N. Willson, Jr., "Design of an improved
interpolation filter using a trigonometric polynomial," in Proc.
Int. Symp. Circuits & Systems, vol. 4, pp. 363-366, May 30-Jun.
3, 1999;
[0137] D. Fu and A. N. Willson, Jr., "A high-speed processor for
rectangular-to-polar conversion with applications in digital
communications," in Proc. GLOBECOM 1999, vol. 4, pp. 2172-2176,
December 1999;
[0138] D. Fu and A. N. Willson, Jr., "A fast synchronizer for burst
modems with simultaneous symbol timing and carrier phase
estimations," in Proc. Int. Symp. Circuits & Systems, vol. 3,
pp. 379-382, May 28-31, 2000; and
[0139] D. Fu and A. N. Willson, Jr., "Optimal interpolator using a
trigonometric polynomial," in Proc. 43rd Midwest Symp. Circuits
& Systems, 4 pages, Aug. 8-11, 2000.
2. Interpolation Using a Trigonometric Polynomial
[0140] As discussed in Section 1, when an analog-to-digital
converter (ADC) is clocked at a fixed rate, the resampler must
provide the receiver with correct samples, as if the sampling is
synchronized to the incoming symbols. How can the resampler recover
the synchronized samples by digital means without altering the
sampling clock? Since the input analog signal to the ADC is
bandlimited, as long as the sampling rate is at least twice the
signal bandwidth, according to the sampling theorem, the sampled
signal carries as much information as the continuous-time signal.
Therefore, the value of the original continuous-time signal at an
arbitrary point can be evaluated by applying an interpolation
filter (e.g., sin c interpolation) to the samples. Hence, the
design of the resampler has been transformed to the design of
effective interpolation filters or, in other words,
fractional-delay filters with variable delay.
[0141] There are numerous methods for designing fractional-delay
filters. These filters have different coefficients for different
delays. Thus, to implement variable-delay interpolation filters,
one could compute one set of coefficients for each quantized delay
value and store them in a memory. Then, in real-time, depending on
the fractional delay extracted from the incoming signal, the
corresponding coefficients could be loaded. However, this method is
likely to result in a large coefficient memory.
[0142] To design low-cost modems, a large coefficient memory is
undesirable. Gardner, et al., have shown that polynomials can be
incorporated to compute the desired samples that are synchronous
with the transmitted samples (Gardner, F. M., IEEE Trans. Comm.
41:502-508 (1993); Erup, L., et al., IEEE Trans. Comm. 41:998-1008
(1993)). In this case, an extensive coefficient memory is not
needed. Moreover, the polynomial-based structure can be implemented
efficiently with a so-called Farrow structure (Farrow, C., "A
continuously variable digital delay element," in Proc. IEEE Int.
Symp. Circuits Syst. (June 1988), pp. 2641-2645). This method is
reviewed in Section 2.1. Although this approach achieves reasonable
performance, the hardware complexity grows rapidly as the number of
samples used to calculate each new sample is increased for better
accuracy. In addition, given a fractional delay .mu., to produce a
new sample using a degree N-1 polynomial, there will be N-1
sequential multiplications that involve .mu. since we must compute
.mu. raised to the (N-1)-th power times a data value. Thus, the
critical data path gets longer as N increases, thereby creating a
limitation on the achievable data rate.
[0143] Starting in Section 2.2, a new approach to interpolation is
introduced. Instead of approximating the continuous-time signal
with a conventional (i.e., algebraic) polynomial, a trigonometric
polynomial is used according to the present invention. First, some
background information is given. Next, the detailed implementation
is discussed. We then evaluate and compare the performance and
computational complexity of the algebraic polynomial interpolation
to that of our method, giving numerical results.
[0144] 2.1 Interpolation Using an Algebraic Polynomial
[0145] To interpolate using function values y(n) at N
equally-spaced sample points, also referred to as a "base point
set," we can fit an algebraic polynomial of degree N-1 to the data,
as in FIG. 2. As explained in (Gardner, F. M., IEEE Trans. Comm.
41:502-508 (1993)), there should be an even number of samples in
the base point set, and the interpolation should be performed only
in the center interval of the base point set. That is, N is
restricted to be even. In other words, given 4 samples points in
FIG. 2 including y(-1), y(0), y(1), and y(2), the interpolation is
performed at offset .mu. between the points y(0) and y(1) to
determine the point 202 on the curve p(t).
[0146] It seems that one would have to solve for the coefficients
of the (N-1)-th degree polynomial from these available samples
before the synchronized (i.e., interpolated) samples can be
computed. However, a method devised by Farrow (Farrow, C., "A
continuously variable digital delay element," in Proc. IEEE Int.
Symp. Circuits Syst. (June 1988), pp. 2641-2645) can compute the
synchronized sample from the available samples efficiently with use
of an algorithm that is well suited for VLSI implementation. To
illustrate, we consider the following example of interpolation
using a cubic Lagrange polynomial. Without loss of generality, let
us assume the sampling interval is T.sub.S=1. Using the Lagrange
formula for N=4, the synchronized samples can be computed as
y(.mu.)=y(-1)C.sub.-1(.mu.)+y(0)C.sub.0(.mu.)+y(1)C.sub.1(.mu.)+y(2)C.sub.-
2(.mu.) (2.1)
[0147] where C.sub.n(.mu.), n=-1, 0, 1, 2, are the third degree
polynomials that are shown in FIG. 3.
[0148] Obviously, 2 C n ( ) = { 1 = nT s , n an integer 0 all other
integers . ( 2.2 )
[0149] Thus, y(.mu.) in (2.1), the sum of polynomials C.sub.n(.mu.)
weighted by the y(n) values, must be a third degree polynomial and
must go through the samples y(-1), y(0), y(1) and y(2). Writing
C.sub.n(.mu.) as 3 C n ( ) = k = 0 3 c nk k ( 2.3 )
[0150] the coefficients C.sub.nk of C.sub.n(.mu.) are fixed
numbers. They are independent of .mu.. We can re-write (2.1) as 4 y
( ) = n = - 1 2 y ( n ) k = 0 3 c nk k = k = 0 3 ( n = - 1 2 y ( n
) c nk ) k = k = 0 3 v ( k ) k where ( 2.4 ) v ( k ) = n = - 1 2 y
( n ) c nk . ( 2.5 )
[0151] To minimize the number of multipliers, we can use a nested
evaluation of (2.4), as
y(.mu.)=((v(3).mu.+v(2)).mu.+v(1)).mu.+v(0). (2.6)
[0152] A Farrow structure 400 (for N=4) that implements equations
(2.5) and (2.6) is shown in FIG. 4. It consists of multiplications
of data by fixed coefficients, and data multipliers, as well as
addition operations.
[0153] 2.2 The Trigonometic Polynomial Method
[0154] To solve the problems discussed in the section 2.1, the
present invention utilizes a trigonometric polynomial to fit the
asynchronous samples in FIG. 2. Using W.sub.N=e.sup.-j2.pi./N
notation, for t.di-elect cons.[-N/2+1, N/2], the polynomial may be
written as: 5 y ( t ) = 1 N ( k = - N 2 + 1 N 2 - 1 c k W N - kt +
1 2 c N 2 W N - ( N 2 ) t + 1 2 c - N 2 W N ( N 2 ) t ) . ( 2.7
)
[0155] The polynomial must cross the N samples. Thus, the
coefficients c.sub.k can be determined by solving the N linear
equations in N unknowns: 6 y ( n ) = 1 N k = - N 2 + 1 N 2 c k W N
- kn , n = - N 2 + 1 , , N 2 ( 2.8 )
[0156] whose solution is 7 c k = n = - N 2 + 1 N 2 y ( n ) W N kn ,
k = - N 2 + 1 , , N 2 . ( 2.9 )
[0157] The expression in (2.9) is simply the N-point discrete
Fourier transform (DFT). This suggests that, given N equally-spaced
samples, we can compute the DFT of these samples as in (2.9) to
obtain the coefficients of the interpolating trigonometric
polynomial in (2.7). Then, for a given offset .mu., the
synchronized sample y(.mu.) can be computed using that polynomial
as: 8 y ( ) = 1 N ( k = - N 2 + 1 N 2 - 1 c k W N - k + c N 2 cos )
. ( 2.10 )
[0158] Since c.sub.k and c.sub.-k are conjugates, this equation can
be simplified as 9 y ( ) = 1 N Re ( c 0 + 2 k = 1 N 2 - 1 c k W N -
k + c N 2 j ) . ( 2.11 )
[0159] Flowchart 500 in FIG. 5 summarizes the interpolation between
two sample points at an offset .mu. using a trigonometric
polynomial, where the two data samples that are to be interpolated
are part of a set of N-data samples (see FIG. 2). The flowchart is
described as follows.
[0160] In step 502, a set of N-data samples are received having the
two data samples that are to be interpolated.
[0161] In step 504, coefficients of a trigonometric polynomial are
determined based on the set of N data samples, according to
equation (2.9). In doing so, the N data samples are multiplied by a
complex scaling factor W.sub.N.sup.kn to generate a k.sup.th
coefficient for the trigonometric polynomial, wherein
W.sub.N=e.sup.-j2.pi./N, and wherein n represents an n.sup.th data
sample of said N data samples.
[0162] In step 506, the trigonometric polynomial is evaluated at
the offset .mu. based on equation (2.10) or (2.11).
[0163] In step 508, the real part of the evaluated trigonometric
polynomial is determined and represents the desired interpolation
value 202 in FIG. 2.
[0164] There are three issues that are to be considered in
evaluating an interpolation scheme: 1) accuracy of interpolation,
2) complexity of implementation and 3) latency. In following
sections, the trigonometric interpolation method is compared to the
traditional polynomial method, particularly the Lagrange
interpolator, in all these categories.
[0165] 2.3 Performance Comparisons
[0166] Let us first derive the impulse responses of the
interpolators. With N samples, N an even integer, the Lagrange
formula (2.1) is 10 y ( ) = n = - N 2 + 1 N 2 y ( n ) C n ( ) . (
2.12 )
[0167] In addition, the interpolation is performed in the center
interval. Thus 0.ltoreq..mu..ltoreq.1. Let us define a new function
.function.(t) such that
.function.(.mu.-n)=C.sub.n(.mu.), 0.ltoreq..mu.<1,
-N/2+1.ltoreq.n<N/2. (2.13)
[0168] Using the example of FIG. 3, defining t=.mu.-n, we have 11 f
( t ) = ( C 2 ( t + 2 ) - 2 t < - 1 C 1 ( t + 1 ) - 1 t < 0 C
0 ( t ) 0 t < 1 C - 1 ( t - 1 ) 1 t < 2 0 otherwise . ( 2.14
)
[0169] Thus, the Lagrange formula becomes 12 y ( t ) = n = - N 2 +
1 N 2 y ( n ) C n ( t ) = n = - .infin. .infin. y ( n ) f ( t - n )
. ( 2.15 )
[0170] Therefore, the approach to reconstruct the continuous signal
using the Lagrange polynomial is in fact equivalent to applying an
interpolation filter .function.(t) to the available samples, with
.function.(t) being a piecewise polynomial. The interpolator's
impulse response .function.(t) obtained from (2.14) is shown in
FIG. 6A.
[0171] Taking the Fourier transform of .function.(t), we obtain its
frequency response. This allows us to evaluate the interpolation
accuracy by examining the frequency response of the interpolation
filter. The frequency response 702 of the Lagrange cubic
interpolator (N=4) is shown in FIG. 7A. The horizontal axis is the
normalized frequency .function./F.sub.S, with F.sub.S=1/T.sub.S. An
ideal frequency response should have value one in the passband
(0.ltoreq..function./F.sub.S<0.5- ) and be zero in the stopband
(.function./F.sub.S.gtoreq.0.5).
[0172] For the interpolator using a trigonometric polynomial, we
can express y(t) in terms of y(n) by substituting (2.9) into
(2.10): 13 y ( ) = 1 N k = - N 2 + 1 N 2 - 1 c k W N - k + 1 N c N
2 cos = 1 N k = - N 2 + 1 N 2 - 1 ( n = - N 2 + 1 N 2 y ( n ) W N k
n ) W N - k + 1 N ( n = - N 2 + 1 N 2 y ( n ) W N ( N 2 ) n ) cos =
1 N n = - N 2 + 1 N 2 y ( n ) ( k = - N 2 + 1 N 2 - 1 W N k ( n - )
+ ( - 1 ) n cos ) = 1 N n = - N 2 + 1 N 2 y ( n ) ( 1 + 2 k = 1 N 2
- 1 cos 2 k N ( - n ) + cos ( - n ) ) . Defining ( 2.16 ) f ( t ) =
( 1 + 2 k = 1 N 2 - 1 cos 2 k N t + cos t t N 2 0 t > N 2 we
have ( 2.17 ) y ( ) = 1 N n = - .infin. .infin. y ( n ) f ( - n ) =
1 N y f . ( 2.18 )
[0173] The impulse response .function.(t) in (2.17) is shown in
FIG. 6B. The corresponding frequency response 704 of the
trigonometric interpolator (for N=4) is shown in FIG. 7A in thin
lines.
[0174] By comparing the frequency responses of the two
interpolators, we can see that the trigonometric interpolation
response 704 has a sharper roll-off in the transition band and more
rapid attenuation in the stopband than the Lagrange response 702.
These traits are enhanced as N increases, as demonstrated in FIG.
7B. For N=32, the trigonometric response 708 has a sharper rolloff
than the Lagrange response 706 as shown in FIG. 7B.
[0175] Next we verify these observations by interpolating the
samples of a practical time-domain signal. As an example, we
interpolate a baseband signal with raised cosine spectrum and
roll-off factor .alpha.=1.0, sampled at two samples per symbol
period, as shown in FIG. 8A.
[0176] The interpolation accuracy here is measured as the
normalized mean-squared difference between the signal interpolated
with an ideal interpolator and the signal interpolated with the
practical interpolator. The normalized mean-squared error (NMSE),
discussed above, is calculated for both the Lagrange interpolator
and the trigonometric interpolator for a range of typical values of
N. The results are plotted in FIG. 8B.
[0177] Our test results demonstrate that the performance is
improved with the trigonometric method. Using the same number of
samples to interpolate, the proposed method produces a smaller
NMSE, and the performance gain becomes greater as the number of
samples increases.
[0178] 2.4 Efficient Implementation Structures
[0179] Recalling from Section 2.2, the trigonometric interpolation
algorithm includes substantially two steps:
[0180] Step 1. Given a number of data samples N, calculate the
Fourier coefficients c.sub.k, k=0, . . . , N/2 using (2.9). In a
preferred embodiment, an even number of N data samples are used. In
other embodiments, an odd number of data samples are used.
[0181] Step 2. Compute the synchronized sample y(.mu.) for any
given .mu. according to (2.11).
[0182] The first step involves multiplying the data samples by
complex scaling factors W.sub.N.sup.kn. Since these factors lie on
the unit circle, the computation in Step 1 can be simplified. Let
us examine the case when N=4:
[0183] Example 2.1: For N=4, the Fourier coefficients are obtained
as:
c.sub.0=y(-1)+y(0)+y(1)+y(2)
c.sub.1=[y(0)-y(2)]+j[-y(1)+y(-1)]
c.sub.2=y(0)-y(1)+y(2)-y(-1) (2.19).
[0184] As seen in (2.19), there is no nontrivial scaling multiplier
required for N=4.
[0185] Example 2.2: We now compute the coefficients c.sub.k in
(2.9) for N=8. Using the trigonometric identities, we can obtain
the following simple form for c.sub.k, k=0, . . . , 4: 14 c 0 = y (
- 3 ) + y ( - 2 ) + y ( - 1 ) + y ( 0 ) + y ( 1 ) + y ( 2 ) + y ( 3
) + y ( 4 ) c 1 = { y ( 0 ) - y ( 4 ) + [ - y ( - 3 ) + y ( 1 ) _ +
y ( - 1 ) - y ( 3 ) _ ] cos ( 4 ) } + j { y ( - 2 ) - y ( 2 ) + [ y
( - 3 ) - y ( 1 ) _ + y ( - 1 ) - y ( 3 ) _ ] cos ( 4 ) } c 2 = { -
y ( - 2 ) + y ( 0 ) - y ( 2 ) + y ( 4 ) } + j { - y ( - 3 ) + y ( -
1 ) - y ( 1 ) + y ( 3 ) } c 3 = { y ( 0 ) - y ( 4 ) + [ y ( - 3 ) -
y ( 1 ) _ + ( - y ( - 1 ) + y ( 3 ) ) _ ] cos ( 4 ) } + j { - y ( -
2 ) + y ( 2 ) + [ y ( - 3 ) - y ( 1 ) _ + ( - y ( - 1 ) - y ( 3 ) )
_ ] cos ( 4 ) } c 4 = - y ( - 3 ) + y ( - 2 ) - y ( - 1 ) + y ( 0 )
- y ( 1 ) + y ( 2 ) - y ( 3 ) + y ( 4 ) . ( 2.20 )
[0186] The only non-trivial scaling multiplications are those
multiplications by cos (.pi./4). It appears that four such
multiplications are needed to compute all the complex coefficients
c.sub.k, k=0, . . . , 4. However, if we examine the data being
multiplied by cos(.pi./4) (those terms embraced by the [ ]
brackets), we observe that they are either the sums or differences
of the [y(-3)-y(1)] and [y(-1)-y(3)] values. Therefore, we can
compute [y(-3)-y(1)]cos(.pi./4) and [y(-1)-y(3)]cos(.pi./4), then
use these two products to generate the c.sub.n coefficients. Thus,
only two scaling multiplications are needed in computing all the
coefficients.
[0187] Having observed the simplicity in the first step, let us
focus on the second step. The second step may look complicated
because of the complex multiplications c.sub.kW.sub.N.sup.-k.mu.
and c.sub.N/2e.sup.j.pi..mu.. However, since
.vertline.W.sub.N.sup.-k.mu..ver-
tline.=.vertline.e.sup.j.pi..mu..vertline.=1, these products are
just rotations of points c.sub.k and c.sub.N/2 in the complex
plane. Furthermore, this is the same type of operation performed in
the phase rotation for carrier recovery by the phase rotator 124
that is shown in FIG. 1D. This suggests that we can reduce the
total complexity of the synchronization circuitry by sharing some
resources needed by the digital resampler 122 and the carrier phase
rotator 124. In one embodiment, a lookup table is utilized to
determine the angle rotation associated with the angle 15 2 N k
[0188] for rotation of the c.sub.k coefficients. In another
embodiment, an angle rotator processor is utilized. Both
embodiments are discussed further below, and the angle rotator
processor is discussed in detail in section 5.
[0189] FIG. 10 illustrates an trigonometric interpolator 1000 that
is one circuit configuration that implements the trigonometric
interpolator equations (2.9)-(2.11), where the number of data
samples is N=4. The interpolator 1000 is not meant to be limiting,
as those skilled in the arts may recognize other circuit
configurations that implement the equations (2.8)-(2.11). These
other circuit configurations are within the scope and spirit of the
present invention.
[0190] The trigonometric interpolator 1000 receives input data
samples having two data samples that are to be interpolated at an
offset .mu. (see FIG. 2). The resulting interpolated value y(.mu.)
represents the interpolated point 202 in FIG. 2. The interpolator
1000 includes a delay module 1004, an adder/subtractor module 1006,
and an angle rotator module 1008, and an adder 1012.
[0191] The delay module 1004 includes one or more delay elements
1012. The delay elements 1012 can be configured using known
components.
[0192] The adder/subtractor module 1006 includes multiple adders
(or subtractors) 1014, where subtraction is indicated by a (-)
signs.
[0193] The angle rotator module includes two angle rotators 1010.
The angle rotators 1010 can be configured using an angle rotator
processor or a table lookup (e.g. read only memory) as discussed
below.
[0194] The operation of the trigonometric interpolator 1000 is
discussed further in reference to the flowchart 1700 in FIG. 17,
which is discussed below.
[0195] In step 1702, the interpolator 1000 receives a set of
N-input data samples. The N-data samples include the two data
samples that are to be interpolated at the offset .mu. relative to
one of the data samples, as shown in the FIG. 2. In FIG. 2, the
interpolation is to be performed between y(0) and y(1) at the
offset .mu. to determine the interpolation value 202.
[0196] In step 1704, the delay module 1004 delays the input data
samples.
[0197] In step 1706, the adder/subtractor module 1006 generates one
or more trigonometric coefficients according to the equation (2.9).
In FIG. 10, the coefficients are represented by C.sub.0, C.sub.1,
and C.sub.2 for N=4, where the coefficients C.sub.1 is a complex
coefficients.
[0198] In step 1708, the angle rotators 1010 rotate appropriate
real and complex coefficients in a complex plane according the
offset .mu., resulting in rotated complex coefficients. More
specifically, the angle rotator 1010a rotates the real coefficient
C.sub.2, and the angle rotator 1010b rotates the complex
coefficient C.sub.1 in the complex plane.
[0199] In embodiments, as discussed herein, the angle rotators 1010
are table look-ups. In which case, a rotation factor is retrieved
from the table lookup based on the offset .mu., where the rotation
factor includes the evaluated cosine and sine functions of 16 2 N
k
[0200] that are shown in the equations (2.21) below. The rotation
factor is then multiplied by the corresponding real or complex
coefficient, to generate the respective rotated complex
coefficient. An interpolator 1800 having a table lookup ROM 1802
and a complex multiplier 1804 are shown in FIG. 18 for
illustration.
[0201] In step 1710, the adder 1012 adds together C.sub.0, a real
part of the rotated coefficient C.sub.1, and a real part of the
rotated coefficient C.sub.2. The adder 1012 also scales the output
of the adder as necessary according to equation 2.10. The resulting
value is the desired interpolation value at the offset .mu., as
represented by point 202 in FIG. 2.
[0202] The trigonometric interpolator is not limited to the 4th
degree embodiment that is shown in FIG. 10. The trigonometric
interpolator can be configured as an N.sup.th degree interpolator
based on N-data points, as represented by the equations
(2.9)-(2.11). These other N.sup.th degree interpolators are within
the scope and spirit of the present invention. For example and
without limitation, FIG. 11 illustrates an interpolator 1100 having
N=8. The trigonometric interpolator 1100 includes: a delay module
1102, an adder/subtractor module 1104 (having two scaling
multipliers having coefficients cos(.pi./4)), an angle rotator
module 1106, and an adder 1108 (having an 1/8 scale factor that is
not shown). The operation of the interpolator 1100 will be
understood by those skilled in the arts based on the discussion
herein.
[0203] 2.4.1 Using a Lookup Table
[0204] For carrier recovery, the phase correction is generally
accomplished by looking up the sine and cosine values corresponding
to the phase, then by multiplying these values with the complex
data. This requires the same operations as the rotation of c.sub.k
by an angle 17 2 N k ,
[0205] that is: 18 Re ( c k W N - k ) = Re ( c k ) cos 2 N k - Im (
c k ) sin 2 N k Im ( c k W N - k ) = Re ( c k ) sin 2 N k + Im ( c
k ) cos 2 N k . ( 2.21 )
[0206] The sine and cosine table can be used for both the
resampler, as in (2.21), and the phase rotator for carrier
recovery. In embodiments, a read only memory (ROM) is utilized as
the lookup table. However, other embodiments could be utilized in
including other types of memories. An interpolator 1800 utilizing a
table lookup ROM 1802 and complex multiplier 1804 are shown in FIG.
18 for illustration. The ROM table access time is insignificant as
compared to the computation time in other parts of the
interpolator. Therefore, this method results in low hardware
complexity and low computational delay. This implementation will be
referred to as the table-lookup method.
[0207] 2.4.2 Using an Angle Rotation Processor
[0208] When a very low complexity implementation is desired at the
expense of a slight increase in computational delay, we propose to
use an efficient structure for angle rotation, which is described
in Section 5. Based on this structure, each angle rotator has a
hardware complexity slightly greater than that of two multipliers.
In addition, a very small ROM is needed. We will subsequently refer
to this particular implementation of our algorithm as the
angle-rotation method.
[0209] Thus, there are at least two choices for implementing (2.21)
as well as the phase rotator for carrier recovery. The trade-off is
between complexity and speed. In a base-station where computation
power is more affordable, the table lookup method might be used. In
a hand-set, where cost is a major concern, an angle rotation
processor might be used for both the resampler and the phase
rotator, multiplexing the operations.
[0210] Now let us compare the complexity of the trigonometric
resampler with that of the Lagrange method. Table 2.1 summarizes
the comparisons for several typical N values. The numbers are based
on the table-lookup method. It indicates that, for the same filter
length N the trigonometric interpolation method needs less
hardware.
2TABLE 2.1 Complexity and latency comparisons. N 4 6 8 10
Nontrivial scaling Lagrange 2 10 25 39 multipliers Trigonometric 0
2 6 16 Data multipliers Lagrange 3 5 7 9 Trigonometric 3 5 7 9
Multiplications in the Lagrange 4 6 8 10 critical path
Trigonometric 1 2 2 2
[0211] 2.5 Delays in the Computation
[0212] The critical path of the Farrow structure 400 (FIG. 2) are
now compared to that of the trigonometric interpolator. The Farrow
structure implements the Lagrange interpolator as discussed above.
The Farrow structure 400 is shown in FIG. 9(or FIG. 4), with the
critical path 902 indicated. The critical path 902 for this
structure includes one scaling multiplier 904 and N-1 data
multipliers 906.
[0213] In contrast, the critical path for the trigonometric
interpolator 1000 is path 1002 and it contains just one angle
rotation 1010, or only one multiplier if the table-lookup method is
employed to replace the angle rotator 1010. Since the angle
rotations for various angles can be carried out in parallel, the
critical path does not lengthen as N increases.
[0214] Table 2.1 compares the computational delays for the
trigonometric interpolator with that of the Lagrange interpolator
for various values of N. The delay data for the trigonometric
interpolator 1000 are based on the table-lookup method. As shown in
FIG. 10, the trigonometric interpolator (for N=4) has only one
multiplier in the critical path, whereas the Lagrange interpolator
has 4 multipliers in the critical path. Therefore, the
trigonometric interpolator has less latency than the Lagrange
interpolator, which is important for voice communications.
[0215] 2.6 Simplifications of the Preliminary Structures
[0216] As mentioned in Section 2.4, to produce each y(.mu.) we
first calculate the Fourier coefficients c.sub.k using existing
samples, according to (2.9). We then compute
Re(c.sub.kW.sub.N.sup.-k.mu.) to be used in (2.11). This is
accomplished either by retrieving W.sub.N.sup.-k.mu. from a lookup
table, followed by two real multiplications, or by an
angle-rotation processor.
[0217] 2.6.1 The Simplified Structure for N=4
[0218] Let us examine the trigonometric interpolator 1000 having
N=4. To compute Re(2c.sub.1W.sub.4.sup.-.mu.) and
Re(c.sub.2W.sub.4.sup.-2.mu.) the system requires either two angle
rotators 1004 or two accesses to a lookup table.
[0219] If the input samples would happen to be such that c.sub.2=0
then one fewer angle rotator, or one fewer ROM access, would be
required. Of course, the original data samples y(-1), y(0), y(1),
and y(2) are not likely to have the special property that
c.sub.2=y(0)-y(1)+y(2)-y(-1)=0. However, if the data samples are
changed, then the modified samples can be determined to satisfy
c.sub.2=0. If the modified samples for interpolation, then the
c.sub.2 angle rotator 1010a can be eliminated. However, the
interpolation result will not then correspond to the original data
samples.
[0220] It seems that the data samples can be changed to attain
savings in hardware, as long as the interpolation result is fixed
so that it corresponds to the original data samples. Of course, it
must also cost less in hardware to "fix" the interpolation result
than is saved in hardware by using the modified samples.
[0221] If the samples y(k) are modified to form {tilde over (y)}(k)
according to:
{tilde over (y)}(-1)=y(-1)-K
{tilde over (y)}(0)=y(0)
{tilde over (y)}(1)=y(1)+K
{tilde over (y)}(2)=y(2)+2K (2.22)
[0222] then the K value can be adjusted to force the {tilde over
(y)}(k) samples to satisfy c.sub.2=0, where K is the slope of a
straight line 1202 in FIG. 12.
[0223] To find K, the c.sub.2 value that corresponds to the
modified samples is determined according to: 19 c 2 = y ~ ( 0 ) - y
~ ( 1 ) + y ~ ( 2 ) - y ~ ( - 1 ) = y ( 0 ) - ( y ( 1 ) + K ) + ( y
( 2 ) + 2 K ) - ( y ( - 1 ) - K ) = 2 K - ( y ( 1 ) + y ( - 1 ) - y
( 0 ) - y ( 2 ) ) . ( 2.23 )
[0224] To force c.sub.2=0, requires: 20 K = 1 2 ( y ( 1 ) + y ( - 1
) - y ( 0 ) - y ( 2 ) ) ( 2.24 )
[0225] Therefore, the c.sub.2 angle-rotator can be eliminated, and
c.sub.0 and c.sub.1 are determined accordingly as,
c.sub.0=2(y(1)+y(-1))
c.sub.1=[2y(0)-y(1)-y(-1)]+j[-2y(1)+y(0)+y(2)]. (2.25)
[0226] Then, the interpolated sample is 21 y ~ ( ) = 1 4 c 0 + 1 2
Re ( c 1 j 2 ) . ( 2.26 )
[0227] However, {tilde over (y)}(.mu.) should be adjusted so that
it corresponds to the original samples. From FIG. 12, the values
expressing the difference between the original and the modified
samples lie on the straight line 1202. From FIG. 13, it follows
that the offset due to the modification of samples is K.mu..
Therefore, the {tilde over (y)}(.mu.) value can be compensated
by:
y(.mu.)={tilde over (y)}(.mu.)-K.mu.. (2.27)
[0228] Using equations (2.25), (2.26) and (2.27) leads to an
interpolator 1400 as shown in FIG. 14. This simplified interpolator
structure is not limited to N=4 configurations. In fact, this
simplification technique can be applied to an interpolator with an
arbitrary N value. To eliminate the angle-rotation needed by
Re(c.sub.N/2e.sup.j.pi..mu.) in (2.11), the samples are modified
according to
{tilde over (y)}(n)=y(n)+nK, n=integer. (2.28)
[0229] Using (2.9), and then applying (2.28), results in 22 c N 2 =
1 N ( n = - N 2 + 1 N 2 ( - 1 ) n y ~ ( n ) ) = 1 N ( n = - N 2 + 1
N 2 ( - 1 ) n y ( n ) ) + 1 2 K . ( 2.29 )
[0230] If we choose 23 K = - 2 N n = - N 2 + 1 N 2 ( - 1 ) n y ( n
) ( 2.30 )
[0231] we can force c.sub.N/2=0.
[0232] Referring to FIG. 14, the interpolator 1400 includes the
delay module 1004, an adder/subtractor module 1402, the angle
rotator 1010b, a multiplier 1404, and an adder 1406. The
interpolator 1400 has a simplified structure when compared the
interpolator 1000 (in FIG. 10), as the interpolator 1400 replaces
the angle rotator 1010b with a multiplier 1404. As discussed above,
this can be done because the coefficient C.sub.N/2=0(C.sub.2=0 for
N=4) by modification of the data samples, and therefore there is no
need to have an angle rotator for C.sub.N/2. The operation of the
interpolator 1400 is further discussed in reference to the
flowchart 1900 that follows.
[0233] In step 1902, the interpolator 1400 receives a set of N
input data samples. The N data samples include two of the data
samples that are to be interpolated at the offset .mu., as shown in
the FIG. 2.
[0234] In step 1904, the delay module 1004 delays the input data
samples.
[0235] In step 1906, the adder/subtractor module 1402 modifies one
or more of the data samples so that a coefficient c.sub.N/2 is 0.
In embodiments the data samples are modified according to
y(n).sub.mod=y(n)+n.multidot.K- , wherein K is determined by the
equation (2.30) above so that c.sub.N/2 is 0, and wherein y(n)
represents the n.sup.th data sample of the N data sample set.
[0236] In step 1908, the adder/subtractor module 1402 generates one
or more trigonometric coefficients according modifications to the
equation (2.8). In the N=4 case, equations 2.25 are implemented by
the module 1402. In FIG. 14, for N=4, the coefficients are
represented by C.sub.0 and C.sub.1, where the coefficient C.sub.1
is a complex coefficients. By comparing with FIG. 10, it is noted
that the C.sub.2 coefficient is 0. Additionally, the
adder/subtractor module 1402 outputs the K value for further
processing. Notice also that in FIG. 14, the output scaling factor
has been changed from 1/4 to 1/2. This reflects several other
straightforward simplfications that have been made to module 1402
and in the angle rotator 1010b. In embodiments, the steps 1906 and
1908 are be preformed simultaneously by the adder/subtractor module
1402, as will be understood by those skilled in the relevant
arts.
[0237] In step 1910, the angle rotator 1010b rotates the complex
coefficient C.sub.1 in a complex plane according the offset .mu.,
resulting in a rotated complex coefficient. In embodiments, as
discussed herein, the angle rotators 1010b is table look-up. In
which case, a complex rotation factor is retrieve from the table
lookup based on the offset .mu., and the resulting rotation factor
is then multiplied by the corresponding complex coefficient, to
generate the respective rotated complex coefficient. The rotation
factor includes the evaluation of the cosine and sine factors that
are shown in equations 2.21. Note that since C.sub.2=0, the angle
rotator 1010a is replaced with the multiplier 1404.
[0238] In the step 1912, the multiplier 1404 multiplies the K
factor by the offset .mu., to produce a K.mu. factor.
[0239] In step 1914, the adder 1406 adds together C.sub.0 and K.mu.
and a real part of the rotated complex coefficient C.sub.1, and
scales the sum by the trivial factor 1/2, to produce the desired
interpolation value. The addition of the K.mu. factor compensates
the desired interpolation value for the modification that was done
to the data samples in order to force cm to zero in step 1906.
[0240] The simplified trigonometric interpolator is not limited to
the four sample embodiment that is shown in FIG. 14. The simplified
trigonometric interpolator can be configured as an N sample
interpolator based on N-data points, as represented by the
equations (2.28)-(2.30). These other N-sample interpolators are
within the scope and spirit of the present invention. For example
and without limitation, an interpolator with N=8 is discussed
below.
[0241] 2.6.2 The Simplified Structure for N=8
[0242] According to (2.30), we choose 24 K = 1 4 ( y ( 1 ) + y ( 3
) + y ( - 3 ) + y ( - 1 ) - y ( 0 ) - y ( 2 ) - y ( 4 ) - y ( - 2 )
) . ( 2.31 )
[0243] The coefficient values can be computed following this
procedure:
p1=(4K+y(1)-y(-3))cos .pi./4
p2=(4K+y(3)-y(-1))cos .pi./4 (2.32)
c.sub.0=2(y(1)+y(3)+y(-3)+y(-1))
c.sub.1=[y(0)-y(4)-4K+p1-p2]+j[y(-2)-y(2)-4K-p1-p2]
c.sub.2=[4K+y(0)+y(4)-y(2)-y(-2)]+j[4K+y(3)+y(-1)-y(1)-y(-3)]
c.sub.3=[y(0)-y(4)-4K-p1+p2]+j[-y(-2)+y(2)+4K-p1-p2]
c.sub.4=0. (2.33)
[0244] The resulting modified structure for N=8 is shown in FIG. 15
as interpolator 1500. Similar to the interpolator 1400, the
interpolator 1500 includes a delay module 1504, an adder/subtractor
module 1506, an angle rotator module 1508, a multiplier 1510, and
an output scaling adder 1512. As in the interpolator 1400, the
multiplier 1510 substantially replaces an angle rotator module. As
in the interpolator 1100 for N=8 (FIG. 11), only two non-trivial
scaling multiplications are needed for the modified structure
1500.
[0245] 2.6.3 Performance Comparisons with Other Structures
[0246] How does the simplified interpolator 1400 (FIG. 14) perform
as compared to the interpolator 1000 (FIG. 16)? FIGS. 16A-C show
the frequency responses, in solid lines, of the Lagrange cubic
interpolator 400 (FIG. 14), the interpolator 1000 (FIG. 10) and the
simplified interpolator 1400 (FIG. 14), respectively. For an input
signal whose spectrum is a raised cosine with .alpha.=0.4, as shown
in dashed lines, the amount of interpolation error corresponds to
the gray areas. Clearly, the interpolator 1400 produces less error
than the Lagrange cubic interpolator 400 and the interpolator 1000
. (FIG. 16D will be discussed in Section 4.)
[0247] Next, let us verify this performance improvement by
interpolating two signals: Signal 1, which is the same as signal
802 in FIG. 8A (.alpha.=1.0), and Signal 2, which is signal 1602 in
FIG. 16 (.alpha.=0.4). Then, the NMSE values are compared. We use
three interpolators: 1) Lagrange cubic interpolator 400; 2) the
trigonometric interpolator 1000; and 3) and the trigonometric
interpolator 1460. Also compared is the number of multipliers.
[0248] The results in Table 2.2 show that the modified structure
for N=4 not only requires less hardware, it also obtains the
highest accuracy among the three methods for these practical
signals used in our simulation.
3TABLE 2.2 Comparison of interpolators for N = 4. Lagrange
Structure in Structure in N = 4 cubic NMSE for Signal 1 in dB
-25.80 -28.45 -29.41 NMSE for Signal 2 in dB -31.08 -31.21 -33.51
Nontrivial scaling 2 0 0 multipliers Data multipliers 3 * *
Multipliers in critical path 4 2 2 * The trigonometric interpolator
1000 employs two one-output angle-rotators, each having the
hardware equivalent of slightly more than two multipliers. The
trigonometric interpolator 1400 employs one such angle-rotator and
one multiplier yielding an equivalent of slightly more than three
multipliers.
[0249] 2.7 Trigonometric Interpolator Application
[0250] An important application of the interpolation method and
apparatus described in this patent is the following. It is often
necessary to increase the sampling rate of a given signal by a
fixed integer factor. For example, a signal received at a rate of
1000 samples per second might need to be converted to one at a rate
of 4000 samples per second, which represents an increase of the
sampling rate by the integer factor four. There are methods in
common practice for doing such a conversion. One method is a very
popular two-step process wherein the first step creates a signal at
the desired higher sampling rate but one where simple zero-valued
samples (three of them in the example situation just mentioned) are
inserted after each input data value. The second step in the
interpolation scheme is to pass this "up-sampled" or "data-rate
expanded" signal through an appropriately designed lowpass digital
filter which, in effect, smoothes the signal by "filling in" data
values at the previously inserted zero-valued samples. In the
process of doing this filtering operation it might or might not be
important that the original data samples remain intact, and when
this is important there exist certain special lowpass filters that
will not alter those samples.
[0251] We can easily adapt the trigonometric interpolator described
herein to efficiently create such a sampling rate conversion
system, but one that does not require such filtering operations. If
we denote the integer factor by which we desire to increase the
data rate as L (in the above example, L=4) we proceed as follows.
We build the system 7800 shown below in FIG. 78A. System 7800
includes a Delay Module 7802 and Add/Subtract Module 7804 (that are
similar to that in FIG. 10), and such that it can accommodate
incoming data at a rate r. We now build L copies of the
Angle-Rotation Module 7806 (similar to that in FIG. 10), with each
one being fed by the same outputs of the Add/Subtract Module.
Within each of these L Angle-Rotation Modules 7806 we fix the .mu.
value; that is, each one has one has a different one of the values:
1/L, 2/L, . . . , (1-L)/L. With such fixed 4 values, each
Angle-Rotation Module 7806 can be constructed as a set of fixed
multipliers (a very special case of the table-lookup method),
although any of the Angle-Rotation Module implementations
previously discussed can be employed.
[0252] As shown in FIG. 78, the L-1 outputs, i.e., the interpolated
samples that are offset by the values 1/L, 2/L, . . . , (L-1)/L
from the first of the two data points (indicated as .mu.=0 and
.mu.=1 in the Delay Module of FIG. A) are routed to a multiplexer
7808, along with the input data point from which all interpolated
samples are offset. The multiplexer 7808 simply selects these
samples, in sequence, and provides them to the output at the
expanded data rate L.times.r.
[0253] A major advantage of the system 7800 is that all of the
system's components are operated at the (slow) input data rate
except the output multiplexer 7808. If desired, it would also be
possible to employ fewer Angle-Rotation Modules 7806, but operating
them at a higher data rate, and using several .mu. values,
sequentially, for each. This would result in a system that employed
less hardware but one that traded off the hardware savings for a
higher data rate operation of such modules.
[0254] 2.8 Trigonometic Interpolator Summary
[0255] In this Section we have described an interpolation method
that we have devised that uses trigonometric series for
interpolation. Comparing the interpolations using the trigonometric
polynomial and the Lagrange polynomial of the same degree, the
trigonometric-based method achieves higher interpolation accuracy
while simultaneously reducing the computation time and the amount
of required hardware. Moreover, the trigonometric-based method
preforms operations that are similar to that of a phase rotator for
carrier phase recovery. This allows a reduction in the overall
synchronization circuit complexity by sharing resources.
[0256] This trigonometric interpolator yields less computational
delay, as compared to algebraic interpolators. To achieve the same
throughput rate, this translates into more savings in hardware
using the proposed structure, because the data registers that are
required by algebraic interpolators to pipeline the computation for
a faster rate would not be needed by our structure.
[0257] We have also introduced two implementations of the
trigonometric interpolation method: one using a lookup table, and
one using an angle rotation processor (to be discussed in Section
5).
[0258] After introducing a first interpolation method, we have
shown that we can trade one angle rotator for a multiplier by
conceptually modifying the input samples, then by "correcting" the
interpolated value obtained from the "modified" samples. Through
this modification, we have obtained a simpler implementation
structure while simultaneously improving the performance when
interpolating most practical signals. This performance improvement
has been demonstrated by comparing the frequency responses of the
interpolators and the mean-squared interpolation errors using these
interpolators. Our discussion of the optimal digital resampler in
Section 4 will be based on this simplified interpolator.
3. Interpolation Filters with Arbitrary Frequency Response
[0259] In Section 2, an interpolation method using a trigonometric
polynomial was introduced, along with an example of such an
interpolation structure of length N=4. In addition to being a very
simple structure, our analyses and simulations also demonstrated
that the trigonometric interpolator outperformed the interpolator
of the same length using a Lagrange polynomial. In this Section, a
more systematic approach will be taken to analyze this method from
the digital signal processing point of view. A formula giving its
impulse response allows us to analyze the frequency response of the
interpolation filter. We then show how to modify the algorithm to
achieve arbitrary frequency response. The discussions in this
Section will provide the framework for the optimal interpolator of
Section 4.
[0260] 3.1 Formulating the Trigonometric Interpolator as an
Interpolation Filter
[0261] We have shown that, given N equally-spaced samples y(n), a
continuous-time signal can be reconstructed as 25 y ( t ) = n = - N
2 + 1 N 2 y ( n ) f ( t - n ) ( 3.1 )
[0262] where .function.(t) is the impulse response of a
continuous-time interpolation filter. As in Section 2, it is
assumed that the sampling period is T.sub.S=1. This assumption
makes the notation simpler and the results can easily be extended
for an arbitrary T.sub.S. In other words, the invention is not
limited to a sampling period of T.sub.S=1, as other sampling
periods could be utilized. In Section 2 we have shown that
.function.(t) can be expressed as: 26 f ( t ) = ( 1 + 2 k = 1 N 2 -
1 cos 2 N kt + cos t t N 2 0 t > N 2 . ( 3.2 )
[0263] FIG. 20 illustrates .function. for the trigonometric
interpolation filters for N=8 and N=16. By computing the Fourier
transform of .function., we obtain the frequency response of the
interpolation filter. The frequency responses for the N=8 and N=16
cases are plotted in FIG. 21. Since .function.(t) is real and
symmetric around t=0, its frequency response has zero phase. In
FIG. 21, the oscillatory behavior near the band edge is quite
obvious. In addition, by comparing FIGS. 21a and b, we can see that
as the filter length is increased from N=8 to N=16 the amount of
ripple does not decrease. Well known as the Gibbs phenomenon, the
magnitude of the ripples does not diminish as the duration of the
impulse response is increased.
[0264] It is apparent that the amount of oscillation cannot be
reduced using the method discussed thus far while only increasing
the filter length N. Moreover, it seems that an arbitrary frequency
response cannot be achieved using this method. To address these
problems, let us examine how the frequency response of this method
is determined.
[0265] 3.2 Analysis of the Frequency Response
[0266] Let us examine how the frequency response of .function. for
the trigonometric interpolator is obtained using the example in
FIGS. 22 with N=8. According to (3.2), the interpolation filter's
impulse response .function.(t) on the interval 27 - N 2 t N 2
[0267] is a weighted sum of cosine functions. We can view the
finite-length filter .function. in (3.2) as having been obtained by
applying a window 2206 according to the following: 28 w ( t ) = { 1
t N 2 0 t > N 2 ( 3.3 )
[0268] to an infinitely-long, periodic function 2204 with period N:
29 f c ( t ) = 1 + 2 k = 1 N 2 - 1 cos 2 N kt + cos t - .infin.
< t < .infin. = k = - N 2 + 1 N 2 - 1 W N kt + 1 2 W N - ( N
2 ) t + 1 2 W N ( N 2 ) t such that ( 3.4 ) f ( t ) = f c ( t ) w (
t ) , - .infin. < t < .infin. . ( 3.5 )
[0269] Thus F the frequency response of .function. can be obtained
by convolving F.sub.c and W the Fourier transforms of
.function..sub.c and w, respectively.
[0270] The Fourier transform of the periodic function 71
.sub.c(t),-.infin.<t<.infin., is 30 F c ( ) = k = - N 2 + 1 N
2 - 1 ( - 2 N k ) + 1 2 ( - ) + 1 2 ( + ) ( 3.6 )
[0271] which consists of a sequence of impulses 2208. We will
subsequently refer to the weights of these impulses as frequency
samples. Denoting the weight of 31 ( - 2 N k )
[0272] by {circumflex over (F)}(k), we have 32 F c ( ) = k = - M M
F ^ ( k ) ( - 2 N k ) ( 3.7 )
[0273] where 33 M N 2
[0274] is an integer. In the case of (3.6), 34 M = N 2 .
[0275] For our particular interpolation filter, according to (3.6),
all in-band frequency samples {circumflex over (F)}(k)=1 for
.vertline.k.vertline.<N/2. For .vertline.k.vertline.>N/2, the
out-of-band samples {circumflex over (F)}(k)=0. The two samples in
the transition band are {circumflex over (F)}(N/2)={circumflex over
(F)}(-N/2)=1/2. The transition bandwidth is determined by the
distance between the last in-band, and the first out-of-band
frequency samples.
[0276] Since w is a rectangular function, W must be a sinc function
2210. Convolving F.sub.c and the sinc function W simply
interpolates the frequency samples to obtain
F(.OMEGA.),-.infin.<.OMEGA.<.infin., shown as response 2212.
(Here we have plotted the symmetric F only on the positive half of
the .OMEGA. axis.) We thus have 35 F ^ ( k ) = F ( ) | = 2 k / N ,
- N 2 k N 2 . ( 3.8 )
[0277] From response 2212, the continuous-frequency response
F(.OMEGA.) is uniquely determined by an infinite number of equally
spaced frequency samples {circumflex over (F)}(k). If we modify the
frequency samples 2214 near the passband edge to let the transition
between the passband and stopband be more gradual, as depicted in
FIG. 23, then the ripple is decreased. FIG. 23 demonstrates
gradually reduced samples 2302, and the reduction of ripples in the
overall response 2304, as compared to the response 2212 in FIG. 22.
The cost of this improvement is, an increased transition bandwidth
in the response 2304, as compared to the response 2212.
[0278] If a narrower transition band is desired, we can increase
the duration of the filter .function.(t). This can be seen by
comparing FIG. 24, where N=16, with response 2304 in FIG. 23, in
which N=8.
[0279] 3.3 Implementing the Modified Algorithm
[0280] By comparing (3.6) and (3.4) we can see that the frequency
sample values, i.e., the weights of the impulses in (3.6), are
determined by the weights in the sum in (3.4).
[0281] We can modify our original interpolation filter in (3.2) for
.vertline.t.vertline..ltoreq.N/2 as 36 f m ( t ) = F ^ ( 0 ) + 2 k
= 1 M F ^ ( k ) cos 2 N k t . ( 3.9 )
[0282] By expressing (3.9) using the W.sub.N notation, for
.vertline.t.vertline..ltoreq.N/2 we have 37 f m ( t ) = F ^ ( 0 ) W
N 0 t + k = 1 M F ^ ( k ) ( W N k t + W N - k t ) ( 3.10 ) = k = -
M M F ^ ( k ) W N - k t .
[0283] and .function..sub.m(t)=0 for .vertline.t.vertline.>N/2.
Substituting this result into (3.1), and re-ordering terms, we have
38 y ( t ) = 1 N k = - M M c ^ k W N - k t ( 3.11 ) where c ^ k = F
^ ( k ) ( n = - N / 2 + 1 N / 2 y ( n ) W N kn ) . ( 3.12 )
[0284] By comparing (3.12) to (2.9), we can see that, for 39 k = -
N 2 + 1 , , N 2 , .sub.k={circumflex over (F)}(k)c.sub.k.
(3.13)
[0285] Thus, a modified algorithm can be implemented as the
following steps:
[0286] Step 1': (Same as Step 1 in Section 2.4): Given an even
number of samples N, calculate the Fourier coefficients c.sub.k,
k=0, . . . , N/2 using (2.9).
[0287] Step 2': Multiply the coefficients c.sub.k by scale factors
{circumflex over (F)}(k) using (3.13).
[0288] Step 3': Given a fractional delay .mu., compute the
synchronized samples using (3.11), which, due to .sub.k*=.sub.-k,
can be simplified as: 40 y ( ) = 1 N Re ( c ^ 0 + 2 k = 1 M c ^ k W
N - k ) . ( 3.14 )
[0289] It seems that, in Step 3', we need coefficients (k) (hence
c.sub.k) not only for k.ltoreq.N/2 but also for k>N/2 while only
c.sub.k values for k>N/2 are computed in Step 1'. However,
c.sub.k values for k>N/2 can be obtained using
c.sub.k=c.sub.k-mN (3.15)
[0290] where m is an integer such that 0.ltoreq.k-mN.ltoreq.N/2. We
have (3.15) because c.sub.k is periodic in k with period N, because
c.sub.k is obtained from the Fourier transform of the discrete-time
signal y(n), -N/2+1.ltoreq.n.ltoreq.N/2.
[0291] At this point, we have shown that the continuous-time
frequency response of the interpolation filter having impulse
response .function.(t) can be improved by modifying the weights
{circumflex over (F)}(k) in (3.10). Now a question arises: the
modification of the weights would alter the shape of the impulse
response of the .function.(t) filter. How do we guarantee that the
resulting filter does not change the original samples?
[0292] 3.4 Conditions for Zero ISI
[0293] In order for .function.(t) not to alter the original samples
when used for interpolation as in (3.1), it must have
zero-crossings at integer multiples of the sampling period: 41 f (
n ) = { 1 n = 0 0 n 0 , n an integer . ( 3.16 )
[0294] The well-known Nyquist condition for zero ISI (Proakis, J.
G., Digital Communications, McGraw-Hill, New York, N.Y. (1993))
states that the necessary and sufficient condition for (3.16) is 42
n = - .infin. .infin. F ( - 2 n ) = 1 - .infin. < < .infin. .
( 3.17 )
[0295] Since the filter's impulse response .function.(t) has a
finite duration, i.e. .function.(t)=0 for
.vertline.t.vertline.>N/2, (3.16) holds if and only if the
frequency samples {circumflex over (F)}(k) satisfy 43 n = - .infin.
.infin. F ^ ( k - N n ) = 1 , k = integer . ( 3.18 )
[0296] The proof is given in Appendix A.
[0297] In summary, we can still guarantee that the modified
interpolation filter .function. does not alter the original samples
as long as the modified weights {circumflex over (F)}(k) (frequency
samples) satisfy (3.18). Using this constraint, we can design the
weights {circumflex over (F)}(k) to meet given frequency response
requirements.
[0298] 3.5 Optimization Algorithm
[0299] Using the approach discussed, one can approximate an
arbitrary frequency response by choosing appropriate weights
{circumflex over (F)}(k). For example, a desired frequency response
F.sub.d(.OMEGA.) for an interpolation filter should be unity in the
passband and be zero in the stopband, as 44 F d ( ) = { 1 - 2 2 0
> . ( 3.19 )
[0300] The interpolation error using our interpolation filter is
defined as
e(.OMEGA.)=W.sub.t(.OMEGA.).vertline.F.sub.d(.OMEGA.)-F(.OMEGA.).vertline.
(3.20)
[0301] where W.sub.t(Q) is a weighting function.
[0302] From Section 3.2 we have F(.OMEGA.)=F.sub.c(.OMEGA.) {circle
over (x)} sin c(.OMEGA.). Thus, we can express F(.OMEGA.) in terms
of {circumflex over (F)}(k), using (3.7), as 45 F ( ) = ( k = - M M
F ^ ( k ) ( - 2 N k ) ) sin c ( ) ( 3.21 ) = k = - M M F ^ ( k )
sin c ( - 2 N k ) .
[0303] An optimal interpolation filter can be designed by choosing
{circumflex over (F)}(k) to minimize the peak interpolation error,
as 46 L .infin. = max { e ( ) } ( 3.22 )
[0304] or the mean-squared interpolation error 47 L 2 = - .infin.
.infin. 2 ( ) ( 3.23 )
[0305] subject to the constraint described by (3.18).
[0306] By examining FIGS. 23 and 24, we can see that, by modifying
only two frequency samples, those nearest the band edge, a
significant improvement is achieved. In these cases we have
{circumflex over (F)}(k)=0 for .vertline.k.vertline.>N/2+1.
[0307] 3.6 Conclusion
[0308] In this Section, an interpolation method was presented that
achieves arbitrary frequency response by modifying the
trigonometric interpolator discussed in Section 2. Using this
approach, the performance of a trigonometric interpolation filter
can be further improved.
[0309] It is interesting to note that this procedure is equivalent
to the well-known filter design method using windows. FIG. 25a
depicts the impulse responses of the original filter (3.2) as the
dashed line, and the modified filter (3.9) as the solid line. By
comparing the two impulse responses, we have found a function
illustrated in FIG. 25b. If we multiply the original impulse
response by this function, we get the impulse response that we
obtained by modifying the frequency samples. Therefore, this
function is equivalent to a window. According to this
interpretation, our frequency domain design method is equivalent to
designing a better window than the rectangular window (3.3) in the
time domain.
4. Design of Optimal Resamplers
[0310] 4.1 Motivation
[0311] We have thus far discussed digital resampling using
interpolation methods. To accomplish this, we conceptually
reconstruct the continuous-time signal by fitting a trigonometric
polynomial to the existing samples and then re-sample the
reconstructed signal by evaluating this polynomial for a given
sampling mismatch (or offset) .mu.. The reconstruction of the
continuous-time bandlimited signal y(t) from existing samples y(m)
using interpolation filter .function.(t), according to (3.1), is 48
y ( t ) = m = - N / 2 + 1 N / 2 y ( m ) f ( t - m ) . ( 4.1 )
[0312] Then y is resampled at t=.mu. as 49 y ( ) = m = - N / 2 + 1
N / 2 y ( m ) f ( - m ) = y f ( 4.2 )
[0313] where .function..sub..mu.(m)=.function.(m-.mu.).
[0314] In the previous sections we approached the problem only from
the point of view of reconstructing the continuous-time signal, as
in (4.1), since we have only examined the frequency response of the
continuous-time filter .function.(t). However, what we actually are
interested in is the new sample y(.mu.) that is obtained by
resampling the continuous-time signal at a new sampling instant,
t=.mu..
[0315] Now, a question arises: Even when the frequency response
F(.OMEGA.) of the continuous-time filter is optimized as in Section
3.5, do we necessarily obtain the minimum error in producing a new
sample y(.mu.) for each .mu. value?
[0316] According to (4.2), the new sample y(.mu.) is actually
obtained by filtering the original samples y(m) using a
discrete-time filter whose impulse response .function..sub..mu.(m)
depends on a particular delay .mu.. What is the desired frequency
response of the .function..sub..mu.(m) filter?
[0317] A digital resampler that is used in timing recovery simply
compensates for the timing offset in sampling the received signal.
Ideally, the .function..sub..mu.(m) filter should not alter the
signal spectrum as it simply delays the existing samples by .mu..
Obviously, the desired frequency response of the discrete-time
.function..sub..mu.(m) filter is
F.sub.d(.omega., .mu.)=e.sup.-j.omega..mu. (4.3)
[0318] where .omega. is the normalized angular frequency. Let us
define the frequency response of .function..sub..mu.(m) as
F.sub..mu.(.omega.). The error in approximating the ideal frequency
response F.sub.d(.omega., .mu.) by F.sub..mu.(.omega.) for a given
.mu. value, is
e(.omega.)=W.sub.t(.omega.).vertline.F.sub.d(.omega.,
.mu.)-F.sub..mu.(.omega.).vertline. (4.4)
[0319] where W.sub.t(.omega.) is a weighting function.
[0320] We now examine how the discrete-time fractional-delay
frequency response F.sub..mu.(.omega.) is obtained. We denote by
F(.OMEGA.) the Fourier transform of the continuous-time filter
.function.(t). Hence, the Fourier transform of .function.(t-.mu.)
must be e.sup.-.OMEGA..mu.F(.OMEG- A.). We know that
.function..sub..mu.(n)=.function..sub..mu.(n-.mu.) are just samples
of .function.(t-.mu.), where -.infin.<t<.infin.. Therefore,
according to the sampling theorem (Proakis, J. G., Digital Signal
Processing, Macmillan, New York, N.Y. (1992)), the Fourier
transform of .function..sub..mu.(n) is 50 F ( ) = k = - .infin.
.infin. - j ( - 2 k ) F ( - 2 k ) ( 4.5 )
[0321] after we replace .OMEGA. on the right-hand-side expression
by the normalized angular frequency variable .omega.
(.omega.=.OMEGA. since T.sub.s=1). This relationship is shown in
FIG. 26, where F(.OMEGA.) corresponds to the N=8 interpolator of
FIG. 11. As discussed in Section 3.2, .function.(t) is symmetric
around t=0. This implies that F(.OMEGA.) has zero phase. To make
the .function.(t) filter physically realizable, of course, we must
introduce a delay of N/2, where N corresponds to the length of the
filter. However, this delay simply "shifts" all input samples by
N/2, which is an integer because N is even, and it does not change
the characteristic of the input signal. Thus, it does not influence
the interpolation accuracy. Therefore, to simplify our notation, we
just use F(.OMEGA.) as a real function. Hence, the phase of the
complex function e.sup.-.function..OMEGA..mu.F(.OMEGA.) is
-.OMEGA..mu. if F(.OMEGA.).gtoreq.0, or --.OMEGA..mu.+.pi. if
F(.OMEGA.)<0--the phase depends on .mu.. FIG. 26 shows that the
frequency response of the discrete-time filter F.sub..mu.(.omega.)
is obtained by first making an infinite number of copies of
e.sup.-.function..OMEGA..mu.F(.OMEGA.) by shifting this function
uniformly in successive amounts of 2.pi., then by adding these
shifted versions to e.sup.-.function..OMEGA..mu.F(.OMEGA.). As a
sum of complex functions, the shape of F.sub..mu.(.OMEGA.) depends
not only on the shape of the continuous-time frequency response
F(.OMEGA.) but also on the value .mu.. The dependence of
F.sub..mu.(.omega.) on .mu. is illustrated in FIG. 27, where
F.sub..mu.(.omega.) is obtained from the function F(.OMEGA.) in
FIG. 26, using .mu.=0.12 and .mu.=0.5. The magnitude of the ideal
fractional-delay frequency response, defined in (4.3), is shown in
both FIG. 27A and FIG. 27B as the dashed lines. It is evident that
the frequency response 2706 is worse for .mu.=0.5 than the
frequency response 2704 is for .mu.=0.12, since the response 2706
deviates more from the ideal frequency response 2702 than does the
frequency response 2704. Hence, the interpolation error is larger
for .mu.=0.5 than for .mu.=0.12. We have observed in our
simulations that the largest interpolation error occurs when
.mu.=0.5, i.e., when the interpolated sample is exactly in the
middle of the two nearest existing samples. As .mu. approaches 0 or
1 (i.e., as the interpolated sample gets closer to an existing
sample), the interpolation error becomes smaller. Moreover, the
interpolation errors obtained for .mu. and 1-.mu. are the same.
[0322] 4.2 Resampler Optimizations
[0323] In Section 3, we analyzed the relationship between the
weights {circumflex over (F)}(k) in (3.9) and the frequency
response of the interpolator. We have shown that we can enhance
F(.OMEGA.) by adjusting the {circumflex over (F)}(k) values. In
FIG. 23 (N=8), for example, the {circumflex over (F)}(3) and
{circumflex over (F)}(5) values correspond to the magnitude of the
pulses 2302 near the band edge. If we adjust {circumflex over
(F)}(3) and {circumflex over (F)}(5) such that the transition
between the passband and stopband is more gradual, we can achieve a
better frequency response.
[0324] To further improve the interpolation performance, we could
take it into account, by optimizing F.sub..mu.(.omega.) for each
.mu. value. As in Section 3, we could adjust {circumflex over
(F)}(k) near the band edge to change the shape of F(.OMEGA.), for
each .mu. value, such that the discrete-time frequency response
F.sub..mu.(.omega.), which is obtained from (4.5), best
approximates the desired response of (4.3).
[0325] As discussed in Section 3, to guarantee that the original
samples are not altered using the modified interpolator, the
weights {circumflex over (F)}(k) should satisfy (3.18). When N=8,
for example, we modify {circumflex over (F)}(3) and {circumflex
over (F)}(5) together in order to satisfy (3.18). Here, however,
our goal of optimization is to obtain the best approximation for
the sample corresponding to a specific .mu.. Hence we need not be
concerned with the issue of altering the original samples in
Section 3, where there is only one set of optimized weights for all
.mu. values.
[0326] Let us demonstrate this method using the example of N=8. We
chose to modify {circumflex over (F)}(3) and {circumflex over
(F)}(4). For each given .mu., we search for the {circumflex over
(F)}(3) and {circumflex over (F)}(4) values such that (4.4) is
minimized. We denote such {circumflex over (F)}(3) and {circumflex
over (F)}(4) values by {circumflex over (F)}.sub..mu.(3) and
{circumflex over (F)}.sub..mu.(4) respectively, since they are now
also dependent on .mu.. FIGS. 28A and B show the modifications to
F(.OMEGA.) for .mu.=0.12 and .mu.=0.5, respectively. The
corresponding optimized F.sub..mu.(.omega.) functions are
illustrated in FIGS. 28C and 28D, respectively.
[0327] To demonstrate the performance improvement, let us use this
example: Given .mu.=0.5, we optimize F.sub..mu.(.omega.) for the
signal whose spectrum is shown in dashed lines 2902 in FIG. 29.
Comparing the un-optimized frequency response 2904 with the
optimized frequency response 2906, the modification clearly
produces a better frequency response. More specifically, the
response 2906 is flat in the frequency band where the power of the
signal 2902 is concentrated, and the deviation from the ideal
response mostly falls in the "don't care" band.
[0328] Similar to Section 3, to implement this improved method, we
first compute the coefficients c.sub.k from the existing samples as
in (2.9). Then, given the .mu. value, we multiply, e.g., for N=8,
the c.sub.3 and c.sub.4 values by {circumflex over (F)}.sub..mu.(3)
and {circumflex over (F)}.sub..mu.(4), respectively. Finally, we
compute the synchronized sample y(.mu.) using (2.11), where c.sub.3
and c.sub.4 are replaced by {circumflex over
(F)}.sub..mu.(3)c.sub.3 and {circumflex over
(F)}.sub..mu.(4)c.sub.4, respectively.
[0329] We can apply similar modifications to the interpolator with
N=4. FIG. 30A show the frequency response of the interpolator 1000,
for .mu.=0.5, while FIG. 30B displays the results of a modified
interpolator 1000, where parameters {circumflex over (F)}(1) and
{circumflex over (F)}(2) are optimized, for .mu.=0.5, to maximize
the interpolation accuracy for the signal whose spectrum is shown
in dashed lines. As can be seen the optimized response 3006 is
flatter in the part of the spectrum of the signal 3002 where most
of its energy is concentrated than is the un-optimized response
3004.
[0330] The flowchart 3400 in FIG. 34 generalizes the optimization
of the trigonometric optimization procedure. The flowchart 3400 is
similar to the flowchart 1700, but includes the additional steps of
3402 and 3404 that are described as follows.
[0331] In step 3402, a factor {circumflex over (F)}.sub..mu. is
determined to adjust the frequency response of the trigonometric
interpolator so that it is consistent with the frequency response
of N-data samples and the offset .mu..
[0332] In step 3404, one or more of the complex coefficients are
multiplied by the {circumflex over (F)}.sub..mu. to modify the
frequency response of the interpolator so that it is consistent
with the input signal and the offset .mu..
[0333] The optimization routine can also be used with K-modified
data samples that leads to the simplified interpolator structures
of FIGS. 14 and 15. The flowchart 3500 illustrates the {circumflex
over (F)}.sub..mu. factor modification in the context of the
flowchart 1900.
[0334] As will be shown in the section that follows, the steps
3402, 3404, and 1708 can be combined into a single step if a table
lookup is used to determine the rotation factor. In other words,
the sine and cosine values can be multiplied by the {circumflex
over (F)}.sub..mu. factor before they are stored in the ROM.
[0335] In Section 2.6, we have presented an efficient algorithm
that eliminates one angle-rotation. For example, for N=4, we can
"modify" the input samples according to (2.22). With this
modification, we can treat the new samples as if the input signal
satisfies c.sub.2=0. The remaining non-zero coefficients are
c.sub.0 and c.sub.1. In the example for N=4 in the previous
Section, two parameters, {circumflex over (F)}.sub..mu.(1) and
{circumflex over (F)}.sub..mu.(2), are optimized to achieve the
best approximation of a desired fractional-delay frequency response
described by (4.3). Now, with c.sub.2=0, we have only one
parameter, {circumflex over (F)}.sub..mu.(1), to choose.
[0336] The impulse response of the simplified interpolation filter
is derived in Appendix B. From the mathematical expression of the
impulse response (B.5), we can obtain the corresponding frequency
response. The frequency responses of the interpolator 1400 (FIG.
14) before and after applying the {circumflex over (F)}.sub..mu.(1)
modification are shown in FIG. 31A-B, respectively. We can see an
improved frequency response 3106 over the response 3104, as the
response 3106 is flatter in the part of the signal 3102 where its
energy is concentrated. Furthermore, it seems that the frequency
response 3106, where only c.sub.1 is modified (c.sub.2=01), is as
good as the modified response 3006 in FIG. 30B where both c.sub.1
and c.sub.2 are modified.
[0337] 4.3 Implementations
[0338] It may appear that additional hardware is needed to
implement the multiplication by, for example, {circumflex over
(F)}.sub..mu.(1) for the simplified N=4 structure. Let us examine
the corresponding computations. As we know, we first compute
coefficients c.sub.0 and c.sub.1 according to (2.25) (c.sub.2=0, of
course). We then compute y(.mu.) using 51 y ( ) = 1 4 ( c 0 ) + 1 2
Re ( F ^ ( 1 ) c 1 j 2 ) - K ( 4.6 )
[0339] according to (2.26) and (2.27), where K is defined in
(2.24). As discussed in Section 2.4, the computation 52 Re ( F ^ (
1 ) c 1 j 2 ) = Re ( c 1 ) Re ( F ^ ( 1 ) j 2 ) - Im ( c 1 ) Im ( F
^ ( 1 ) j 2 ) ( 4.7 )
[0340] can be accomplished by retrieving the 53 F ^ ( 1 ) j 2
[0341] value from a ROM lookup table and then multiplying
Re(c.sub.1)+j Im(c.sub.1) by the retrieved value, since both
{circumflex over (F)}.sub..mu.(1) and 54 j 2
[0342] can be pre-determined for all .mu. values.
[0343] If the angle-rotation method is used, we can use a lookup
table to store the {circumflex over (F)}.sub..mu.(1) values. In
real-time, after computing 55 c 1 j 2
[0344] using an angle-rotation processor, we can multiply the
result by {circumflex over (F)}.sub..mu.(1). In this case, if
{circumflex over (F)}.sub..mu.(1) is allowed to be a complex number
in optimizing performance, we then need two real multiplications to
obtain 56 Re ( F ^ ( 1 ) c 1 j 2 )
[0345] in (4.6). However, if we restrict {circumflex over
(F)}.sub..mu.(1) to be a real number, we can use just one real
multiplication as 57 Re ( F ^ ( 1 ) c 1 j 2 ) = F ^ ( 1 ) Re ( c 1
j 2 ) ( 4.8 )
[0346] According to Table 4.1, the NMSE using complex and real
{circumflex over (F)}.sub..mu.(1) values are -37.41 dB and -37.08
dB, respectively. Therefore, the performance degradation caused by
restricting {circumflex over (F)}.sub..mu.(1) to be a real number
is insignificant.
[0347] When the table-lookup method is employed, the implementation
structure for the optimal interpolator is the same as that for the
interpolator 1400, except for the coefficient c.sub.1 which is now
multiplied by 58 F ^ ( 1 ) j 2
[0348] instead of 59 j 2 .
[0349] The table should therefore contain the 60 Re ( F ^ ( 1 ) j 2
) and Im ( F ^ ( 1 ) j 2 )
[0350] values, rather than the 61 sin 2 and cos 2
[0351] values used by the interpolator 1400. We now show that the
size of the table is the same as the one storing the sine and
cosine values.
[0352] Let us examine the contents of the lookup table. FIG. 32
displays the 62 Re ( F ^ ( 1 ) j 2 ) and Im ( F ^ ( 1 ) j 2 )
[0353] values, used by (4.7), where the real values are represented
by curve 3202, and the imaginary values are represented by the
curve 3204. These values are monotonic with respect to .mu., just
like the 63 sin 2 and cos 2
[0354] values for 0.ltoreq..mu..ltoreq.1. Moreover, simulations
show that, when optimal values of {circumflex over (F)}.sub..mu.(1)
are reached, the real and imaginary components of 64 F ^ ( 1 ) j
2
[0355] display the following complementary relationship: 65 Im ( F
^ ( 1 ) j 2 ) = Re ( F ^ 1 - ( 1 ) j 2 ( 1 - ) ) . ( 4.9 )
[0356] Therefore, we need only store one of the 66 Re ( F ^ ( 1 ) j
2 ) and Im ( F ^ ( 1 ) j 2 )
[0357] values. The other can be obtained by looking-up the value
corresponding to 1-.mu.. This is the same technique used in storing
and retrieving the sine and cosine values with the purpose of
reducing the table size.
[0358] Various circuit implementations of optimized interpolators
having N=4 are illustrated in FIGS. 36-37. These circuit
implementations are presented for example purposes only and are not
meant to be limiting, as those skilled in the arts will recognize
other circuit implementation based on the discussion given herein,
including interpolator configurations having different N
values.
[0359] FIG. 36 illustrates an optimized interpolator 3600 that is
based on the simplified interpolator 1400 (FIG. 14). The
interpolator 3600 includes an {circumflex over (F)}.sub..mu. ROM
3602 and a multiplier 3604. The ROM 3600 stores the appropriate
{circumflex over (F)}.sub..mu. value indexed by A. The multiplier
3604 multiples the complex coefficient C.sub.1 by the appropriate
{circumflex over (F)}.sub..mu.0 value, and therefore optimizes the
frequency response of the interpolator 3600. As discussed above,
the order of the angle rotator 1010b and the multiplier 3604 can be
interchanged so that the rotated complex coefficient is modified by
the {circumflex over (F)}.sub..mu. value.
[0360] FIG. 37 illustrates an optimized interpolator 3700 that is
similar to the optimized interpolator 3600, except that the
{circumflex over (F)}.sub..mu. ROM 3602, the multiplier 3604, and
the angle rotator 1010b are combined into to a single ROM 3702,
that stores the 67 Re ( F ^ ( 1 ) j 2 ) and Im ( F ^ ( 1 ) j 2
)
[0361] values. Therefore, coefficient optimization and angle
rotation are performed in a simultaneous and efficient manner.
[0362] It will be apparent to those skilled in the arts that the
combined {circumflex over (F)}.sub..mu. and angle rotator ROM 3702
can be implemented for interpolator configurations that include
more than N=4 elements, based on the discussions herein.
[0363] 4.4 Simulation Results
[0364] We have verified the new design with the following
simulation. A baseband signal, shown in FIG. 33, with raised cosine
spectrum, two samples per symbol and 40% excess bandwidth was
generated. Table 4.1 compares the result for N=4 using four
interpolation structures: 1) the Lagrange cubic interpolator, 2)
the interpolator 1000, 3) the interpolator 1400, 4) the optimal
resampler using a complex {circumflex over (F)}.sub..mu.(1) value,
and 5) the optimal resampler employing a real {circumflex over
(F)}.sub..mu.(1) value.
[0365] Using the optimal structure, the NMSE is reduced by 4 dB
over the method without optimization (FIG. 14 structure). The
performance is improved by more than 6 dB compared to the Lagrange
cubic interpolator, while the hardware is reduced. Comparing the
optimal structure to the FIG. 14 structure, a 4 dB performance gain
was obtained without increasing the amount of hardware.
4TABLE 4.1 Comparison of interpolators for N = 4. Lagr. Optimal
Optimal cubic Struct. in Struct. in table- angle- N = 4 struct.
lookup.sup.a rotat..sup.b NMSE in dB -31.08 -31.21 -33.51 -37.41
-37.08 Scaling multipliers 2 0 0 0 0 Data multipliers 3 3 3 3 4
Multipliers in 4 2 2 2 3 critical path 68 a Complex F ^ ( 1 )
values are used . The table stones the F ^ ( 1 ) j 2 values . 69 b
Real F ^ ( 1 ) values are used . The output of the angle-rotator Re
( c , i j 2 ) is multiplied by {circumflex over (F)}.sub..mu.(1).
Thus, one more real multiplexer is needed.
[0366] The frequency response of an optimized interpolator 1400
(FIG. 14) using a lookup table is shown in FIG. 16D. Also shown in
FIG. 16A-D are the frequency responses of the Lagrange cubic
interpolator 400, the interpolator 1000, and the interpolator 1400
without optimization. The signal spectrum of FIG. 33 is shown in
FIG. 16D in dashed lines 1604. The interpolation error corresponds
to the gray area 1606. From FIG. 16D, the performance improvement
achieved by the optimal interpolator is evident because the gray
area 1606d is a lower amplitude than the corresponding gray areas
1606a-c. In addition, these improvements are accomplished without
increasing the amount of hardware.
[0367] For a high-performance interpolator, we now turn to the
structure described in Section 2.6.2, for N=8. Applying a similar
approach for N=4, as just discussed, to the N=8 interpolator of
Section 2.6.2, we can multiply the c.sub.3 coefficient by
{circumflex over (F)}.sub..mu.(3), whose value optimizes the
frequency response F.sub..mu.(.omega.) of a fractional-delay filter
with delay .mu..
[0368] In designing the proposed N=8 interpolator, only one
parameter {circumflex over (F)}.sub..mu.(3) was adjusted to
minimize the MSE in (3.22).
[0369] Next, three interpolators of length N=8 were used to
interpolate the signal in FIG. 33: 1) a Lagrange polynomial
interpolator, 2) a Vesma-Saramki optimal polynomial interpolator
(Vesma, J., and Saramki, T., "Interpolation filters with arbitrary
frequency response for all-digital receivers," in Proc. 1996 IEEE
Int. Symp. Circuits Syst. (May 1996), pp. 568-571) (with length 8
but a third degree polynomial) and, 3) the proposed
interpolator.
[0370] Table 4.2 shows the simulation results. These results
demonstrate that our method has an NMSE more than 16 dB lower than
the Lagrange interpolator, and more than 4 dB lower than the
Vesma-Saramki polynomial interpolator in (Vesma, J., and Saramki,
T., "Interpolation filters with arbitrary frequency response for
all-digital receivers," in Proc. 1996 IEEE Int. Symp. Circuits
Syst. (May 1996), pp. 568-571).
5TABLE 4.2 Performance comparison. Lagrange Vesma-Saramki Proposed
N = 8 interpolator interpolator interpolator NMSE in dB -45.29
-57.34 -62.17 Scaling multipliers 25 16 2 Multipliers 7 3 7
[0371] 4.5 Conclusion
[0372] Instead of optimizing F(.OMEGA.), the frequency response of
the continuous-time interpolation filter, we could optimize
F.sub..mu.(.omega.) of the fractional-delay filter for each .mu.
value. By doing this, better interpolation performance can be
achieved, as demonstrated by the simulations.
[0373] As for the implementation complexity, when a table-lookup
method is employed, the optimal interpolator does not require
additional hardware, just table values that implement the
coefficient optimization and angle rotation. When the angle
rotation method is used, one additional real multiplier is
needed.
[0374] For N=4, the optimal interpolator attained a 6 dB lower NMSE
than the Lagrange cubic interpolator, while requiring less
hardware.
5. A High-Speed Angle Rotation Processor
[0375] In previous Sections, an interpolation method and apparatus
for timing recovery using a trigonometric polynomial has been
discussed. The major computation in this method is the angle
rotation, such as angle rotator 1010 (in FIG. 10 and FIG. 14). As
mentioned in Section 2.4, these operations, together with the phase
rotator for carrier recovery, can be implemented by table-lookup in
a ROM containing pre-computed sine and cosine values, followed by
four real multipliers to perform the angle rotation (see FIG. 18).
Herein, going forward, this approach will be referred to as the
single-stage angle rotation. Although fast angle rotation can be
achieved with efficient multiplier design techniques, for practical
precision requirements, the ROM table can be quite large. For
applications where low complexity and low power are the major
concern, can we further reduce the amount of hardware for angle
rotation with slightly more computational delay?
[0376] There are various hardware designs that accomplish angle
rotations, notably the CORDIC processors (Ahn, Y., et al., "VLSI
design of a CORDIC-based derotator," in Proc. 1998 IEEE Int. Symp.
Circuits Syst., Vol. II (May 1998), pp. 449-452; Wang, S., et al.,
"Hybrid CORDIC algorithms," IEEE Trans. Comp. 46:1202-1207 (1997)),
and, recently, an angle-rotation processor (Madisetti, A., et al.,
"A 100-MHz, 16-b, direct digital frequency synthesizer with a
100-dBc spurious-free dynamic range," IEEE J. Solid-State Circuits
34:1034-1043 (1999)). These algorithms accomplish the rotation
through a sequence of subrotations, with the input to each
subrotation stage depending on the output of the previous stage. In
these cases, the latency is proportional to the precision of the
angle.
[0377] We now propose a different approach for angle rotation. Here
the rotation is partitioned into just two cascaded rotation stages:
a coarse rotation and a fine rotation. The two specific amounts of
rotation are obtained directly from the original angle without
performing iterations as does CORDIC. The critical path is
therefore made significantly shorter than that of the CORDIC-type
methods. In addition, only a small lookup table is needed.
[0378] In this Section, methods and apparatus for two-stage angle
rotation will be described. These method and apparatus are meant
for example purposes only, and are not meant to be limiting. Those
skilled in the arts will recognize other methods and apparatus for
two stage angle rotation based on the discussion given herein.
These other methods and apparatus for angle rotation are within the
scope and spirit of the present invention.
[0379] It will be shown that more precision and less hardware can
be obtained using the two stage angle rotator compared to the
single-stage angle rotator, with slightly more computational delay.
We will then show that, given an overall output precision
requirement, various simplifications can be applied to the
computations within the two stages to reduce the total
hardware.
[0380] 5.1 The Angle Rotation Problem
[0381] If we rotate a point in the X-Y plane having coordinates
(X.sub.0, Y.sub.0) counterclockwise, around the origin, by the
angle .phi., a new point having coordinates (X, Y) is obtained. It
is related to the original point (X.sub.0, Y.sub.0) as:
X=X.sub.0 cos .phi.-Y.sub.0 sin .phi.
Y=Y.sub.0 cos .phi.+X.sub.0 sin .phi. (5.1)
[0382] 5.1.1 Single-Stage Angle Rotation
[0383] The operation in (5.1) is found in many communication
applications, notably in digital mixers which translate a baseband
signal to some intermediate frequency and vice versa. In addition
to accomplishing (5.1) with CORDIC, a very common implementation is
to store pre-computed sine/cosine values in a ROM (Tan, L. and
Samueli, H., "A 200-MHz quadrature frequency synthesizer/mixer in
0.8-.mu.m CMOS," IEEE J. Solid-State Circuits 30:193-200 (1995)).
Then, in real-time, the computation in (5.1) is accomplished with a
ROM access for each given .phi. followed by four real
multiplications. This method avoids the excessive latency of the
iterations performed by CORDIC and can yield lower latency than the
angle-rotation method (Madisetti, A., "VLSI architectures and IC
implementations for bandwidth efficient communications," Ph.D.
dissertation, University of California, Los Angeles (1996)).
Furthermore, a very fast circuit can be built, based on efficient
multiplier design techniques. However, since the size of the ROM
grows exponentially with the precision of .phi., a rather large ROM
is required to achieve accurate results.
[0384] ROM compression can be achieved by exploiting the
quarter-wave symmetry of the sine/cosine functions and such
trigonometric identities as sin .theta.=cos(.pi./2-.theta.). The
angle .phi. in the full range [0,2.pi.] can be mapped into an angle
.theta..di-elect cons.[0,.pi./4]. This is accomplished by
conditionally interchanging the input values and X.sub.0 and
Y.sub.0, and conditionally interchanging and negating the output X
and Y values (Madisetti, A., "VLSI architectures and IC
implementations for bandwidth efficient communications," Ph.D.
dissertation, University of California, Los Angeles (1996)). Thus,
we will focus only on .theta..di-elect cons.[0.pi./4] and replace
.phi. by .theta. in (5.1). Defining .theta.=(.pi./4){overscore
(.theta.)}, we must have {overscore (.theta.)}.di-elect
cons.[0,1].
[0385] Obviously, the sine/cosine ROM samples must be quantized
because of the limited storage space for sine/cosine samples. This
produces an error in the ROM output when compared to the true
(unquantized) sine/cosine value, which will subsequently be
referred to as the ROM quantization error. Next we examine how this
error affects the output. Let cos .theta. and sin .theta. be
quantized to N bits, to become [cos .theta.] and [sin .theta.],
respectively. We have
cos .theta.=[cos .theta.]+.DELTA. .sub.cos .theta.
sin .theta.=[sin .theta.]+.DELTA. .sub.sin .theta. (5.2)
[0386] where .DELTA. .sub.cos .theta. and .DELTA. .sub.sin .theta.
are the ROM quantization errors, which satisfy .vertline..DELTA.
.sub.cos .theta..vertline.<2.sup.-N and .vertline..DELTA.
.sub.sin .theta..vertline.<2.sup.-N. The error in X due to the
ROM quantization is the difference between X calculated using
infinite-precision sine/cosine values and the quantized values,
that is 70 X = ( X 0 cos - Y 0 sin ) - ( X 0 [ cos ] - Y 0 [ sin ]
) ( 5.3 ) = X 0 cos - Y 0 sin .
[0387] Its upper bound is
.vertline..DELTA..sub.X.vertline.<(.vertline.X.sub.0.vertline.+.vertlin-
e.Y.sub.0.vertline.)2.sup.-N. (5.4)
[0388] 5.1.2 Rotation by a Small Angle
[0389] If the rotation angle happens to be so small that
.vertline..theta..vertline.<2.sup.-N/3 (5.5)
[0390] then its sine/cosine values can be approximated as
sin .theta..apprxeq..theta. (5.6)
cos .theta..apprxeq.1-(.theta..sup.2/2). (5.7)
[0391] For such .theta. no table is needed. Next, we show how
accurate (5.6) and (5.7) are by estimating their approximation
errors.
[0392] The Taylor expansion of sin .theta. near .theta.=0 yields 71
sin = - sin ' 6 3 ( 5.8 )
[0393] where .xi.=h.theta., 0.ltoreq.h.ltoreq.1. Thus, since
.vertline.sin' .xi..vertline.=.vertline.cos .xi..vertline..ltoreq.1
(5.9)
[0394] and in view of (5.5), an error bound on (5.6) is
.vertline..DELTA. .sub.sin .theta..vertline.=.vertline.sin
.theta.-.theta..vertline..ltoreq..vertline..theta..sup.3/6.vertline.<2-
.sup.-N/6. (5.10)
[0395] Similarly, the Taylor expansion of cos .theta. yields 72 cos
= 1 - 1 2 2 + cos 24 4 . ( 5.11 )
[0396] Thus, an error bound on (5.7) is
.vertline..DELTA. .sub.cos .theta..vertline.=.vertline.cos
.theta.-(1-.theta..sup.2/2).vertline..ltoreq..vertline..theta..sup.4/24.v-
ertline. (5.12)
[0397] which is negligible in comparison to the bound on
.vertline..DELTA. .sub.sin .theta..vertline..
[0398] 5.1.3 Partitioning into Coarse and Fine Rotations
[0399] While it is unlikely that (5.5) is satisfied for a given
.theta..di-elect cons.[0,.pi./4], i.e., {overscore
(.theta.)}.di-elect cons.[0,1], if we let B>N/3 be the number of
bits in {overscore (.theta.)}, then we can express {overscore
(.theta.)}={overscore (.theta.)}.sub.M+{overscore (.theta.)}.sub.L
where
{overscore (.theta.)}.sub.M=d.sub.12.sup.-1+ . . .
+d.sub.N/32.sup.-N/3 (5.13)
{overscore (.theta.)}.sub.L=d.sub.N/3+12.sup.-N/3-1+ . . .
d.sub.B2.sup.-B (5.14)
[0400] with d.sub.i.di-elect cons.{0,1}. Next we define
.theta..sub.M=(.pi./4){overscore (.theta.)}.sub.M and
.theta..sub.L=(.pi./4){overscore (.theta.)}.sub.L. Clearly, with
.pi./4<1 and from (5.14), .theta..sub.L satisfies (5.5).
[0401] If we substitute .theta.=.theta..sub.M+.theta..sub.L for
.phi. in (5.1) and expand cos(.theta..sub.M+.theta..sub.L) and
sin(.theta..sub.M+.theta..sub.L), we obtain:
X=X.sub.1 cos .theta..sub.L-Y.sub.1 sin .theta..sub.L
Y=Y.sub.1 cos .theta..sub.L+X.sub.1 sin .theta..sub.L (5.15)
and
X.sub.1=X.sub.0 cos .theta..sub.M-Y.sub.0 sin .theta..sub.M
Y.sub.1=Y.sub.0 cos .theta..sub.M+X.sub.0 sin .theta..sub.M.
(5.16)
[0402] Now the rotation (5.1) is decomposed into two stages: a
coarse rotation (5.16) by .theta..sub.M followed by a fine rotation
(5.15) by .theta..sub.L. With this partitioning (5.5) and (5.6) can
be applied to the fine stage:
X=X.sub.1(1-.theta..sub.L.sup.2/2)-Y.sub.1.theta..sub.L
Y=Y.sub.1(1-.theta..sub.L.sup.2/2)+X.sub.1.theta..sub.L. (5.17)
[0403] A benefit of this partitioning is that the functions cos
.theta..sub.M and sin .theta..sub.M in (5.16) depend only on the
N/3 most significant bits of the angle {overscore (.theta.)}, where
.theta.=(.pi./4){overscore (.theta.)}. They can be stored in a
small lookup table. This results in a significant ROM size
reduction. However, the approximation (5.6) introduces additional
error. We now seek to achieve an overall precision comparable to
that in the implementation having one stage and a large ROM
table.
[0404] Defining the approximation errors .DELTA. .sub.sin
.theta..sub..sub.L=sin .theta..sub.L-.theta..sub.L and .DELTA.
.sub.cos .theta..sub..sub.L=cos
.theta..sub.L-(1-.theta..sub.L.sup.2/2), and neglecting terms that
are products of error terms or products of an error term and sin
.theta..sub.L, which is always small, we calculate the total error
in X as the difference between X calculated using (5.15) and (5.16)
and X calculated using quantized sin .theta..sub.M and cos
.theta..sub.M, and in (5.16) and using (5.17) instead of (5.15). We
obtain:
.DELTA..sub.X=X.sub.0(.DELTA. .sub.cos .theta..sub..sub.M cos
.theta..sub.L+.DELTA. .sub.cos .theta..sub..sub.L cos
.theta..sub.M-.DELTA. .sub.sin .theta..sub..sub.L sin
.theta..sub.M)-Y.sub.0(.DELTA. .sub.sin .theta..sub..sub.M cos
.theta..sub.L+.DELTA. .sub.cos .theta..sub..sub.L sin
.theta..sub.M-.DELTA. .sub.sin .theta..sub..sub.L cos
.theta..sub.M) (5.18)
[0405] Comparing this error estimate with (5.3) and (5.4) it is
evident that, so long as the errors due to .DELTA. .sub.cos
.theta..sub..sub.L and .DELTA. .sub.sin .theta..sub..sub.L are
sufficiently small, the error .DELTA..sub.X in (5.18) can be made
comparable to that of (5.4) by reducing the .DELTA. .sub.cos
.theta..sub..sub.M and .DELTA. .sub.sin .theta..sub..sub.M values,
i.e., by increasing the number of bits in the sine/cosine samples
stored in the ROM. For example, if we add one more bit to the
sine/cosine samples, then .vertline..DELTA. .sub.cos
.theta..sub..sub.M.vertline.<2.sup.-N-1 and .vertline..DELTA.
sin .theta..sub..sub.M.vertline.<2.sup.-N-1. Therefore, from
(5.18), we have 73 x < X 0 ( 2 - N - 1 + 2 - 4 N / 3 / 24 + 2 -
N / 6 ) + Y 0 ( 2 - N - 1 + 2 - 4 N / 3 / 24 + 2 - N / 6 ) = ( X 0
+ Y 0 ) 2 - N ( 1 / 2 + 1 / 6 + ( 1 / 24 ) 2 - N / 3 ) ( 5.19 )
[0406] which is smaller than (5.4). A similar relationship can be
found for .DELTA..sub.Y. This demonstrates that, if we add one more
bit of precision to the ROM for the coarse stage, we can achieve
the same precision as that in the one-stage case, but with a
significantly smaller ROM.
[0407] A straightforward implementation of this method is
illustrated by the angle rotator 3800 in FIG. 38. The angle rotator
3800 includes a ROM 3802, butterfly circuits 3806 and 3810, and
fine adjustment circuit 3804.
[0408] The ROM 3802 stores the cos .theta.M and sin .theta..sub.M
values, where .theta..sub.M is the most significant part of the
input angle .theta.. In embodiments the input angle .theta. is
normalized and represented by a binary number, so that
.theta..sub.M is the most significant word of the binary number,
and .theta..sub.L is the least significant word of the binary
number.
[0409] The first butterfly circuit 3806 multiplies the input
complex number 3812 by the (cos .theta..sub.M).sup.+ and the (sin
.theta..sub.M).sup.+ to perform a coarse rotation, where the (
).sup.+ denotes that the appropriate ROM quantization errors have
been added to the cos .theta..sub.M and sin .theta..sub.M, by the
adders 3814.
[0410] The fine adjustment circuit 3804 generates a fine adjust
value (1-.theta..sub.L.sup.2), where .theta..sub.L is the least
significant word of the input angle .theta..
[0411] The second butterfly circuit 3810 multiples the output of
circuit 3806 by .theta..sub.L.sup.+ and the fine adjustment value
from circuit 3804, to perform a fine rotation that results in the
rotated complex number 3814. The .sup.+ on the .theta..sub.L.sup.+
denotes the an error value .DELTA. .sub.sin .theta..sub..sub.L has
been added to improve the accuracy of the fine rotation.
[0412] The three error sources .DELTA. .sub.cos .theta..sub..sub.M,
.DELTA. .sub.sin .theta..sub..sub.M and .DELTA. .sub.sin
.theta..sub..sub.L are shown . The much smaller error source
.DELTA. .sub.cos .theta..sub..sub.L has been neglected. The thick
line depicts the path along which the ROM quantization error
.DELTA. .sub.cos .theta..sub..sub.M propagates to X. The error
.DELTA. .sub.cos .theta..sub..sub.M is multiplied by X.sub.0 and
then by cos .theta..sub.L as it propagates along this path to
become .DELTA. .sub.cos .theta..sub..sub.MX.sub.0 cos .theta..sub.L
when it reaches the output. This matches the error term in (5.18)
obtained from our calculation. In subsequent discussions we will
use this graphical approach to find the error at the output due to
various error sources.
[0413] The ROM table 3802 in the rotator 3800 contains many fewer
sine/cosine samples in comparison to the number of samples needed
to implement (5.1) using a conventional (single stage) table-lookup
approach. Although the approximation (5.6) introduces additional
error, so long as that error is smaller than the conventional ROM
quantization error, we can increase the precision of the samples in
our small ROM table such that, overall, precision is not
sacrificed. In principle, we can reduce the hardware complexity
significantly in one block of our structure, with the corresponding
accuracy loss compensated by higher precision from another block,
and at the cost of a slight increase in the complexity of that
block. As a result, the complexity of the overall structure is
reduced without loss of accuracy. We will now exploit this idea
again to further reduce the computational complexity.
[0414] 5.2 Simplification of the Coarse Stage
[0415] The coarse stage, according to (5.16), involves
multiplications of input data X.sub.0 and Y.sub.0 by the cos
.theta..sub.M and sin .theta..sub.M values. Writing sin
.theta..sub.M as the binary number
sin .theta..sub.M=0.b.sub.1 . . . b.sub.N/3b.sub.N/3+1 . . .
(5.20)
[0416] where b.sub.n.di-elect cons.{0,1}, we now round sin
.theta..sub.M upward, to obtain an (N/3+1)-bit value [sin
.theta..sub.M], as
[sin .theta..sub.M]=0.b.sub.1 . . .
b.sub.N/3b.sub.N/3+1+2.sup.-(N/3+1), (5.21)
[0417] where N represents the number of bits in the real part and
the imaginary part of the input complex number. In other words, the
real part has N bits, and the imaginary part has N bits.
[0418] Letting .theta..sub.1 be the angle for which
sin .theta..sub.1=[sin .theta..sub.M] (5.22)
[0419] we must have .theta..sub.1.gtoreq..theta..sub.M. Next we can
compute the corresponding cos .theta..sub.1 value. Using sin
.theta..sub.1=[sin .theta..sub.M] and cos .theta..sub.1 values, we
actually rotate the point having coordinate (X.sub.0, Y.sub.0) by
.theta..sub.1, instead of .theta..sub.M, as
X.sub.1=X.sub.0 cos .theta..sub.1-Y.sub.0 sin .theta..sub.1
Y.sub.1=Y.sub.0 cos .theta..sub.1+X.sub.0 sin .theta..sub.1.
(5.23)
[0420] Since .theta..sub.1=arcsin([sin .theta..sub.M]) and, of
course, .theta..sub.M=arcsin(sin .theta..sub.M), applying the mean
value theorem we have 74 1 - M [ sin M ] - sin M = a sin ' = 1 1 -
2 ( 5.24 )
[0421] where .xi. satisfies sin
.theta..sub.M.ltoreq..xi..ltoreq.[sin .theta..sub.M]. Since sin
.theta..sub.M.ltoreq.1/({square root}{square root over (2)}),
according to (5.21) we must have 75 [ sin M ] sin M + 2 - ( N / 3 +
1 ) 1 2 + 2 - ( N / 3 + 1 ) . ( 5.25 )
[0422] For most applications, N.gtoreq.9. Thus, according to
(5.25), we have .xi..ltoreq.0.7696.
[0423] Applying this value to (5.24), 76 1 - M 1 1 - 0.7696 2
.times. ( [ sin M ] - sin M ) 1.566 .times. 2 - ( n / 3 + 1 ) . (
5.26 )
[0424] Because 0<.theta..sub.M.ltoreq..pi./4, we have, for
N.gtoreq.12, that
0.ltoreq..theta..sub.1.ltoreq.0.0978+.pi./4=0.8832. (5.27)
[0425] The resulting fine-stage angle is .theta.-.theta..sub.1,
instead of .theta..sub.L=.theta.-.theta..sub.M. Thus, as in
(Madisetti, A., "VLSI architectures and IC implementations for
bandwidth efficient communications," Ph.D. dissertation, University
of California, Los Angeles (1996)), a modified fine-stage angle
compensates for a simplified coarse-stage angle. Since sin
.theta..sub.1=[sin .theta..sub.M], by rotating by .theta..sub.1,
the (N/3+1)-bit number sin .theta..sub.1 decreases the number of
partial products needed in computing X.sub.0 sin .theta..sub.1 and
Y.sub.0 sin .theta..sub.1 to just over a third of those needed for
X.sub.0 sin .theta..sub.M and Y.sub.0 sin .theta..sub.M. This
simplifies the computation in (5.23). However, if we can also
reduce the multiplier size in computing X.sub.0 cos .theta..sub.1
and Y.sub.0 cos .theta..sub.1, we can further simplify (5.23).
Certainly, truncating the cos .theta..sub.1 value would reduce the
number of partial products in computing X.sub.0 cos .theta..sub.1
and Y.sub.0 cos .theta..sub.1. Let us truncate cos .theta..sub.1 to
2N/3 bits to obtain [cos .theta..sub.1]. Then,
0.ltoreq..DELTA. .sub.cos .theta..sub..sub.1=cos .theta..sub.1-[cos
.theta..sub.1]<2.sup.-2N/3 (5.28)
We now have
X.sub.1=X.sub.0[cos .theta..sub.1]-Y.sub.0 sin .theta..sub.1
Y.sub.1=Y.sub.0[cos .theta..sub.1]+X.sub.0 sin .theta..sub.1.
(5.29)
[0426] Apparently, by truncating cos .theta..sub.1, smaller
multipliers are needed. But the amount of rotation is no longer
.theta..sub.1. We now examine the effect on .theta..sub.1 of using
the truncated value [cos .theta..sub.1] instead of cos
.theta..sub.1 as 77 m = a tan sin [ cos 1 ] . ( 5.30 )
[0427] First, we determine how .theta..sub.m is different from
.theta..sub.1 due to the truncation of cos .theta..sub.1. Letting
cos .theta..sub.1 and [cos .theta..sub.1] denote specific values of
a variable z we consider the function 78 ( z ) = a tan sin 1 z . (
5.31 )
[0428] Hence, .theta..sub.1 and .theta..sub.m are the .THETA.(z)
values corresponding to z.sub.1=cos .theta..sub.1 and z.sub.2=[cos
.theta..sub.1], i.e., .theta..sub.1=.THETA.(z.sub.1) and
.theta..sub.m=.THETA.(z.sub.2). According to the mean value
theorem, we have 79 ( z 1 ) - ( z 2 ) z 1 - z 2 = ' ( ) ( 5.32 ) 1
- m cos 1 - [ cos 1 ] = a tan ' sin 1 ( 5.33 )
[0429] where [cos .theta..sub.1.ltoreq..xi..ltoreq.cos
.theta..sub.1. The negation of the derivative 80 a tan ' A
[0430] satisfies 81 - a tan ' sin 1 = - sin 1 2 1 + ( sin 1 ) 2 =
sin 1 ( sin 1 ) 2 + 2 sin 1 ( sin 1 ) 2 + [ cos 1 ] 2 . ( 5.34
)
[0431] According to (5.27), for N.gtoreq.9 we have
0.ltoreq.sin .theta..sub.1.ltoreq.0.7727
0.6347.ltoreq.cos .theta..sub.1.ltoreq.1
0.6406<[cos .theta..sub.1].ltoreq.1. (5.35)
[0432] Since [cos .theta..sub.1]=cos .theta..sub.1-.DELTA. .sub.cos
.theta..sub..sub.1 from (5.28) we also have 82 ( sin 1 ) 2 + [ cos
1 ] 2 = ( sin 1 ) 2 + ( cos 1 - cos 1 ) 2 = 1 - 2 cos 1 cos 1 + cos
1 2 > 1 - 2 - N / 3 + 1 ( 5.36 )
[0433] Combining (5.35) and (5.36), one can verily, for N.gtoreq.9,
that (5.34) satisfies 83 - a tan ' sin 1 < 0.7976 . ( 5.37 )
[0434] Thus, according to (5.28) and (5.33), we have
.theta..sub.m-.theta..sub.1<(cos .theta..sub.1-[cos
.theta..sub.1]).times.0.7976.ltoreq.0.7976.times.2.sup.-2N/3.
(5.38)
[0435] Combining (5.26) and (5.38), and for N.gtoreq.9, we have
0.ltoreq..theta..sub.m-.theta..sub.M.ltoreq.1.566.times.2.sup.-(N/3+1)+0.7-
976.times.2.sup.-2N/3<0.8827.times.2.sup.-N/3. (5.39)
[0436] This is the amount of coarse rotation error, due to
coarse-stage simplifications, that a modified fine stage must
compensate. Let us examine the bound on the fine-stage angle.
[0437] Now, the fine rotation angle is
.theta..sub.l=.theta.-.theta..sub.m instead of .theta..sub.L. If
.theta..sub.l satisfies
.vertline..theta..sub.l.vertline.<2.sup.-N/3 (5.40)
[0438] then we have .vertline.sin
.theta..sub.l-.theta..sub.l.vertline.<- ;2.sup.-N/6. That is,
the approximations sin .theta..sub.l.apprxeq..theta.- .sub.l and
cos .theta..sub.l.apprxeq.1-.theta..sub.l.sup.2/2 can be applied as
discussed in Section 5.1. Let us now examine the bound on
.theta..sub.l. By definition, 84 0 L = 4 _ L < 0.7854 .times. 2
- N / 3 . ( 5.41 )
[0439] Therefore, subtracting (5.39) from (5.41) yields
-0.8827.times.2.sup.-N/3<.theta..sub.L-(.theta..sub.m-.theta..sub.M)<-
;0.7854.times.2.sup.-N/3 (5.42)
[0440] which implies (5.40) because
.theta..sub.l=.theta.-.theta..sub.m=.theta..sub.M+.theta..sub.L-.theta..su-
b.m=.theta..sub.L-(.theta..sub.m-.theta..sub.M). (5.43)
[0441] Hence, no lookup table is needed for the fine stage.
[0442] Next, we examine the magnitude of the complex input sample
after rotation. One can verify from (5.29) that
X.sub.1.sup.2+Y.sub.1.sup.2=(X.sub.0.sup.2+Y.sub.0.sup.2)([cos
.theta..sub.1].sup.2+(sin .theta..sub.l).sup.2). (5.44)
[0443] Since [cos .theta..sub.1] is obtained by truncating cos
.theta..sub.1, we must have 0.ltoreq.[cos .theta..sub.1].ltoreq.cos
.theta..sub.1, thus
[cos .theta..sub.1].sup.2+(sin .theta..sub.1).sup.2.ltoreq.(cos
.theta..sub.1).sup.2+(sin .theta..sub.1).sup.2=1. (5.45)
Therefore,
X.sub.1.sup.2+Y.sub.1.sup.2.ltoreq.X.sub.0.sup.2+Y.sub.0.sup.2.
(5.46)
[0444] To maintain the magnitude, the result X.sub.1 and Y.sub.1
must then be multiplied by 1/{square root}{square root over ([cos
.theta..sub.1].sup.2+(sin .theta..sub.1).sup.2.)} We define a new
variable .delta..sub.[cos .theta..sub..sub.1.sub.] such that 85 1 [
cos 1 ] 2 + ( sin 1 ) 2 = 1 + [ cos 1 ] . ( 5.47 )
[0445] Since {square root}{square root over ((cos
.theta..sub.1).sup.2+(si- n .theta..sub.1).sup.2)}=1, and [cos
.theta..sub.1] is very close to cos .theta..sub.1 because of
(5.28), we have that {square root}{square root over ([cos
.theta..sub.1].sup.2+(sin .theta..sub.1).sup.2 )} is very close to
1. Thus, the .delta..sub.[cos .theta..sub..sub.1.sub.] value must
be very small. We now examine the bound on .delta..sub.[cos
.theta..sub..sub.1.sub.]. We can write 1/{square root}{square root
over ([cos .theta..sub.1].sup.2+(sin .theta..sub.1).sup.2)} as 86 1
[ cos 1 ] 2 + ( sin 1 ) 2 = ( cos 1 ) 2 + ( sin 1 ) 2 [ cos 1 ] 2 +
( sin 1 ) 2 . ( 5.48 )
[0446] Substituting (5.28) into (5.48), we have 87 1 [ cos 1 ] 2 +
( sin 1 ) 2 = 1 + 2 [ cos 1 ] cos 1 + cos 1 2 [ cos 1 ] 2 + ( sin 1
) 2 . ( 5.49 )
[0447] Because (5.28) and (5.35) imply that .DELTA. .sub.cos
.theta..sub..sub.1<<[cos .theta..sub.1], we have .DELTA.
.sub.cos .theta..sub..sub.1.sup.2<<[cos .theta..sub.1].DELTA.
.sub.cos .theta..sub..sub.1.sup.2, hence we can omit .DELTA.
.sub.cos .theta..sub..sub.1.sup.2 in (5.49). Defining 88 = [ cos 1
] cos 1 [ cos 1 ] 2 + ( sin 1 ) 2 ( 5.50 )
[0448] then (5.49) becomes {square root}{square root over
(1+2.delta.)}. From (5.28) and (5.35) we must have
.delta..ltoreq.0.
[0449] Applying the mean-value theorem to {square root}{square root
over (1.degree.2.delta.)}, we have 89 1 + 2 - 1 + 0 - 0 = 1 1 + 2 1
( 5.51 )
[0450] where 0.ltoreq..zeta..ltoreq..delta.. Hence,
{square root}{square root over (1+2.delta.)}.ltoreq.1+.delta.
(5.52)
[0451] According to (5.35), 90 [ cos 1 ] [ cos 1 ] 2 + ( sin 1 ) 2
< 1 ( 5.53 )
[0452] and therefore, from (5.28) and (5.50), we have
0.ltoreq..delta.<2.sup.-2N/3.
[0453] By definition, in (5.47), we have {square root}{square root
over (1+2.delta.)}=1+.delta..sub.[cos .theta..sub..sub.1.sub.].
Thus .delta..sub.[cos .theta..sub..sub.1.sub.].ltoreq..delta.,
Hence .delta..sub.[cos .theta..sub..sub.1.sub.] is bounded by
0.ltoreq..delta..sub.[cos .theta..sub..sub.1.sub.]<2.sup.-N/3.
(5.54)
[0454] According to (5.40) and (5.54), instead of storing the sin
.theta..sub.M and cos .theta..sub.M values in ROM, we may store sin
.theta..sub.1, which has N/3+1 bits for each sample, and [cos
.theta..sub.1], which has 2N/3 bits. Given {overscore
(.theta.)}.sub.M, the sin .theta..sub.1 and [cos .theta..sub.1]
values are retrieved from the ROM to be used in performing the
coarse rotation. Since the actual angle .theta..sub.m differs from
the .theta..sub.M=(.pi./4){overscore (.theta.)}.sub.M, we must also
store the .theta..sub.M-.theta..sub.m values, so that the fine
stage can compensate for the difference. The approximations (5.6)
and (5.7) still apply to .theta..sub.l, in view of (5.40). In
addition, the change of magnitude in the rotation using the sin
.theta..sub.1 and [cos .theta..sub.1] values, as seen in (5.45),
must also be compensated. Therefore we store the .delta..sub.[cos
.theta..sub..sub.1.sub.] values in order to scale the coarse-stage
output by 1+.delta..sub.[cos .theta..sub..sub.1].
[0455] We can now implement the coarse rotation stage as in (5.29).
Later we will show that the scale factor 1+.delta..sub.[cos
.theta..sub..sub.1.sub.] can be combined with the scaling that will
be done for the second stage (i.e., the fine stage) at its
output.
[0456] To compute .theta..sub.l we must first convert the
normalized {overscore (.theta.)}.sub.L value to the radian value
.theta..sub.L, which involves a multiplication by .pi./4. Since
.pi./4=2.sup.31 1+2.sup.-2+2.sup.-5+2.sup.-8+2.sup.-13+ . . . , if
we multiply 0.ltoreq.{overscore (.theta.)}.sub.L<2.sup.-N/3 by
(2.sup.-1+2.sup.-2+2.sup.-5+2.sup.-8), this product and (.pi./4)
{overscore (.theta.)}.sub.Ldiffer by no more than
2.sup.-12.times.2.sup.-- N/3=2.sup.-(N/3+12), which is sufficiently
small for a 12-bit system. (And two more bits would suffice for
building a 16-bit system.)
[0457] 5.3 Reduction of Multiplier Size in the Fine Stage
[0458] In the fine rotation stage, the computations involved in
generating X.sub.2 are 91 X 2 = X 1 ( 1 - l 2 / 2 ) - Y 1 l = X 1 (
l 2 / 2 ) X 1 - Y 1 l . ( 5.55 )
[0459] Since .vertline..theta..sub.l.vertline.<.sup.2-N/3 it
follows that .theta..sub.l can be expressed as
.theta..sub.l=s.s . . . s.theta..sub.N/3+1 . . .
.theta..sub.2N/3.theta..s- ub.2N/3+1 . . . (5.56)
[0460] where s is the sign bit. The N/3 MSBs do not influence the
result. This property helps to reduce the size of the multipliers
that implement (5.55). Even more savings in hardware can be
achieved by further reducing multiplier size, with just a small
loss of accuracy.
[0461] Let [Y.sub.1] represent the 2N/3 MSBs of Y.sub.1 as in
Y.sub.1=s.y.sub.1 . . . y.sub.2N/3 y.sub.2N/3+1 . . .
=[Y.sub.1]+.DELTA..sub.Y.sub..sub.1. (5.57)
[0462] Then we must have
.vertline..DELTA..sub.Y.sub..sub.1.vertline.<2- .sup.-2N/3. The
error contributed to the product Y.sub.1.theta..sub.l by using
[Y.sub.1] instead of Y.sub.1 is
.vertline.Y.sub.1.theta..sub.l-[Y.sub.1].theta..sub.l.vertline.=.vertline.-
.DELTA..sub.Y.sub..sub.1.theta..sub.l.vertline.<2.sup.-N.
(5.58)
[0463] Therefore, for N-bit precision, the multiplication
Y.sub.1.theta..sub.l can be accomplished with a (2N/3).times.(2N/3)
multiplier.
[0464] This method can be applied to the computation of
.theta..sub.l.sup.2/2. Defining [.theta..sub.l] as the 2N/3 MSBs of
.theta..sub.l, and letting .DELTA..sub..theta..sub..sub.l denote
the remaining LSBs, we have
[.theta..sub.l]=s.s . . . s.theta..sub.N/3+1 . . . .theta..sub.2N/3
(5.59)
and .DELTA..sub..theta..sub..sub.l=.theta..sub.2N/3+1 . . .
(5.60)
[0465] The error in calculating .theta..sub.l.sup.2/2 using
[.theta..sub.l] instead of .theta..sub.l is
.vertline.([.theta..sub.l]+.DELTA..sub..theta..sub..sub.l).sup.2-[.theta..-
sub.l].sup.2.vertline./2.apprxeq..vertline.[.theta..sub.l].DELTA..sub..the-
ta..sub..sub.l.vertline.<2.sup.-N. (5.61)
[0466] Thus .theta..sub.l.sup.2 can be implemented with an
(N/3).times.(N/3) multiplier, since the N/3 MSBs of [.theta..sub.l]
are just sign-bits.
[0467] 5.4 Scaling Multiplier Simplification
[0468] As mentioned in Section 5.2, the scale factor
1+.delta..sub.[cos .theta..sub..sub.l.sub.] can be applied at the
output of the fine stage. A straightforward implementation would
use the full wordlength of 1+.delta..sub.[cos
.theta..sub..sub.l.sub.] in the product
X=X.sub.2(1+.delta..sub.[cos .theta..sub..sub.l.sub.], which would
require a multiplier of size N.times.N. But this multiplier's size
can be reduced as follows: According to (5.54),
0.ltoreq..delta..sub.[cos .theta..sub..sub.l.sub.]<2.sup.-2n/3.
Moving the factor 1+.delta..sub.[cos .theta..sub..sub.l.sub.] into
the fine stage, we have 92 X 2 = X 1 ( 1 - l 2 / 2 ) ( 1 + [ cos 1
] ) - Y 1 l ( 1 + [ cos 1 ] ) ( 5.62 ) = X 1 + X 1 ( [ cos 1 ] - l
2 / 2 ) - Y 1 l . ( 5.63 )
[0469] The only significant error in approximating (5.62) by (5.63)
is the absence of the .theta..sub.l.delta..sub.[cos
.theta..sub..sub.l.sub.] term in the factor multiplying Y.sub.1.
But this is tolerable since, according to (5.54) and (5.40),
.vertline..theta..sub.l.delta..sub.[cos
.theta..sub..sub.l.sub.].vertline.- <2.sup.-N. (5.65)
[0470] In view of (5.40) we have
0.ltoreq.-.theta..sub.l.sup.2<2.sup.-2- N/3 which, combined with
(5.54), yields
.vertline..delta..sub.[cos
.theta..sub..sub.1.sub.]-.theta..sub.l.sup.2/2.-
vertline.<2.sup.-2N/3. (5.65)
[0471] Thus, if we truncate .delta. .sub.cos
.theta..sub..sub.m-.theta..su- b.l.sup.2/2 to N bits, only the
least significant N/3 bits in the truncated result will be non-sign
bits. Therefore, in our computation of X.sub.1(.delta. .sub.cos
.theta..sub..sub.m-.theta..sub.l.sup.2/2) in (5.63), if we truncate
X.sub.1 to N/3 bits, we can use an (N/3).times.(N/3) multiplier,
with the product's error bound being
.vertline..delta..sub.[cos
.theta..sub..sub.l.sub.]-.theta..sub.l.sup.2/2.-
vertline.2.sup.-N/3<2.sup.-N. (5.66)
[0472] By merging the scale factor of the coarse stage into the
fine stage, we thus replace multiplications by the scale factor by
additions. The final architecture is shown in FIG. 39, where the
size of the multipliers is shown in FIG. 40.
[0473] FIG. 39 illustrates an angle rotator 3900 according to
embodiments of the invention. The angle rotator 3900 includes a ROM
3902, a fine adjustment circuit 3904, a first butterfly circuit
3908, and a second butterfly circuit 3910. The angle rotator 3900
rotates an input complex signal 3906 according to angle .theta. to
produce a rotated complex output signal 3912. The angle .theta. can
be broken down into a most significant portion (or word)
.theta..sub.m and least significant portion (word) .theta..sub.L.
Note that normalized angle values are shown in FIG. 39, as
represented by the {overscore (.theta.)} nomenclature. However
normalized angle values are not required, as will be understood by
those skilled in the arts.
[0474] The ROM 3902 stores the following for each corresponding
{overscore (.theta.)}: sin .theta..sub.1, [cos .theta..sub.1],
.delta..sub.[cos .theta..sub..sub.1.sub.], and
.theta..sub.M.theta..sub.m, where all of these values have been
exactly defined in preceding sections. To summarize, the sin
.theta..sub.1 and [cos .theta..sub.1] values are MSBs of sin
.theta..sub.M and cos .theta..sub.1, respectively. The
.delta..sub.[cos .theta..sub..sub.1.sub.] error value represents
the difference between the cos .theta..sub.M and the [cos
.theta..sub.1] value. (The exact definition for .delta..sub.[cos
.theta..sub..sub.1.sub.- ] is given in (5.47)) Likewise, the
(.theta..sub.M-.theta..sub.m) error value represents the difference
between sin .theta..sub.M and the sin .theta..sub.1 value. (The
exact definition for .theta..sub.m is given in equation (5.30))
[0475] The butterfly circuit 3908 includes multiple multipliers and
adders as shown. The implementation of these multipliers and adders
is well known to those skilled in the arts. In embodiments, the
sizes of the multipliers and adders are as shown in FIG. 40. Note
that savings are obtained on the size of the multipliers because of
bit truncated approximations that are described above. This
produces a faster and more efficient angle rotator compared to
other angle rotator schemes.
[0476] The operation of the angle rotator 3900 is further described
in reference to the flowchart 4100. As with all flowcharts herein,
the order of the steps is not limiting, as one or more steps can be
performed simultaneously (or in a different order) as will be
understood by those skilled in the arts.
[0477] In step 4102, the input complex signal is received.
[0478] In step 4104, the sin .theta..sub.1, cos [.theta..sub.1],
.delta..sub.[cos .theta..sub..sub.1.sub.], and
.theta..sub.M-.theta..sub.- m values are retrieved from the ROM
3902, based on the rotation angle .theta..
[0479] In step 4106, the butterfly circuit 3908 multiplies the
input complex signal by the sin .theta..sub.1 and [cos
.theta..sub.1] values to perform a coarse rotation of the input
complex signal, resulting in an intermediate complex signal at the
output of the butterfly circuit 3908.
[0480] In step 4108, the adder 3914 adds the .theta..sub.L value to
the error value .theta..sub.M-.theta..sub.m to produce a
.theta..sub.l angle.
[0481] In step 4110, a fine adjustment circuit 3904 generates a
fine adjust value (.delta..sub.[cos
.theta..sub..sub.1.sub.]-.theta..sub.l.sup- .2) based on the
.theta..sub.l angle and .delta..sub.[cos
.theta..sub..sub.1.sub.].
[0482] In step 4112, the butterfly circuit 3910 multiplies the
intermediate complex signal by the .theta..sub.l angle, and the
fine adjustment value (.delta..sub.[cos
.theta..sub..sub.1.sub.]-.theta..sub.l- .sup.2) to perform a fine
rotation of the intermediate complex number, resulting in the
output complex signal 3912.
[0483] In embodiments, the ROM 3902 storage space is 2.sup.N/3
words, where N is the bit size of the real or imaginary input
complex number 3906. Therefore, the overall size of the ROM 3902
can be quite small compared with other techniques. This occurs
because of the two-stage coarse/fine rotation configuration of the
angle rotator 3900, and saving of storing sin .theta..sub.1, [cos
.theta..sub.1] instead of sin .theta. and cos .theta.. Also, there
is another advantage to having a small ROM: in certain technologies
it is awkward to implement a ROM. Thus, if only a small ROM is
needed, it is possible to implement the ROM's input/output
relationship by combinatorial logic circuits instead of employing a
ROM. Such circuits will not consume an unreasonable amount of chip
area if they need only be equivalent to a small ROM.
[0484] 5.5 Computational Accuracy and Wordlength
[0485] In this section we study the effect of quantization errors
on the final output's computational accuracy and the most efficient
way to quantize the data for a given accuracy.
[0486] In our algorithm, the errors can be classified into three
categories. The first category is the quantization of the values in
the ROM table. The second category is the error due to the
truncation of data before multiplications, to reduce multiplier
size. The third type of error is that resulting from approximating
sin .sub.l by .theta..sub.l. Quantization errors are marked in FIG.
40 with an .epsilon. marker as shown. The total error can be
obtained by combining the errors propagated from each source. To
calculate the propagated error at the output with a given error at
the source, we can first identify all paths by which the error
reaches the output and then use the approach discussed in Section
5.1.3. Let us first examine all the error sources and determine
their effects on X, which is the real component of the output
complex signal 3912. Table 5.1 displays this information. (Similar
results apply to Y.)
[0487] The values stored in the ROM are sin .theta..sub.1, [cos
.theta..sub.1], .theta..sub.M-.theta..sub.m and .delta..sub.[cos
.theta..sub..sub.1.sub.], where sin .theta..sub.1 and [cos .theta.]
are MSBs of sin .theta..sub.M and cos .theta..sub.1, respectively.
A loss of precision due to ROM quantization error depends only on
the number of bits used in representing .theta..sub.M-.theta..sub.m
and .delta..sub.[cos .theta..sub..sub.1.sub.].
6TABLE 5.1 Effect of errors at the X output Error at the Error
source output .epsilon..sub.1 quantizing (.theta..sub.M -
.theta..sub.m) in ROM Y.sub.1.epsilon..sub.1 .epsilon..sub.2
quantizing (.delta..sub.[cos.theta..sub.1]) in ROM
Y.sub.1.epsilon..sub.2 .epsilon..sub.3 truncating Y.sub.1 for
Y.sub.1.theta..sub.1 -.theta..sub.l.epsilon..sub.3 .epsilon..sub.4
93 truncating X 1 for [ X 1 ( [ cos 1 ] - l 2 2 ) ] 94 ( [ cos l ]
- l 2 2 ) 2 .epsilon..sub.5 truncating .theta..sub.l for
.theta..sub.l.sup.2 -.theta..sub.lX.sub.1.epsilon..sub.- 5
.epsilon..sub.6 95 truncating l 2 for [ X 1 ( [ cos 1 ] - l 2 2 ) ]
Y.sub.1.epsilon..sub.6 .epsilon..sub.7 96 quantizing L = ( 4 ) _ L
Y.sub.1.epsilon..sub.7 .epsilon..sub.8 quantizing X at the output
.epsilon..sub.8 .epsilon..sub.9 approximating sin .theta..sub.l by
.theta..sub.l 97 Y 1 ( l 3 6 ) .epsilon..sub.10 neglecting
.delta..sub.[cos.theta..sub..sub.1.sub.].theta..sub.l in (5.62)
Y.sub.1
(-.delta..sub.[cos.theta..sub..sub.1.sub.].theta..sub.l)
[0488] The total error in X can be obtained by combining all the
terms in the third column of Table 5.1: 98 Y 1 ( 1 + 7 ) + Y 1 l (
l 2 6 - [ cos 1 ] ) + X 1 ( 2 + 6 ) - X 1 l 5 - l 3 + ( cos m - l 2
2 ) 4 + 8 ( 5.67 )
[0489] Since .epsilon..sub.6 in Table 5.1 is a truncation error, we
have .epsilon..sub.6.gtoreq.0. If we quantize .delta..sub.[cos
.theta..sub..sub.1.sub.] by rounding it upward before storing it in
ROM, then .epsilon..sub.2.ltoreq.0. This way such errors tend to
cancel each other. Cancelling errors are grouped together in (5.67)
since the magnitude of their combined error is no greater than the
larger of the two. This yields seven terms in (5.67), each
contributing a maximum possible error of 2.sup.-N. If the
multiplier sizes are as indicated in FIG. 40, the total error in X
is bounded by 7.times.2.sup.-N.
[0490] From the above analysis it can be seen that the computation
errors resulting from hardware reduction have similar magnitudes
and no particular source dominates. This seems to provide the best
trade-off between the hardware complexity and the accuracy of the
entire system.
[0491] According to (5.67), the total output error can be reduced
by increasing the internal data wordlength and the wordlength of
each sample in the ROM. For each bit increase, we get one more bit
of precision at the output. Therefore, we can design the processor
to have the minimum hardware for a given precision requirement.
Next, we give a simulation example to illustrate this method.
[0492] Example: A cosine waveform with an error less than 2.sup.-12
is specified. According to (5.67), we chose N=15, as indicated in
FIG. 40. We obtained the maximum error to be approximately
5.times.10.sup.-5, which is considerably smaller than 2.sup.-12
.
[0493] In FIG. 40, the ROM is shown as having 99 2 N 3
[0494] words to achieve no more than a total error of
7.times.2.sup.-N in the X output. If N is not a multiple of 3, we
can choose the smallest N'>N that is a multiple of 3. Having 100
2 N ' 3
[0495] words in ROM, of course, suffices to achieve the required
precision. As discussed before, the total output error is a
combination of errors from various sources, such as from quantizing
the data before multiplications and from approximating sin
.theta..sub.l by .theta..sub.l, etc. However, our error bound
estimation is rather conservative. Hence, the ROM size can be
perturbed to determine the minimum size to satisfy a specific
precision requirement. Our experience in designing the
angle-rotation processor has shown that, even by rounding 101 N
3
[0496] down to determine the ROM size, the total error is still
less than 7.times.2.sup.-N.
[0497] 5.6 Comparison with the Single-Stage Mixer
[0498] As mentioned earlier, the main advantage of the two-stage
angle rotator is that it requires only a small ROM 3902. For the
single-stage angle rotation, the ROM size grows exponentially with
the precision of the angle. Thus, our two-stage method is
well-suited for applications where more than 14 bits in the input
angle are required. In this case, the sine lookup table for the
single-stage angle-rotator, even with compression, is too large for
high-speed operations (Vankka, J., "Methods of mapping from phase
to sine amplitude in direct digital synthesis," IEEE Trans.
Ultrasonics, Ferroelectronics and Freq. Control 44:526-534 (1997)).
However, the following comparison of our method to a well-known
single-stage method with 14-bit input angle shows that even in this
case our method has advantages, and this is true even when the
single-stage method is optimized for that particular precision
requirement.
[0499] To compare, we use the quadrature direct digital frequency
synthesizer/mixer (QDDFSM) with 14-bit input angle and 12-bit input
data that is reported in (Tan, L. and Samueli, H., "A 200-MHz
quadrature frequency synthesizer/mixer in 0.8-.mu.m CMOS," IEEE J.
Solid-State Circuits 30:193-200 (1995)). It achieves 84.3 dB
spurious free dynamic range (SFDR). According to this method, the
sine and cosine values are generated using a DDFS, which employes
lookup tables for these values. To reduce the ROM size, ROM
compression techniques are used. The DDFS is followed by four
12.times.12 real multiplications.
[0500] For our structure, we chose the internal wordlengths and
multiplier sizes as indicated in FIG. 42. The phase-accumulator
that generates .theta., as well as the circuit that maps an angle
in the range [0, 2.pi.] into [0, .pi./4], are described in
(Madisetti, A., "VLSI architectures and IC implementations for
bandwidth efficient communications," Ph.D. dissertation, University
of California, Los Angeles (1996)). These structures are also
employed here in our test. Truncating the 32-bit phase word to 14
bits, this structure has achieved a SFDR of 90.36 dB, as shown in
FIG. 43. This is 6 dB better than the single-stage method.
[0501] The integrated circuit that implements this structure is
currently being built. A preliminary estimation of its hardware
complexity yields a similar transistor count as that of (Tan, L.
and Samueli, H., IEEE J. Solid-State Circuits 30:193-200 (1995)).
Thus, using approximately the same number of transistors, our
structure achieves a 6 dB performance gain. Our structure requires
a much smaller ROM (17.times.25=425 bits) in comparison to the
single-stage method, which needs a 3072-bit ROM when the ROM
compression technique is employed. Since the ROM access is hard to
pipeline, it is usually the bottleneck in the data path, thereby
limiting the achievable data rate. Hence, one pronounced benefit of
having a much smaller ROM would be the much faster ROM access.
Also, there is another advantage to having a small ROM: in certain
technologies it is awkward to implement a ROM. Thus, if only a
small ROM is needed, it is possible to implement the ROM's
input/output relationship by combinatorial logic circuits instead
of employing a ROM. Such circuits will not consume an unreasonable
amount of chip area if they need only be equivalent to a small
ROM.
[0502] 5.7 A Modified Structure When Only One Output is Needed
[0503] In some applications, such as the implementation of the
trigonometric interpolator discussed in the previous sections, only
one output, say X is needed. In such cases, obviously, we can
eliminate certain computations used to generate Y. However, using
the angle rotator 3900, only those generating Y in the fine stage
are subject to deletion, while the coarse stage must remain the
same, since we need both X.sub.1 and Y.sub.1 to generate the X
output. Let us seek to further simplify the coarse stage by
attempting to eliminate one multiplication by cos
.theta..sub.M.
[0504] 5.7.1 Modifications to the Coarse Stage
[0505] If we factor out the cos .theta..sub.M term of the coarse
stage in (5.16), we can then apply the factor cos .theta..sub.M to
the output of the second stage in (5.17), because the two
operations (scaling and rotation) are permutable, to obtain 102 X 1
= X 0 - Y 0 tan M ( 5.68 ) Y 1 = Y 0 - X 0 tan M X = cos M ( X 1 (
1 - L 2 2 ) - Y 1 L ) ( 5.69 ) Y = cos M ( Y 1 ( 1 - L 2 2 ) + X 1
L )
[0506] In this case, we have only two multipliers in the coarse
stage (5.68), and the multiplications by the scale factor
.theta..sub.M are applied to the output of the fine stage (5.69).
Unlike the situation in (5.16) and (5.17), if only one output from
the angle rotator, say X, is needed, we can also eliminate one more
multiplier--the one that multiplies the coarse stage output with
the cos .theta..sub.M factor. As in Section 5.2, we now seek to
simplify the coarse stage in (5.68).
[0507] Let tan .theta..sub.m be tan .theta..sub.M rounded upward at
the (N/3)-rd bit. In other words, writing .theta..sub.M as the
binary number
tan .theta..sub.M=0.b.sub.1 . . . b.sub.N/3b.sub.N/3+1 . . . ,
(5.70)
[0508] where b.sub.n.di-elect cons.{0, 1}, tan .theta..sub.m is
obtained from tan .theta..sub.M according to
tan .theta..sub.m=0.b.sub.1 . . . b.sub.N/3+2.sup.-N/3. (5.71)
Obviously,
0.ltoreq.tan .theta..sub.m-tan .theta..sub.M.ltoreq.2.sup.-N/3.
(5.72)
[0509] The N/3-bit number tan .theta..sub.m decreases the number of
partial products needed in computing X.sub.0 tan .theta..sub.m and
Y.sub.0 tan .theta..sub.m to at most a third of those needed for
X.sub.0 tan .theta..sub.m and Y.sub.0 tan .theta..sub.M.
[0510] The resulting fine-stage angle is
.theta..sub.1=.theta.-.theta..sub- .m. Thus, as in Section 5.2, a
modified fine-stage angle compensates for a simplified coarse-stage
angle. If .theta..sub.1 satisfies (5.40), we then have
.vertline.sin .theta..sub.l-.theta..sub.l.vertline.<2.sup.-N/6.
That is, the approximations sin .theta..sub.l=.theta..sub.l and cos
.theta..sub.l=1-.theta..sub.l.sup.2/2 can be applied. The proof
that (5.40) holds is as follows:
[0511] Proof: According to the mean value theorem 103 tan m - tan M
m - M = tan ' ( 5.73 )
[0512] where .xi.=.theta..sub.M+(.theta..sub.m-.theta..sub.m)h,
0.ltoreq.h.ltoreq.1. The derivative tan' .xi. satisfies
tan' .xi.=1+(tan .xi.).sup.2.gtoreq.1, for every .xi.. (5.74)
[0513] Re-arranging (5.73), and using (5.74), we have 104 m - M =
tan m - tan M tan ' tan m - tan M . ( 5.75 )
[0514] Hence, according to (5.72),
0.ltoreq..theta..sub.m-.theta..sub.M.ltoreq.2.sup.N/3. (5.76)
By definition,
0.ltoreq..theta..sub.L<2.sup.-N/3. (5.77)
[0515] Therefore, subtracting (5.76) from (5.77) yields
-2.sup.-N/3<.theta..sub.L-(.theta..sub.m-.theta..sub.M)<2.sup.-N/3.
(5.78)
[0516] which is exactly (5.40) because
.theta..sub.l=.theta.-.theta..sub.m=.theta..sub.M+.theta..sub.L-.theta..su-
b.m=.theta..sub.L-(.theta..sub.m-.theta..sub.M). (5.79)
[0517] This concludes our proof.
[0518] This indicates that, instead of storing the tan
.theta..sub.M values in the ROM, we may store tan .theta..sub.m,
which has N/3 bits for each sample, and we may store
.theta..sub.m-.theta..sub.M. This results in a reduction of the
multiplier size in the coarse stage. The difference between
.theta..sub.m and .theta..sub.M can be compensated in the following
fine rotation stage. Furthermore, the approximations (5.6) and
(5.7) still apply to .theta..sub.l, in view of (5.40).
[0519] We can now implement the coarse rotation stage as
follows:
X.sub.1=X.sub.0-Y.sub.0 tan .theta..sub.m
Y.sub.1=Y.sub.0+X.sub.0 tan .theta..sub.m. (5.80)
[0520] Accordingly, the scale-factor at the output of the fine
stage is cos m instead of cos .theta..sub.M. Since .theta..sub.l
satisfies (5.40), the fine stage simplification is similar to the
method described in Section 5.3. Next we examine how the
multiplications of the fine-stage output by cos .theta..sub.m can
be simplified.
[0521] 5.7.2 Scaling Multiplier Simplification
[0522] A straightforward implementation would use the full
wordlength of cos .theta..sub.m in the product X=X.sub.2 cos
.theta..sub.m, which would require a multiplier of size N.times.N.
But this multiplier's size can be reduced as follows: By defining
[cos .theta..sub.m] as the 2N/3+1 MSBs of cos .theta..sub.m the
scale factor can be written as 105 cos m = [ cos m ] + cos m = [
cos m ] ( 1 + cos m [ cos m ] ) . ( 5.81 )
[0523] Let us define 106 cos m = cos m [ cos m ]
[0524] and, since 0.ltoreq..theta..sub.m.ltoreq./4, we surely have
[cos .theta..sub.m]>0.5, and hence
0.ltoreq..delta. .sub.cos
.theta..sub..sub.m<2.times.2.sup.-2N/3-1. (5.82)
[0525] Moving the factor 1+.delta. .sub.cos .theta..sub..sub.m into
the fine stage, we have 107 X 2 = X 1 ( 1 - l 2 / 2 ) ( 1 + cos m )
- Y 1 l ( 1 + cos m ) ( 5.83 ) = X 1 + X 1 ( cos m - l 2 / 2 ) - Y
1 l . ( 5.84 )
[0526] The only significant error in approximating (5.83) by (5.84)
is the absence of the .theta..sub.l.delta. .sub.cos
.theta..sub..sub.m term in the factor multiplying Y.sub.1. But this
is tolerable since, according to (5.40) and (5.82),
.vertline..theta..sub.l.delta. .sub.cos
.theta..sub..sub.m.vertline.<2.- sup.-N. (5.85)
[0527] In view of (5.40) we have
0.ltoreq..theta..sub.l.sup.2<2.sup.-2N- /3 which, combined with
(5.82), yields
.vertline..delta. .sub.cos
.theta..sub..sub.m-.theta..sub.l.sup.2/2.vertli- ne.<2.sup.-N/3.
(5.86)
[0528] Thus, if we truncate .delta. .sub.cos
.theta..sub..sub.m-.theta..su- b.l.sup.2/2 to N bits, only the
least significant N/3 bits in the truncated result will be non-sign
bits. Therefore, in our computation of X.sub.1(.delta. .sub.cos
.theta..sub..sub.m-.theta..sub.l.sup.2/2) in (5.84), if we truncate
X.sub.1 to N/3 bits, we can use an (N/3).times.(N/3) multiplier,
with the product's error bound being
.vertline..delta. .sub.cos
.theta..sub..sub.m-.theta..sub.l.sup.2/2.vertli-
ne.2.sup.-N/3<2.sup.-N. (5.87)
[0529] The factorization of cos .theta..sub.m in (5.81) allows a
reduction of the multiplier to approximately 2/3 its original size.
In this case, the values of [cos .theta..sub.m] and .delta.
.sub.cos .theta..sub..sub.m are stored in the ROM instead of cos
.theta..sub.m.
[0530] The modified structure for one output is illustrated as
angle rotator 4400 in FIG. 44. The angle rotator 4400 includes a
ROM 4402, a fine adjustment circuit 4404, a first butterfly circuit
4408, and a second butterfly circuit 4410. The angle rotator 4400
rotates an input complex signal 4406 according to angle .theta. to
produce a rotated complex output signal 4412. As with the rotator
3900, the angle .theta. can be broken down into a most significant
portion (or word) .theta..sub.M and least significant portion
(word) .theta..sub.L. Note that normalized angle values are shown
in FIGS. 39, 40, 42, and 44, as represented by the {overscore
(.theta.)} nomenclature. However normalized angle values are not
required, as will be understood by those skilled in the arts.
[0531] The ROM 4402 stores the following for each corresponding
normalized .theta.: tan .theta..sub.m, [cos .theta..sub.m], .delta.
.sub.cos .theta..sub..sub.m, and .theta..sub.M-.theta..sub.m, where
all of these values have been exactly defined in preceding
sections.
[0532] In the butterfly circuit 4410, the arithmetic units that are
encircled by the line 4418 can be eliminated when only the X output
is needed in the output signal 4412. This may be desirable for
applications where only one output from the angle rotator 4400 is
needed, such as when implementing a trigonometric interpolator,
such as interpolator 1000 in FIG. 10 or interpolator 1400 in FIG.
14.
[0533] The operation of the angle rotator 4400 is further described
in reference to the flowchart 4500 in FIG. 45. As with all
flowcharts herein, the order of the steps is not limiting, as one
or more steps can be performed simultaneously (or in a different
order) as will be understood by those skilled in the arts.
[0534] In step 4502, the input complex signal 4406 is received.
[0535] In step 4504, the tan .theta..sub.m, [cos .theta..sub.m],
.delta. .sub.cos .theta..sub..sub.m, and
.theta..sub.M-.theta..sub.m values are retrieved from the ROM 4402,
based on the rotation angle .theta. (or the normalized value
{overscore (.theta.)}).
[0536] In step 4506, the butterfly circuit 4408 multiplies the
input complex signal 4406 by tan .theta..sub.m to perform a coarse
rotation of the input complex number, resulting in an intermediate
complex signal at the output of the butterfly circuit 4408.
[0537] In step 4508, the adder 4414 adds the .theta..sub.L value to
the error value .theta..sub.M-.theta..sub.m to produce a
.theta..sub.l angle.
[0538] In step 4510, a fine adjustment circuit 4404 generates a
fine adjust value (.delta. .sub.cos
.theta..sub..sub.m-.theta..sub.l.sup.2) based on the .theta..sub.l
angle and .delta. .sub.cos .theta..sub..sub.m.
[0539] In step 4512, the butterfly circuit 4410 multiplies the
intermediate complex signal by the .theta..sub.l angle, and the
fine adjustment value (.delta. .sub.cos
.theta..sub..sub.m-.theta..sub.l.sup.2- ) to perform a fine
rotation of the intermediate complex signal, resulting in the
output complex signal.
[0540] In step 4514, the X value for the output complex signal is
scaled by the [cos .theta..sub.m] value, resulting in the output
complex number 4412. As discussed above, the elements inside the
outline 4418 can be eliminated if only the X value of signal 4412
is desired. Alternatively, similar elements could be eliminated
from the butterfly circuit 4410 if only the Y value of signal 4412
was desired.
[0541] 5.8 Application of Angle Rotation Processors
[0542] This subsection describes exemplary applications for angle
rotator processors. These applications are provided for example
purposes only and are not meant to be limiting, as those skilled in
the arts will recognize other applications based on the discussions
given herein. These other applications are within the scope and
spirit of the present invention.
[0543] One application for the angle rotation processor is the
Quadrature Direct Digital Frequency Synthesizer/Mixer (QDDFSM),
including a few special cases that are candidates for the angle
rotator algorithm. One is the case when only one of the outputs (X
or Y) is desired, as shown by angle rotator 4400 (FIG. 44). As
shown in FIG. 44, this is accomplished by simply deleting the
hardware required for the computation of the unused output. Yet
another special case of QDDFSM is the Direct Digital Frequency
Synthesizer (DDFS). In DDFS configuration we simply fix the input
vector (X.sub.0, Y.sub.0) to be (1,0). This enables the complete
elimination of the coarse stage by taking advantage of the fact
that 1.times.A=A and 0.times.A=0. In the following section we will
concentrate our discussion on the QDDFSM, since it is the general
case, while keeping in mind the special cases and the associated
hardware reductions mentioned above.
[0544] 5.8.1 Using the Angle Rotation Processor in a Quadrature
Direct Digital Frequency Synthesizer/Mixer
[0545] The frequency synthesis and mixing operation can be
described with the following pair of equations, which relate an
input with x-y coordinates (X.sub.0, Y.sub.0) and a frequency
control word (fcw) for the synthesizer, to an output with new x-y
coordinates (X, Y). The following pair of equations establishes the
relationship between (X.sub.0, Y.sub.0), fcw, and (X, Y).
X=[X.sub.0.times.cos(fcw.times.n)]-[Y.sub.0.times.sin(fcw.times.n)]
Y=[Y.sub.0.times.cos(fcw.times.n)]+[X.sub.0.times.sin(fcw.times.n)]
(5.88)
[0546] where n is the time index
[0547] Per (5.88), since the sine and cosine functions are periodic
with period 2.pi. (i.e.,
fcw.times.n=<fcw.times.n>.sub.2.pi.=.phi., where <>is a
modulo operator) an overflowing adder is used as a phase
accumulator to compute .phi. from the input fcw, as shown by the
the phase accumulator 4600 in FIG. 46.
[0548] Now, for any given time instance n, we have a corresponding
angle 4 from the phase accumulator, hence the original pair of
equations (5.88) for QDDFSM can be rewritten in terms of the angle
.phi. as follows.
X=[X.sub.0.times.cos .phi.]-[Y.sub.0.times.sin .phi.]
Y=[Y.sub.0.times.cos .phi.]+[X.sub.0.times.sin .phi.] (5.89)
[0549] Note that the expressions (5.89) are exactly those of an
angle rotator expressed by equations (5.1). By applying a phase
accumulator fed by an fcw, we have converted the QDDFSM into an
angle rotation application. The only conflict between the above
expressions and the angle rotation processor is that the angle
rotation processor takes an angle .theta. in the range [0, .pi./4],
while the angle .phi. in the above expressions is in the interval
[0,2.pi.).
[0550] 5.8.1.1 A General Angle Rotator for Arbitrary Input
Angles
[0551] Let us consider the changes necessary to make the angle
rotation processor use an input angle .phi. that may lie outside
the [0, .pi./4) range. Fortunately, a simple interchange operation
at the input of the coarse stage, and an interchange/negate
operation at the output of the fine stage is all we need in order
to map .phi. into an angle .theta. in the range [0, .pi./4] and use
it as the input to the angle rotator. Even though the input angle
.theta. is in the range [0, .pi./4], the rotation by .theta. along
with the interchange and interchange/negate operations make the
overall rotation of the input (X.sub.0, Y.sub.0) equivalent to a
rotation by the original angle .phi. in the full range [0,2.pi.).
The latter is possible because of the convenient symmetry
properties of sine and cosine functions over the range [0,
2.pi.].
[0552] For example, sin .phi.=-sin(.phi.-.pi.) and cos
.phi.=-cos(.phi.-.pi.), while sin .phi.=cos(.phi..pi./2) and cos
.phi.=-sin(.phi.-n/2), and finally, for 0<.phi.<.pi./4, if we
write .pi./4+.phi. for .phi. then
sin(.pi./4+.phi.)=cos(.pi./4-.phi.) and
cos(.pi./4+.phi.)=sin(.pi./4-.phi.). Using the first pair of
trigonometric identities, we can map .phi. into the range [0,
.pi.]) by simply performing a negate operation at the output of the
angle rotator. Using the second pair of identities along with the
first pair enables one to map .phi. into the range [0, .pi./2) by
performing negate and interchange operations at the output of the
angle rotator. Finally, using all three pairs of identities, the
angle .phi. can be mapped into the range [0, .pi./4) by performing
an interchange operation at the input of the angle rotator, along
with interchange and negate operations at the output of the angle
rotator. Note that all of these interchange and negate operations
are conditioned only on the issue of which octant .phi. is in. This
means that if .phi. is a normalized angle, then the interchange and
negate decisions depend only on the top three MSB bits of 4. The
following tables show the interchange and negate operations
required for all eight octants (specified by the three MSB bits of
.phi.). It is evident, as well, that other interchange and negate
criteria for the input and output would also be suitable.
[0553] This table indicates when an interchange operation is
required at the input and when an interchange operation is required
at the output of the angle rotator.
7 Octant of .phi. (3 MSBs of .phi.) Output Interchange Input
Interchange 1-st octant (000) 2-nd octant (001) Interchange inputs
3-rd octant (010) Interchange outputs 4-th octant (011) Interchange
outputs Interchange inputs 5-th octant (100) 6-th octant (101)
Interchange inputs 7-th octant (110) Interchange outputs 8-th
octant (111) Interchange outputs Interchange inputs
[0554] The following table indicates when a negation operation is
required at the output of the angle rotator.
8 Octant of .phi. Negation Negation (3 MSBs of .phi.) of output X
of output Y 1-st octant (000) 2-nd octant (001) Negate output X
3-rd octant (010) Negate output Y 4-th octant (011) Negate output X
Negate output Y 5-th octant (100) Negate output X Negate output Y
6-th octant (101) Negate output Y 7-th octant (110) Negate output X
8-th octant (111)
[0555] Note that the flag for input interchange is simply the 3rd
MSB bit of .phi., while the flag for output interchange is just the
2nd MSB bit of .phi.. Finally, to produce the remapped angle
.theta. in the range [0, .pi./4) for the angle rotation processor,
we simply take the remaining bits of .phi. after stripping the top
two MSBs and performing a conditional subtract operation to produce
.theta.. More specifically, if the MSB bit (after stripping the two
MSB bits) is low, i.e., the angle is in an even octant (numbering
them 0, . . . , 7), we pass the angle unchanged, otherwise we
perform a "two's-complement type" inversion of the angle. Note here
that after such remapping operation, the MSB bit of 0 is set to one
only in the case when .theta.=.pi./4. This fact is useful in
determining the required amount of lookup table in the angle
rotation processor. In other words, even though the MSB bit of
.theta. is an address to the lookup table, since we know that when
it is `1` the remaining bits have to all be `0` we only need to
allocate a single address for that case (as opposed to increasing
the size of the lookup table by an entire factor of two).
[0556] 5.8.1.2 Adapting the General Angle Rotator to Make a
QDDFSM
[0557] The structure of the QDDFSM using an angle rotation
processor 3900 is depicted in the FIG. 4700. It simply requires the
employment of a phase accumulator 4702 and a conditional subtract
4704 to provide an input angle from the input frequency control
word fcw. We refer to the system of FIG. 47 with the phase
accumulator excluded as a General Angle Rotator. It has the
capability to receive an angle in the interval [0, 2.pi.) and to
perform an angle rotation of the input data (X.sub.0, Y.sub.0) by
that angle. We show a general angle rotator in FIG. 48, but one in
which further structural simplification has been made. The method
of performing these simplifications will be discussed next.
[0558] 5.8.2 How to Use the Conditionally Negating Multipliers in
the General Angle Rotator
[0559] For a moment assume we have a powerful technique for making
conditionally negating multipliers. What we mean by that is a
multiplier which takes a negate flag to produce an output depending
on that negate flag as follows: The output is simply the product of
the input signals if the flag is low (0) and the output is the
negative of the product of the input signals if the flag is high
(1).
[0560] Each one of the two outputs in the coarse and fine stages is
computed with two multipliers and one adder as shown in FIG. 47.
These multipliers and the adder are implemented in a single
Carry-Save Adder (CSA) tree, with the partial products being
generated from Booth decode modules corresponding to the two
multipliers. This technique of employing a single tree eliminates
the need for intermediate carry propagation from each multiplier
and makes the propagation delay of each rotation stage very short.
Note that the single CSA tree implementation is possible since the
multipliers are operating in parallel. Furthermore, because the
structure that is needed to compute one output of a rotation stage
is identical to the structure required by the other output (with
the exception of the minus sign), a single CSA tree can easily be
interleaved between the two outputs for a significant amount of
hardware savings. The minus sign at the output of the multiplier
can be implemented very efficiently by the technique described in
the following sections (using the conditionally negating
multiplier). The negation or non-negation of the multiplier output
can be controlled with a flag that changes between the two cycles
of the interleave operation.
[0561] The angle at the output of the conditional subtract module
4704 in FIG. 47 is in the range [0, .pi./4]. As already discussed,
the outputs for the angles outside this range are constructed by
mapping the angle into the range [0, .pi./4) while conditionally
interchanging the inputs (inputs to the coarse stage) and
conditionally interchanging and negating the outputs (outputs of
the fine stage) of the angle rotator. A negation at the output of
the fine stage simply means changing the output signs of the
multipliers and negating the input of the adder coming from the
input of the fine rotation stage. Changing the output signs of the
multipliers is once again accomplished by using conditionally
negating multipliers. The negation of the input to the fine
rotation stage can easily be implemented with XOR gates and a
conditional high or low bit insertion into the CSA tree at the
position corresponding to the LSB location of the input. Since this
conditional high or low bit is inserted in the CSA tree, there is
no additional carry propagation introduced for the negation of the
input. Note that the latter technique eliminates any circuitry
required to implement the conditional negation of the outputs, and
hence eliminates any carry propagations associated with two's
complement numbers.
[0562] Furthermore, the conditional interchange of the outputs can
be implemented by conditionally interchanging the inputs of the
fine rotation stage and appropriately controlling the signs of the
multiplier outputs in the fine stage. The conditional interchange
of the fine stage inputs can be propagated to the inputs of the
coarse stage with the same line of reasoning. Remember that the
inputs to the coarse stage were conditionally interchanged
according to the three MSBs of the input angle anyway. In
conclusion, the conditional interchange and negation operations of
the outputs can be implemented by modifying only the condition of
the interchange at the inputs of the coarse stage and appropriately
controlling the multiplier output signs by using conditionally
negating multipliers (which we had to do for interleaving anyway).
This eliminates the conditional negate and interchange block at the
output of the fine stage entirely (i.e., it eliminates muxes and
two's complement negators), and also eliminates the need for
storing and pipelining control signals (i.e., it eliminates
registers) to perform the conditional interchange and negation
operations at the output. The resulting General Angle Rotator 4800
is now depicted in the following FIG. 48.
[0563] 5.8.2.1 Booth Multiplier
[0564] There are many algorithms for digital multiplication. One of
the most popular is the Booth multiplier. The, essence of the Booth
multiplier is in the decoding scheme performed on the multiplicand
to reduce the number of partial products which, when added
together, produce the desired product. For an N.times.M Booth
multiplier, where N is the wordlength of the multiplier, and M is
the wordlength of the multiplicand, there will be ceiling(N/2)
Booth decoders. Each Booth decoder will take three bits from the
multiplier (with one bit overlapping the decoders on both sides)
and will manipulate the multiplicand according to the Booth
decoding table 5000 shown in FIG. 50. Some relevant details for a
10.times.M Booth multiplier are depicted in FIG. 49, especially how
the multiplier bits feed into the Booth decoders to produce the
five partial products which, when added, compute the result (the
product of the multiplier and the multiplicand).
[0565] 5.8.2.2 How to Make a Negating Booth Multiplier
[0566] Suppose we wish to make a multiplier that produces the
negative of the product. More specifically, suppose we wish to
multiply two signals N and M and get -C=-(N.times.M). The latter
can be accomplished in a number of different ways. The most obvious
is perhaps to use a regular multiplier to produce the product
C=(N.times.D) and then negate C to achieve -C=-(N.times.M). In case
of two's complement representation, this approach requires an
additional carry propagation chain through the negator, which is
costly in terms of speed and additional hardware associated with a
negating circuit. Another approach, described below, is more
favorable in a few key aspects.
[0567] The product C is essentially the result of adding a number
of partial products, which are generated by the Booth decode blocks
as described in the previous section. Therefore, we can write the
following sum expression for C: 108 C = i = 1 n p i ( 5.90 )
[0568] where p.sub.i are then (in the 10.times.M example above n=5)
partial products generated from the n Booth decoders. Note that, in
order to negate C, we can negate all of the partial products and
proceed with the summation of the negated partial products to
produce -C. The expression for -C is then the following: 109 - C =
i = 1 n - p i , ( 5.91 )
[0569] where -p.sub.i are the negated n partial products generated
from the n Booth decoders. Let us investigate how the Booth
decoders need to change to produce the desired negated partial
products. All we need to do is to change the decoding table 5000
from that of FIG. 50, to the decoding table 5100 in FIG. 51. Note
that the difference between the tables is only in the partial
product columns and, more specifically, the partial product column
5102 of table 5100 is the negative of the partial product column
5002 of table 5000. This means that by simply modifying the Booth
decode table to the negating Booth decode table shown in FIG. 51,
the result will be the negative of the product, as desired, with
absolutely no additional hardware and absolutely no speed penalty.
An example for a 10.times.M negating Booth multiplier 5200 is shown
in FIG. 52.
[0570] 5.8.2.3 How to Make a Conditionally Negative Booth
Multiplier
[0571] A particularly interesting case arises when one wishes the
multiplier product to be negated sometimes, and normal
(non-negated) the other times. One can extend the idea presented in
the previous section to craft the following powerful technique. Let
us investigate the original Booth decode table 5000 depicted in
FIG. 50 and the negating Booth decode table 5100 of FIG. 51 a bit
more closely. Note the horizontal line of symmetry that runs
through the midline of both decoding tables. This line of symmetry
suggests that we can create the negating Booth decode table 5100
from the original Booth decode table 5000 by simply inverting the
three bits (b2 b1 b0). For example, if the three bits (b2 b1 b0)
are (0 1 0), then, according to the original Booth decode table,
the corresponding partial product is A, where A is the
multiplicand. If we invert the three bits (b2 b1 b0) as suggested
above, we will have (1 0 1) and the corresponding partial product
will be -A, exactly what is needed for a negated partial
product.
[0572] Given a signal F which specifies when the output of the
multiplier should be negated and when not (F=0 implies regular
multiplication, F=1 implies negating multiplication), F can simply
be XORd with the three bits (b2 b1 b0) at the input of the regular
Booth decoders to make a new conditionally negating Booth decoder,
hence a conditionally negating multiplier. The details of a
conditionally negating Booth decoder 5300 are captured in FIG. 53.
Note that with a minimal amount of hardware (N XOR gates for an
N.times.M multiplier, which is insignificant compared to the
hardware cost of the entire multiplier), we have the means to
control the sign of the multiplier product. Also note that the
overall latency of the multiplier is increased insignificantly
since the latency through a single XOR gate is much smaller than
the latency through the entire multiplier. Furthermore, the latency
of a single XOR gate is much smaller than the latency associated
with a carry propagation chain that would be necessary if one built
such a circuit with a two's complement negator. A 10.times.M
conditionally negating multiplier 5400 is shown in FIG. 54.
[0573] 5.8.3 Using the Angle Rotation Processor in a Quadrature
Direct Digital Frequency Synthesizer
[0574] As mentioned above, the angle rotator is useful in
implementing various forms of direct digital synthesizers. In this
case, all starting points for the angle rotations are X.sub.0=1,
Y.sub.0=0 (with, of course, the various usual interchange/negation
requirements). FIG. 55 shows a quadrature direct digital
synthesizer (QDDS) 5500, a system having two outputs, one being
samples of a cosine waveform and the other being samples of a sine
waveform. An exact 90-degree phase offset between the two waveforms
is obtained by the QDDS, and numerous applications for such a
device are well known. No X.sub.0 and Y.sub.0 input samples are
shown in the FIG. 55 system. These fixed values have been "built
in" and used to greatly simplify the coarse rotation stage.
[0575] Notice that the angle rotator 5502 is preceded by a system
5504 that generates a data stream of input rotation angles, a
so-called overflowing phase accumulator 5506, and its input is a
single fixed data word that precisely controls the frequency of the
output sample waveforms. The three MSBs of each phase accumulator
output word, of course, assess the approximate size of the angle
that is being used as a rotation angle (i.e., these three bits show
how many octants the rotation angle spans), and they are stripped
off to control the interchange/negation operations that are
appropriate for obtaining the desired output samples. Also, the
third MSB is used, as described previously, to determine whether or
not to perform a "two's complement type" inversion of the LSBs. One
other operation is required by the "Conditional Subtract" module
5508 shown in FIG. 55; in addition to stripping off the three MSBs,
it appends one MSB having the value zero except in the case where a
rotation angle of exactly .pi./4 is required. In that case, the
appended MSB is one and all other ROM-address bits are zero.
[0576] A special case of the QDDS system, one having only a single
output data stream, which could be either of the two, but which we
call the "cosine-only" case, is also useful for various well-known
applications. FIG. 56 and FIG. 57 show two specializations of the
angle rotator circuits previously discussed to implement the
cosine-only DDS. The system 5600 in FIG. 56 results from
specializing the angle-rotation system 3900 in FIG. 39. The system
5700 in FIG. 57 is a specialization of the angle rotator 4400 in
FIG. 44.
[0577] 5.9 Conclusion
[0578] Based on the design method discussed, for a given accuracy
requirement, an architecture with the least amount of hardware is
produced by balancing the precision of intermediate computations
and the complexity of each arithmetic block, while keeping the
output error within the specified bound. Furthermore, our
architecture consolidates all operations into a small number of
reduced-size multipliers. This permits us to take advantage of many
efficient techniques that have been developed for multiplier
implementation, such as Booth encoding, thereby yielding a smaller
and faster circuit than those previously proposed.
[0579] Simulations and preliminary complexity estimation show that,
even comparing to the method of (Tan, L. and Samueli, H., IEEE J.
Solid-State Circuits 30:193-200 (1995)) that is optimized for a
14-bit input angle, our method achieved 6 dB more SFDR while using
approximately the same number of transistors as those needed by
(Tan, L. and Samueli, H., IEEE J. Solid-State Circuits 30:193-200
(1995)). In addition, since our structure employs only a small ROM,
it overcomes the problem of slow access time that occurs when large
ROMs are used, thereby facilitating a higher data rate. Using the
two-stage method, when a higher precision is needed, it is very
straightforward to satisfy such a requirement, since more accurate
results can be attained simply by increasing the wordlength and the
multiplier size. For the single-stage method, however, when high
precision is desired, the required lookup table is likely to be too
large to be practical, particularly for high-speed operation.
6. Symbol Synchronization for Bursty Transmissions
[0580] We have thus far discussed methods that provide efficient
implementations of the resampler for symbol synchronization in a
digital receiver using trigonometric interpolation as well as the
phase rotator for carrier recovery. To produce the correct samples,
a timing recovery circuit must supply the resampler with symbol
timing information, as shown in FIG. 1D. We will now consider how
this can be accomplished.
[0581] 6.1 Initial Parameter Estimations for Burst Modems
[0582] There are many methods to derive timing information from the
received signal. According to their topologies, synchronization
circuits can be divided into two categories: there are feedback and
feedforward schemes. Feedback structures usually have very good
tracking performance, and they work quite well in continuous mode
transmissions. For packet data systems used by third-generation
mobile communications, where the transmission is bursty, it is
essential to acquire initial synchronization parameters rapidly
from the observation of a short signal-segment.
[0583] A typical packet format is shown in FIG. 58. It includes a
short preamble 5802 followed by user data 5804. The preamble 5802
is a set of known modulation symbols added to the user data packet
at the transmitter with the intention of assisting the receiver in
acquisition.
[0584] There are many approaches to burst demodulation, depending
on the specific system requirements. In one approach (S. Gardner,
"Burst modem design techniques, part 1," Electron. Eng. 71:85-92
(September 1999); Gardner, S., "Burst modem design techniques, part
2," Electron. Eng. 71:75-83 (December 1999)), the receiver first
detects the presence of the preamble, using a correlator, whose
output should produce a large magnitude when the preamble is
present. It then estimates the symbol timing. If the sampling
frequency error is small, the total change of the timing phase from
the start of the short preamble to the end is negligible. Next, it
estimates the initial carrier frequency and phase. The above steps
assume that the impairment caused by the channel is small enough
that the modem can successfully track the timing carrier phase
prior to equalization. Otherwise, equalizer training prior to the
timing and carrier recovery is needed.
[0585] With a typical preamble of 8 to 32 symbols, depending on the
required system performance, for QPSK modulation, rapid acquisition
is desired. Feedforward timing estimation is known to have rapid
acquisition, since it produces a one-shot estimate instead of
tracking the initial timing through a feedback loop.
[0586] A well-known method, digital square timing recovery (Oerder
M., and Meyr, H., IEEE Trans. Comm. 36:605-612 (1988)), has shown
rapid acquisition, but it requires oversampling of the signal at,
typically, four times the symbol rate, which imposes a demand for
higher processing speed on the subsequent digital operations.
Moreover, it does not work well for signals employing small excess
bandwidth. However, pulses with small excess bandwidth are of
interest for bandwidth-efficient modulation.
[0587] For applications where low power and low complexity are the
major requirements, such as in personal communications, it is
desirable to sample the signal at the lowest possible rate and to
have the synchronizer be as simple as possible. In this section, a
synchronizer is proposed that needs just two samples per symbol
period. In addition, it has been shown to work well for small
excess bandwidth, which is important for spectral efficiency. Using
this method, the estimations of the symbol timing and the carrier
phase can be carried out independently of each other. Hence, they
can be carried out in parallel. Using the proposed structure, the
timing and carrier-phase estimators can be implemented efficiently
by means of direct computation (instead of a search, as is
employed, for example, by (Sabel, L., and Cowley, W., "A recursive
algorithm for the estimation of symbol timing in PSK burst modems,"
in Proc. Globecom 1992, vol. 1 (1992), pp. 360-364) using an
efficient rectangular-to-polar converter (to be discussed in
Section 7). This yields a very small computation load. Thus, this
structure is well suited for low-power, low-complexity and
high-data-rate applications, such as those in multimedia mobile
communications.
[0588] 6.2 Background Information
[0589] The system model 5900 used in developing the symbol timing
and carrier phase recovery algorithm described in this section is
shown in FIG. 59.
[0590] Here h(t) is a real-valued, unit-energy square-root Nyquist
pulse and w(t) is complex white Gaussian noise with independent
real and imaginary components, each having power spectral density
N.sub.0/2.
[0591] As mentioned in Section 6.1, a typical data packet for a
burst modem consists of a short preamble 5802 followed by user data
5804. According to the approach of (Gardner, S., Electron. Eng.
71:85-92 (September 1999)), the matched filter output is sampled
every T.sub.s=T/2 seconds, i.e., at twice the symbol rate. The
receiver then detects the presence of the preamble in the received
signal by correlating the conjugate of the known preamble sequence
a.sub.m, whose length is L, with the sampled data x(nT.sub.s) as
110 r xx ( n ) = m = 0 L - 1 a m * x ( nT s + 2 mT s ) . ( 6.1
)
[0592] The correlator output r.sub.xx(n) should produce a large
magnitude .vertline.r.sub.xx(n).vertline. when the preamble is
encountered. It then estimates the initial synchronization
parameters, namely the symbol timing and the carrier phase,
assuming the transmitter/receiver frequency mismatches are
insignificant.
[0593] The complex envelope x(t) of the received signal, after the
matched filter, is 111 x ( t ) = j k = - .infin. .infin. a k g ( t
- kT - ) + v ( t ) ( 6.2 )
[0594] where {a.sub.k} is a sequence of independent
equally-probable symbols with
E[.vertline.a.sub.k.vertline..sup.2]=1. We also have that
v(t)=w(t){circle over (x)}h(-t) and that g(t)=h(t){circle over
(x)}h(-t) is a Nyquist pulse. The time delay .tau. and the carrier
phase .theta. are both unknown.
[0595] To estimate the data sequence a.sub.k we want sample values
of x(t) at t=mT+.tau., with m an integer, whereas only the samples
x(nT.sub.s) are available after sampling x(t) by a fixed clock.
[0596] Now let us examine how the correlator output relates to
symbol timing and carrier phase. Inserting (6.2) into (6.1) yields
112 r xx ( n ) = m = 0 L - 1 k = - .infin. .infin. a m * a k g ( nT
s + 2 mT s - kT - ) j + m = 0 L - 1 a m * v ( nT s + 2 mT s ) . (
6.3 )
[0597] Since the data are independent, and they are independent of
the noise, we have 113 E [ a m * a k ] = { 1 k = m 0 k m ( 6.4 ) E
[ a m * v ( nT s + 2 mT s ) ] = 0. ( 6.5 )
[0598] According to (6.4) and (6.5), and because T=2T.sub.s the
expectation of r.sub.xx(n) with respect to the data and the noise
is (for simplicity, we omit the constant real scale factor L)
E[r.sub.xx(n)]=e.sup.j.theta.g(nT.sub.s-.tau.). (6.6)
[0599] Thus, the mean value of the complex preamble correlator
output actually equals the sample of the delayed signaling pulse
g(t), with delay being .tau., rotated by the angle .theta.. This is
shown in FIG. 60 for .theta.=0, where g(t) is a raised cosine
Nyquist pulse with .alpha.=0.35. The total timing delay .tau. can
be expressed as
.tau.=n.sub.0T.sub.s+.mu. (6.7)
[0600] where the integer no represents the portion of .tau. that
corresponds to an integer multiple of the sampling interval T.sub.s
and 0.ltoreq..mu.<T.sub.s is the sampling time mismatch.
[0601] Most practical signaling pulses g(t) are symmetrical and
their peak value occurs at g(0). If .theta. is known, using these
properties, we can estimate the sampling time mismatch .mu. from
the correlator output r.sub.xx(n). In the next section we will
discuss such an algorithm. We will derive this algorithm by first
assuming that .theta.=0. Then we will discuss how the method can be
carried out independently of the carrier phase. Simultaneously, we
also derive a phase estimation algorithm that is independent of the
symbol timing.
[0602] 6.3 Symbol Timing Estimation Assuming .theta.=0
[0603] From (6.6), with .theta.=0, we have
E[r.sub.xx(n)]=g(nT.sub.s-.tau.). (6.8)
[0604] According to (6.7) and (6.8), if the transmission delay
.tau. is exactly an integer multiple of T.sub.s we must have
.mu.=0, and thus r.sub.xx(n.sub.0) must correspond to the peak
g(0). Otherwise, we have .mu..noteq.0, with r.sub.xx(n.sub.0) and
r.sub.xx(n.sub.0+1) being the two correlator output values nearest
the peak value g(0), as shown in FIG. 60. That is,
r.sub.xx(n.sub.0) and r.sub.xx(n.sub.0+1) must be the two largest
correlator outputs. Therefore, once the largest correlator output
is located, we can obtain n.sub.0, the integer part of .tau..
[0605] We now turn to finding .mu.. Without loss of generality, let
us assume T.sub.s=1. Replacing n by n.sub.0+n we have, according to
(6.8) and (6.7),
E[r.sub.xx(n.sub.0+n)]=g((n.sub.0+n)-.tau.)=g(n-.mu.). (6.9)
[0606] For simplicity in our discussion on finding the fractional
delay .mu., we henceforth drop the index n.sub.0, which corresponds
to an integer multiple of sample delays, from our notation. Next we
define R(e.sup.j.omega.) as the Fourier transform of r.sub.xx(n):
114 R ( j ) = n = - .infin. .infin. r xx ( n ) j n . ( 6.10 )
[0607] The expectation of R(e.sup.j.omega.) can be expressed as
E[R(e.sup.j.omega.)]=E[FT(r.sub.xx(n))]=FT(E[r.sub.xx(n)]).
(6.11)
[0608] According to (6.9), and (C.4) in Appendix C, we have
E[R(e.sup.j.omega.)]=FT(g(n-.mu.))=e.sup.j.omega..mu.G(e.sup.j.omega.)
(6.12)
[0609] where G(e.sup.j.omega.) is the Fourier transform of g(n).
Since g(n) is symmetrical, G(e.sup.j.omega.) must have zero phase.
Thus, according to (6.12),
arg(E[R(e.sup.j.omega.)])=arg(e.sup.j.omega..mu.G(e.sup.j.omega.))=.omega.-
.mu.. (6.13)
[0610] Evaluating (6.13) at .omega.=.pi./2, we can obtain an
estimate of .mu. as 115 = 2 arg ( R ( j / 2 ) ) . ( 6.14 )
[0611] Therefore, the unknown sampling mismatch .mu. can be
obtained by taking the Fourier transform of r.sub.xx(n) and
evaluating the phase corresponding to .omega.=.pi./2.
[0612] To make the implementation of (6.14) realistic, we should
truncate the sequence r.sub.xx(n) before taking its Fourier
transform. For example, using only the four samples r.sub.xx(-1),
r.sub.xx(0), r.sub.xx(1), and r.sub.xx(2), we have
R.sub.T(e.sup.j.pi./2)=[r.sub.xx(0)-r.sub.xx(2)]+j[r.sub.xx(-1)-r.sub.xx(1-
)]. (6.15)
[0613] Using the correlator output, the .mu. value can be obtained
by first computing R.sub.T(e.sup.j.pi./2) according to (6.15), and
then from the following: 116 = 2 arg ( R T ( j / 2 ) ) . ( 6.16
)
[0614] For low precision requirements, this operation can be
accomplished using a small CORDIC processor (Chen, A., et al.,
"Modified CORDIC demodulator implementation for digital IF-sampled
receiver," in Proc. Globecom 1995, vol. 2 (November 1995), pp.
1450-1454) or a ROM lookup table (Boutin, N., IEEE Trans. Consumer
Electron. CE-38:5-9 (1992)). With high accuracy requirements,
however, the CORDIC processor will have long delays, while the
table-lookup method will certainly require a very large ROM. In
this case, we propose to use the rectangular-to-polar converter
which will be discussed in Section 7. This rectangular-to-polar
converter requires two small ROMs and it consolidates the
operations into small array-multipliers, which can yield a smaller
and faster circuit using well-known efficient multiplier
implementation techniques.
[0615] A synchronizer 6100 for implementing the synchronization
scheme described above is illustrated in FIG. 61. The synchronizer
6100 includes a correlator 6102, a Fourier Transform module 6104,
and a rectangular-to-polar converter 6106. The Fourier transform
module 6104 includes various delays and adders that are known to
those skilled in the arts. The rectangular-to-polar converter is
described further in Section 7.
[0616] The synchronizer 6100 receives data samples associated with
sampling one or more received symbols and determines an offset
.pi..mu./2, where .mu. represents a synchronization offset of the
data samples relative to the incoming symbols. The operation of
synchronizer 6100 is described in reference to the flowchart 6200,
as follows.
[0617] In step 6202, a set of complex data samples is received.
[0618] In step 6204, the correlator 6102 correlates the complex
data samples with a complex conjugate of a preamble data set
(a.sub.m*), resulting in correlated complex data samples.
[0619] In step 6206, the Fourier transform module 6104 determines
the Fourier transform of the correlated data samples signal,
according to equations (6.10)-(6.13) and related equations;
[0620] In step 6208, the Fourier transform module 6104 evaluates
the Fourier transform of the correlated data samples at .pi./2,
generating a complex signal representing a complex number;
[0621] In step 6210, the rectangular-to-polar converter 6106
determines an angle in a complex plane associated with the complex
number of step 6210, where the angle represents synchronization
between the data samples and the incoming symbols.
[0622] In step 6212, the angle from step 6210 is scaled by 2/.pi.
to determine the synchronization offset.
[0623] 6.4 Bias in Symbol Timing Estimation due to Truncating the
Sequence
[0624] By truncating the sequence r.sub.xx(n) before taking the
Fourier transform, we have produced a very simple structure to
compute .mu.. However, since R.sub.T(e.sup.j.omega.) differs from
R(e.sup.j.omega.) we must determine how this difference would
affect the estimated .mu. value. The truncated sequence r.sub.T(n)
is related to the original sequence r.sub.xx(n) as
r.sub.T(n)=r.sub.xx(n)w(n) (6.17)
[0625] where w(n) is a rectangular function whose Fourier transform
W(e.sup.j.omega.) is a sinc function. Thus,
R.sub.T(e.sup.j.omega.)=R(e.sup.j.omega.){circle over
(x)}W(e.sup.j.omega.). (6.18)
[0626] Taking the expectation of (6.18) we have 117 E [ R T ( j ) ]
= E [ R ( j ) W ( j ) ] = E [ R ( j ) ] W ( j ) . ( 6.19 )
[0627] Obviously, the .mu. value obtained using
R.sub.T(e.sup.j.omega.) in (6.16) would be different from that
obtained using R(e.sup.j.omega.). This will introduce a non-zero
timing-jitter mean (bias) to the .mu. value obtained using
R.sub.T(e.sup.j.omega.) instead of R(e.sup.j.omega.). But the phase
difference of the expected values of R.sub.T(e.sup.j.pi./2) and
R(e.sup.j.pi./2) can be computed for a given g(t). The procedure is
as follows:
[0628] 1. Given the pulse waveform g(t), obtain, for each value
.mu., the samples g(n-.mu.), n=-1, . . . , 2.
[0629] 2. Compute R.sub.T(e.sup.j.pi./2) using these samples
g(n-.tau.) according to (6.15).
[0630] 3. Find the value {circumflex over (.mu.)} according to
(6.16). The difference between the desired value .mu. and the value
{circumflex over (.mu.)} computed using finite samples g(n-.mu.),
n=-1, . . . , 2, is the bias.
[0631] This bias is illustrated in FIG. 63, where g(t) is a raised
cosine Nyquist pulse with rolloff factor .alpha.=0.1.
[0632] From FIG. 63, the bias is a function of .mu. and it can be
precalculated and stored in a ROM in the receiver. Although the
size of the ROM depends on the precision of .mu., for typical
precision requirements on .mu. the ROM can be quite small. Let us
illustrate this point using an example: If an 8-bit accuracy is
desired for the bias, the bias value corresponding to the three
most significant bits (MSBs) in .mu. is indistinguishable from that
corresponding to the full-precision .mu. value. Hence, we can use
only the 3 MSBs in .mu. to determine the bias, thereby needing only
8 words in the bias lookup table.
[0633] Thus, for each of our symbol timing detector output samples,
we can obtain the corresponding bias value from the ROM table, then
subtract this bias from the original timing estimate to obtain an
unbiased estimate.
[0634] We have thus far restricted our discussion to the timing
recovery algorithm for .theta.=0. We now consider how this
algorithm can be made to accommodate an arbitrary carrier phase
.theta..
[0635] 6.5 Carrier-Independent Symbol Timing Recovery
[0636] According to (6.6), with the T.sub.s=1 normalization, the
complex correlator output r.sub.xx(n) is dependent on .theta..
Although the expectation of its magnitude
E[.vertline.r.sub.xx(n).vertline.]=.vertline.g(n-.tau.).vertline.
(6.20)
[0637] does not depend on .theta., it is non-trivial to compute the
magnitude of r.sub.xx(n) from its real and imaginary components.
Expressing r.sub.xx(n) in terms of its real and imaginary
components, according to (6.6), we have
E[r.sub.xx(n)]=g(n-.mu.)cos .theta.+jg(n-.mu.)sin .theta..
(6.21)
Thus,
E[Re[r.sub.xx(n)]]=g(n-.mu.)cos .theta. (6.22)
E[Im[r.sub.xx(n)]]=g(n-.mu.)sin .theta.. (6.23)
[0638] Since the carrier phase .theta. does not depend on .mu. we
can treat it as a constant scale factor in Re[r.sub.xx(n) and
Im[r.sub.xx(n)] when we are only concerned with extracting the
timing information.
[0639] Clearly, therefore, instead of using the magnitude of the
complex r.sub.xx(n) value, we can use one of its real and imaginary
parts, which are available at the output of the preamble
correlator.
[0640] We, of course, must decide which of Re[r.sub.xx(n)] and
Im[r.sub.xx(n)] to use. If the unknown phase .theta. is such that
cos .theta..apprxeq.0 it is certainly desirable to use
Im[r.sub.xx(n)] instead of Re[r.sub.xx(n)], and vise versa. But we
don't know the .theta. value thus far. How do we decide which one
to use?
[0641] From (6.22) and (6.23) we can see that the relative
magnitudes of cos .theta. and sin .theta. can be obtained from the
real and imaginary components of r.sub.xx(n). For example, if
.vertline.Re[r.sub.xx(n)].vert-
line.>.vertline.Im[r.sub.xx(n)].vertline. we certainly have that
.vertline.cos .theta..vertline.>.vertline.sin .theta..vertline.,
thus we should use the real part of the correlator output to find
.mu.. Henceforth we denote the appropriate (real or imaginary) part
of r.sub.xx(n) by {circumflex over (r)}.sub.xx(n).
[0642] 6.6 Carrier Phase Computation
[0643] Next, let us examine the problem of extracting the carrier
phase. From (6.6) we can see that the phase of the complex number
E[r.sub.xx(n)] does not depend on .mu.. Moreover, the carrier phase
can simply be obtained by extracting the phase of r.sub.xx(n). In
order to achieve the best precision, it is desirable to choose the
r.sub.xx(n) value with the largest magnitude for carrier phase
estimation. For example, if r.sub.xx(n.sub.0) is the correlator
output with largest squared-magnitude, we choose r.sub.xx(n.sub.0)
to compute
.theta.=arg(r.sub.xx(n.sub.0)). (6.24)
[0644] One advantage of this approach is that the symbol timing and
carrier phase estimations are independent of each other. They can
thus be carried out in parallel.
[0645] As for symbol timing estimation in (6.16), the computation
in (6.24) can be accomplished efficiently using the
rectangular-to-polar converter to be discussed in Section 7.
[0646] A synchronizer 6400 for determining timing and phase offsets
is shown in FIG. 64. Similar to synchronizer 6100, the synchronizer
6400 receives data samples associated with sampling one or more
received symbols and determines a timing offset .pi..mu./2, where
.mu. represents a synchronization offset between the data samples
and the incoming symbols. Additionally, the synchronizer 6400
determines a carrier phase offset represented by .theta.. The
synchronizer 6400 includes the correlator 6102, sample selectors
6404 and 6406, the Fourier transform module 6104, and two
rectangular-to-polar converters 6106. The operation of synchronizer
6400 is described in reference to the flowchart 6500 in FIGS.
65A-B, as follows. The order of the steps in flowchart 6500 is not
limiting, as one or more steps can be performed simultaneously or
in a different order, as will be understood by those skilled in the
relevant arts.
[0647] In step 6502, a set of complex data samples is received.
[0648] In step 6504, the correlator 6102correlates the complex data
samples with a complex conjugate of a preamble data set (a.sub.m*),
resulting in correlated complex data samples. Each correlated
complex data sample includes a real sample and an imaginary
sample.
[0649] In step 6506, the sample set selector 6404 selects either
the set of real correlated samples or the set of imaginary
correlated samples. In embodiments, the set with the larger
magnitude is selected.
[0650] In step 6508, the Fourier transform module 6104 determines
the Fourier transform of the selected real or imaginary data
samples, according to equations (6.10)-(6.13) and related
equations;
[0651] In step 6510, the Fourier transform module 6104 evaluates
the Fourier transform at .pi./2, generating a complex signal
representing a complex number;
[0652] In step 6512, the rectangular-to-polar converter 6106a
determines an angle in a complex plane associated with the complex
number of step 6510, where the angle represents synchronization
between the data samples and the incoming symbols.
[0653] In step 6514, the angle from step 6512 is scaled by 2/.pi.
to determine the synchronization offset.
[0654] In step 6516, the selector 6406 selects the largest
correlator complex output. This selection can be based on an
examination of one of the parts (real, imaginary) of the data
sequence.
[0655] In step 6518, the rectangular-to-polar converter 6106b
determines an angle in a complex plane associated with complex
output of step 6516, where the angle represents the carrier phase
offset .theta..
[0656] 6.7 Simulation Result
[0657] We have used the above procedures to estimate the timing
delay and the carrier phase of binary PAM symbols. The pulse shape
was raised cosine with rolloff factor .alpha.=0.4. The block size
was L=32 preamble symbols. To demonstrate its performance for
signals with small excess bandwidth, we also tested this method
with .alpha.=0.1. For a carrier phase offset .theta.=45.degree., we
ran the simulation repeatedly using the synchronizer 6400, each
time using .mu. value randomly chosen between 0 and 1.
[0658] In addition to synchronizer 6400, we have also used the
following two well-known methods to estimate the sampling
mismatch:
[0659] 1) the DFT-based square-timing recovery (Oerder M., and
Meyr, H., IEEE Trans. Comm. 36:605-612 (1988)),
[0660] 2) the method of (Gardner, S., Electron. Eng. 71:75-83
(December 1999)) that maps
r.sub.xx(n.sub.0+1)/r.sub.xx(n.sub.0)--the ratio of the two
correlation values nearest the peak (see FIG. 60)--to the sampling
mismatch value .mu..
[0661] The variances of the timing jitter using these estimation
methods for .alpha.=0.4 and .alpha.=0.1 are plotted in FIG. 66 and
FIG. 67, respectively. The corresponding Cramer-Rao bounds
(CRB)--the theoretical lower bounds of estimation errors (Meyr, H.,
et al., Digital Communication Receivers. Synchronization, Channel
Estimation and Signal Processing, Wiley, New York, N.Y. (1998))are
also shown. We can see that, in both cases, the timing-jitter
variance using the proposed synchronizer is quite close to the
theoretical bound. It clearly outperforms the other two methods,
even for signals employing small excess bandwidth, as seen in FIG.
67.
[0662] The variance of the phase estimation error is depicted in
FIG. 68. It shows that, using the proposed method, the phase
estimation error agrees quite well with the theoretical bound.
[0663] 6.8 Conclusion
[0664] A synchronizer for initial symbol timing and carrier phase
estimation using preambles has been presented. This synchronizer
requires just two samples per symbol. Since the two estimations are
independent of each other, they can be carried out simultaneously.
These characteristics would ease the demand for computational speed
for high-data-rate applications. Moreover, this synchronizer has
demonstrated very good timing estimation performance even for
signals with small excess bandwidth, which is essential for
bandwidth efficient communications. The parameter estimations can
be implemented very efficiently using the synchronizer 6400. Due to
its simplicity, this method is attractive for applications where
low power and low complexity are desired, such as in a hand-held
transceiver.
7. A High-Speed Processor for Rectangular-to-Polar Conversion
[0665] As discussed previously, the rapid acquisition
characteristic of feedforward symbol synchronizers is essential to
symbol synchronization for burst modems. Many feedforward
structures require the evaluation of the phase of a complex number.
That is, an efficient implementation of the phase extraction
process is crucial. In order to handle a wide range of
communications problems (Section 8), a general rectangular-to-polar
conversion problem is considered.
[0666] There are several well-known implementations for a
rectangular to polar coordinate conversion, i.e. obtaining the
magnitude and phase of a complex number. One method uses a ROM
lookup table with both the real and imaginary components as input.
This is practical only for low bit-accuracy requirements, as the
ROM size grows exponentially with an increasing number of input
bits. To reduce the ROM size, we can first divide the imaginary by
the real component, then use the quotient to index the lookup
table. But the hardware for a full-speed divider is very
complicated and power consuming. An iterative divider implemented
using shifting and subtraction requires less hardware, but it is
usually quite slow. Recently, CORDIC has been applied in this
coordinate conversion (Chen, A., and Yang, S., "Reduced complexity
CORDIC demodulator implementation for D-AMPS and digital IF-sampled
receiver," in Proc. Globecom 1998, vol. 3 (1998), pp. 1491-1496).
However, due to the sequential nature of CORDIC, it is difficult to
pipeline, thus limiting the throughput rate.
[0667] In burst-mode communication systems, rapid carrier and clock
synchronization is crucial (Andronico, M., et al., "A new algorithm
for fast synchronization in a burst mode PSK demodulator," in Proc.
1995 IEEE Int. Conf. Comm., vol. 3 (June 1995), pp. 1641-1646).
Therefore, a fast rectangular-to-polar conversion is desired. In
this section, we present an apparatus and method that implements
the angle computation for rectangular-to-polar conversion with low
latency and low hardware cost. This processor and the
polar-to-rectangular processor presented in Section 5 (See rotator
3900 in FIG. 39), together, can perform the M-ary PSK modulation
devised in (Critchlow, D., "The design and simulation of a
modulatable direct digital synthesizer with non-iterative
coordinate transformation and noise shaping filter," M.S. thesis,
University of California, San Diego (1989)).
[0668] 7.1 Partitioning the Angle
[0669] FIG. 69 displays a point in the Cartesian X-Y plane having
coordinates (X.sub.0, Y.sub.0), wherein X.sub.0 and Y.sub.0
represent the real and imaginary parts of an input complex signal.
The angle .phi. can be computed as
.phi.=tan.sup.-1(Y.sub.0/X.sub.0). (7.1)
[0670] In deriving the core of our algorithm, we assume the
dividend and divisor satisfy
X.sub.0.gtoreq.Y.sub.0.gtoreq.0. (7.2)
[0671] We will discuss how to extend the result to arbitrary values
in Section 7.4. To achieve the highest precision for given
hardware, the inputs X.sub.0 and Y.sub.0 should be scaled such
that
1.ltoreq.X.sub.0<2. (7.3)
[0672] A straightforward method for fast implementation of (7.1)
can be devised as follows:
[0673] 1) Obtain the reciprocal of X.sub.0 from a lookup table.
[0674] 2) Compute Y.sub.0.times.(1/X.sub.0) with a fast
multiplier.
[0675] 3) Use this product to index an arctangent table for
.phi..
[0676] However, the size of the two tables grows exponentially with
increased precision requirements on .phi., and rather large tables
would be required to achieve accurate results. Therefore, for
high-precision applications, such an implementation seems
impractical.
[0677] If we approximate 1/X.sub.0 by the reciprocal of the most
significant bits (MSBs) of X.sub.0, denoted by [X.sub.0], then the
required reciprocal table is much smaller. We can then multiply the
table output by Y.sub.0 to yield Y.sub.0/[X.sub.0], which is an
approximation of Y.sub.0/X.sub.0. This quotient can then be used to
index an arctangent table. Similar to the reciprocal table, a much
smaller arctangent table is needed if we use only the MSBs of
Y.sub.0/[X.sub.0], denoted by [Y.sub.0/[X.sub.0]], to address the
table, which returns .phi..sub.1=tan.sup.-1([Y.sub.0/[X.sub.0]]).
Obviously, this result is just an approximation to .phi.. We will
subsequently refer to the computation of .phi..sub.1 as the coarse
computation stage.
[0678] Let .phi..sub.2 be the difference between .phi. and
.phi..sub.1. Using the trigonometric identity
tan .phi..sub.2=tan (.phi.-.phi..sub.1)=(tan .phi.-tan
.phi..sub.1)/(1+tan .phi..times.tan .phi..sub.1) (7.4)
[0679] and the definitions tan .phi.=Y.sub.0/X.sub.0 and tan
.phi..sub.1=[Y.sub.0/[X.sub.0]], we have 118 tan 2 = Y 0 / X 0 - [
Y 0 / [ X 0 ] ] 1 + ( Y 0 / X 0 ) .times. [ Y 0 / [ X 0 ] ] = Y 0 -
X 0 .times. [ Y 0 / [ X 0 ] ] X 0 + Y 0 .times. [ Y 0 / [ X 0 ] ] .
( 7.5 )
[0680] Using this relationship, .phi..sub.2 can be determined from
[Y.sub.0/[X.sub.0]], the coarse computation results. Therefore, the
desired result .phi. can be obtained by adding the fine correction
angle .phi..sub.2 to the coarse approximation .phi..sub.1. This
procedure of finding .phi..sub.2 will subsequently be referred to
as the fine computation stage.
[0681] By partitioning the computation of (7.1) into two stages,
the table size in the coarse stage can be reduced significantly at
the expense of additional computations, which are handled by the
fine stage. Let us now examine the complexity of the fine stage. To
find .phi..sub.2, we can first compute
X.sub.1=X.sub.0+Y.sub.0.times.[Y.sub.0/[X.sub.0]]
Y.sub.1=Y.sub.0-X.sub.0.times.[Y.sub.0/[X.sub.0]] (7.6)
and then find .phi..sub.2 as
.phi..sub.2=tan.sup.-1(Y.sub.1/X.sub.1). (7.7)
[0682] The computation in (7.6) involves only adders and
multipliers, while (7.7) requires lookup tables. Moreover, it seems
we can't use the same coarse-stage tables because they have low
resolution and thus can't satisfy the high precision requirements
for the fine angle .phi..sub.2. Now let us analyze .phi..sub.2 to
see if there is any property that can help in this situation.
[0683] If .phi..sub.1 is a good approximation of .phi., then
.phi..sub.2=.phi.-.phi..sub.1 is close to zero. In view of (7.7),
Y.sub.1/X.sub.1 should be very small too. This property helps us in
two respects: 1) The difference between Y.sub.1/X.sub.1 and
Y.sub.1/[X.sub.1] is much smaller than that between 1/X.sub.1 and
1[X.sub.1]. This suggests that if we use the same low resolution
reciprocal table as in the coarse stage, the error contributed to
the final result will be very small. We will demonstrate this in
the next section. 2) If Y.sub.1/X.sub.1 is sufficiently small to
satisfy
.vertline.Y.sub.1/X.sub.1.vertline.=.vertline.tan
.phi..sub.2.vertline.<- ;2.sup.-N/3 (7.8)
[0684] where N denotes the desired number of bits in .phi.,
then
.phi..sub.2=tan.sup.-1(Y.sub.1/X.sub.1).apprxeq.Y.sub.1/X.sub.1
(7.9)
[0685] and we can compute .phi..sub.2 without using an arctangent
table. This is explained as follows:
[0686] From the Taylor expansion of tan.sup.-1(Y.sub.1/X.sub.1)
near Y.sub.1/X.sub.1=0, we obtain
tan.sup.-1(Y.sub.1/X.sub.1)=Y.sub.1/X.sub.1-(Y.sub.1/X.sub.1).sup.3/3+O((Y-
.sub.1/X.sub.1).sup.5). (7.10)
[0687] Since O((Y.sub.1/X.sub.1).sup.5) is negligible in comparison
to (Y.sub.1/X.sub.1).sup.3/3, it can be omitted. Therefore, if
Y.sub.1/X.sub.1 is used to approximate tan.sup.-1(Y.sub.1/X.sub.1),
an error
.DELTA.
.sub.tan=tan.sup.-1(Y.sub.1/X.sub.1)-Y.sub.1/X.sub.1=-(Y.sub.1/X.s-
ub.1).sup.3/3 (7.11)
[0688] will occur. However, according to (7.8), .DELTA..sub.tan is
bounded by
.vertline..DELTA. .sub.tan.vertline.<2.sup.-N/3 (7.12)
[0689] which is very small. This indicates that the approximation
(7.9) is quite accurate if (7.8) is satisfied.
[0690] From the above analysis, no additional tables are needed for
the fine stage if .phi..sub.1 is sufficiently close to .phi.. On
the other hand, the better that .phi..sub.1 approximates .phi., the
larger the tables required for its computation become. As mentioned
previously, table size grows exponentially as the precision
increases. A good trade-off is obtained when the result .phi..sub.1
of the coarse stage is just close enough to .phi. that (7.8) is
satisfied, thereby eliminating the additional tables in the fine
stage. A detailed description of a rectangular-to-polar converter
that implements the algorithm follows.
[0691] FIG. 71 illustrates a rectangular-to-polar converter 7100
that implements the coarse and fine rotation described in section 7
herein, including equation (7.1)-(7.53). The converter 7100
receives a complex input signal 7102 (that represents a complex
number having X.sub.0 and Y.sub.0 components) and determines the
angle .phi., which represents the position of the complex signal
7102 in the complex plane. In doing so, the converter 7100
determines a coarse angle computation that is represented by the
angle .phi..sub.1, and performs a fine angle computation
represented by the angle .phi..sub.2. Once .phi..sub.1 is
determined, the input complex number 7102 is conceptually rotated
back toward the X-axis to an intermediate complex signal 7115 as
represented in FIG.72, and .phi..sub.2 is determined from
intermediate complex signal 7115. The angles .phi..sub.1 and
.phi..sub.2 are added together to determine .phi..
[0692] The converter 7100 includes: an input mux 7104, reciprocal
ROM 7106, output demux 7108, an arctan ROM 7110, a multiplier 7112,
a butterfly circuit 7114, a scaling shifter 7116, a fine angle
computation stage 7124, and an adder 7126. The fine angle
computation includes a multiplier set 7118, a one's complementer
7120, and a multiplier 7122.
[0693] The ROM 7106 stores reciprocal values of [X.sub.0], wherein
[X.sub.0] is defined as the most significant bits (MSB) of X.sub.0
of the input signal 7102. The reciprocal of [X.sub.0] is
represented as Z.sub.0, for ease of reference. As will be shown,
the ROM 7106 is re-used to determine the reciprocal of [X.sub.1],
where X.sub.1 is the real part of the intermediate complex number
7115 shown in FIG. 71 and FIG. 72. The reciprocal of [X.sub.1] is
represented as Z.sub.1, for ease of reference. In embodiments, the
ROM 7106 has 2.sup.N/3+1 storage spaces, where N is the number of
bits that represents X.sub.0 (and Y.sub.0) of the input signal
7102.
[0694] The input mux 7104 chooses between [X.sub.0] and [X.sub.1]
as an input to the reciprocal ROM 7106, according to the control
7128. The output demux 7108 couples an output of the ROM 7106 to
Z.sub.0 or Z.sub.1 according to the control 7128. The control 7128
assures that Z.sub.0 receives the stored reciprocal value for
[X.sub.0], and that Z.sub.1 receives the stored reciprocal value
for [X.sub.1].
[0695] The arctan ROM 7110 stores the coarse approximation angle
.phi..sub.1 based on a [Y.sub.0Z.sub.0] input. Therefore, a coarse
stage can be described as including the ROM ROM 7110, the ROM 7106,
and the multiplier 7112, as they are used in the coarse angle
computation.
[0696] The operation of the converter 7100 is described further
with reference to the flowchart 7300, as follows. The order of the
steps in the flowchart 7300 is not limiting as one or more of the
steps can be performed simultaneously, or in different order.
[0697] In step 7302, the input complex signal 7102 having a X.sub.0
component and a Y.sub.0 component is received. In embodiments, the
X.sub.0 and Y.sub.0 components are N-bit binary numbers.
[0698] In step 7304, the control 7128 causes Z.sub.0 to be
retrieved from the ROM 7106, where Z.sub.0 represents 1/[X.sub.0],
and wherein [X.sub.0] is the MSBs of X.sub.0.
[0699] In step 7306, the multiplier 7112 multiplies Y.sub.0 of the
input complex number 7102 by Z.sub.0, resulting in a
[Z.sub.0Y.sub.0] component. The [Z.sub.0Y.sub.0] component is an
approximation of Y.sub.0/X.sub.0.
[0700] In step 7308, the coarse angle .phi..sub.1 is retrieved from
the ROM 7110 based on [Z.sub.0Y.sub.0], and is sent to the adder
7126. Note that the coarse stage can be described as including the
ROM 7110, the ROM 7106, and the multiplier 7112, as they are used
in the coarse angle computation.
[0701] In step 7310, the butterfly circuit 7114 multiplies the
input complex signal 7102 by [Z.sub.0Y.sub.0]. This causes the
input complex signal 7102 to be rotated in the complex plane toward
the real axis to produce the intermediate complex signal 7115
(representing an intermediate complex number), having a real
X.sub.1 component and an imaginary Y.sub.1 component.
[0702] In step 7312, the scaler 7116 scales the X.sub.1 component
of the intermediate complex signal so that it is compatible with
the reciprocal values stored in the ROM 7106. The scaler also
scales the Y.sub.1 component by the same amount.
[0703] In step 7314, the control 7128 causes Z.sub.1 to be
retrieved from the ROM 7106 based on [X.sub.1], where Z.sub.1
represents 1/[X.sub.1], and wherein [X.sub.1] is the MSBs of
X.sub.1. Note, that the ROM 7106 is efficiently used twice to
calculate two different reciprocals Z.sub.0 and Z.sub.1, thereby
reducing overall memory size.
[0704] In step 7316, the fine angle computation stage 7124
determines the fine angle .phi..sub.2 based on Z.sub.1 and the
scaled intermediate complex number 7115. In doing so the
Newton-Raphson method is emulated in hardware to estimate
.phi..sub.2, which is the arctan of Z.sub.1Y.sub.1. More
specifically, multiplier set 7118 multiples X.sub.1, Y.sub.1 by
Z.sub.1. The ones' (approximating two's) complement 7120 is then
determined for X.sub.1Z.sub.1. After which, the multiplier 7127
multiplies (2-X.sub.1Z.sub.1) by Y.sub.1Z.sub.1, to determine tan
.phi..sub.2. Since .phi..sub.2 is a small angle, the value tan
.phi..sub.2 is used as an approximation of .phi..sub.2.
[0705] In step 7316, the .phi..sub.1 and .phi..sub.2 are added
together to get .phi..
[0706] A more detailed description of the algorithm follows.
[0707] 7.2 The Two-Stage Algorithm
[0708] In this section we first analyze how the coarse
approximation error .phi..sub.2=.phi.-.phi..sub.1 depends upon the
precision of the tables 7106 and 7110, in order to determine the
amount of hardware that must be allocated to the coarse stage. Next
we explore ways to simplify the computations in the fine stage.
[0709] 7.2.1 Simplification in the Coarse Computation Stage
[0710] The main concern in the coarse stage design is how the
lookup table values are generated to produce as precise results as
possible for a given table size. As mentioned previously, there are
two lookup tables:
[0711] 7.2.1.1 The Reciprocal Table 7106
[0712] The input to this table, 1.ltoreq.X.sub.0<2, can be
expressed as
X.sub.0=1.x.sub.1x.sub.2 . . . x.sub.m . . . x.sub.N (7.13)
[0713] where only bits x.sub.1 through x.sub.m are used to index
the table. To generate the table value, if we merely truncate
X.sub.0 as
[X.sub.0]=1.x.sub.1x.sub.2 . . . x.sub.m (7.14)
[0714] then the quantization error
.DELTA..sub.X.sub..sub.0=X.sub.0-[X.sub- .0] is bounded by
0<.DELTA..sub.X.sub..sub.0<2.sup.-m. (7.15)
[0715] Thus, the difference between the table value and
1/X.sub.0,
1/X.sub.0-1/[X.sub.0]=([X.sub.0]-X.sub.0)/([X.sub.0]X.sub.0).apprxeq.-.DEL-
TA..sub.X.sub..sub.0/X.sub.0.sup.2 (7.16)
is bounded by
-2.sup.-m<1/X.sub.0-1/[X.sub.0].ltoreq.0. (7.17)
[0716] But if we generate the table value corresponding to
[X.sub.0]=1.x.sub.1x.sub.2 . . . x.sub.m1 (7.18)
[0717] with a bit "1" appended as the LSB, then the quantization
error in (7.15) is centered around zero:
-2.sup.-m-1<.DELTA..sub.X.sub..sub.0<2.sup.-m-1 (7.19)
[0718] hence, the error in the reciprocal is also centered around
zero:
-2.sup.-m-1<1/X.sub.0-1/[X.sub.0].ltoreq.2.sup.-m-1. (7.20)
[0719] Comparing (7.20) to (7.17), the maximum absolute error is
reduced. This is the technique introduced in (Fowler, D. L., and
Smith, J. E., "An accurate high speed implementation of division by
reciprocal approximation," in Proc. 9th Symp. on Computer
Arithmetic (1989), pp. 60-67).
[0720] Since the output of the table will be multiplied by Y.sub.0,
the fewer the bits in the table value, the smaller the required
multiplier hardware. Let the table value Z.sub.0 be generated by
rounding 1/[X.sub.0] to n bits:
Z.sub.0=0.1z.sub.2z.sub.3 . . . z.sub.n. (7.21)
[0721] The quantization error
.DELTA..sub.Z.sub..sub.0=1/[X.sub.0]-Z.sub.0 is then bounded by
-2.sup.-n-1<.DELTA..sub.Z.sub..sub.0<2.sup.-n-1. (7.22)
[0722] Once we have obtained Z.sub.0 from the reciprocal table, we
can get an approximation to the quotient Y.sub.0/X.sub.0 by
computing Y.sub.0Z.sub.0. This result is then used to address the
arctangent table for .phi..
[0723] 7.2.1.2 The Arctangent Table 7110
[0724] In order to use a very small table, Y.sub.0Z.sub.0 is
rounded to k bits to the right of the radix point to become
[Y.sub.0Z.sub.0], with the rounding error bounded by
-2.sup.-k-1<.DELTA..sub.Y.sub..sub.0.sub.Z.sub..sub.0=Y.sub.0Z.sub.0-[Y-
.sub.0Z.sub.0]<2.sup.-k-1. (7.23)
[0725] Then, [Y.sub.0Z.sub.0] is used to index the arctangent
table, which returns the coarse angle
.phi..sub.1=tan.sup.-1([Y.sub.0Z.sub.0]).
[0726] Now we must determine the minimum m, n and k values such
that (7.8) is satisfied. First, let us examine X.sub.1 and Y.sub.1
which are computed using [Y.sub.0Z.sub.0] as
X.sub.1=X.sub.0+Y.sub.0[Y.sub.0Z.sub.0] (7.24)
Y.sub.1=Y.sub.0-X.sub.0[Y.sub.0Z.sub.0]. (7.25)
[0727] Dividing (7.25) by (7.24), and then dividing both the
numerator and denominator by X.sub.0, we have 119 Y 1 / X 1 = ( Y 0
/ X 0 - [ Y 0 Z 0 ] ) / ( 1 + ( Y 0 / X 0 ) [ Y 0 Z 0 ] ) Y 0 / X 0
- [ Y 0 Z 0 ] . ( 7.26 )
[0728] The inequality is true because
X.sub.0.gtoreq.Y.sub.0.gtoreq.0 and [Y.sub.0Z.sub.0].gtoreq.0.
Taking into account all the quantization errors in (7.20), (7.22)
and (7.23), we can express Y.sub.0//X.sub.0 in terms of [Y.sub.0
Z.sub.0] as 120 Y 0 ( 1 / X 0 ) Y 0 ( 1 / [ X 0 ] - X 0 / X 0 2 ) =
Y 0 ( ( Z 0 + Z 0 ) - X 0 / X 0 2 ) = Y 0 Z 0 + Y 0 Z 0 - Y 0 ( X 0
/ X 0 2 ) = [ Y 0 Z 0 ] + Y 0 Z 0 + Y 0 Z 0 - Y 0 ( X 0 / X 0 2 ) .
( 7.27 )
[0729] Substituting this result into (7.26), we have
.vertline.Y/X.sub.1.vertline..ltoreq..vertline..DELTA..sub.Y.sub..sub.0.su-
b.Z.sub..sub.0+Y.sub.0.DELTA..sub.Z.sub..sub.0-Y.sub.0(.DELTA..sub.X.sub..-
sub.0/X.sub.0.sup.2).vertline.. (7.28)
[0730] Since
Y.sub.0(.DELTA..sub.X.sub..sub.0/X.sub.0.sup.2)=(Y.sub.0/X.su-
b.0)(.DELTA..sub.X.sub..sub.0/X.sub.0), from (7.2) and (7.19),
-2.sup.-m-1<Y.sub.0(.DELTA..sub.X.sub..sub.0/X.sub.0.sup.2)<2.sup.-m-
-1. (7.29)
[0731] Also, according to (7.2) and (7.22), we have
-2.sup.-n<Y.sub.0.DELTA..sub.Z.sub..sub.0<2.sup.-n.
(7.30)
[0732] Applying (7.23), (7.29) and (7.30) to (7.28), we obtain
.vertline.Y.sub.1/X.sub.1.vertline.<2.sup.-m-1+2.sup.-n+2.sup.-k-1.
If we choose m.gtoreq.N/3+1, n.gtoreq.N/3+2 and k.gtoreq.N/3+1,
then
.vertline.Y.sub.1/X.sub.1.vertline.<0.75.times.2.sup.-N/3.
(7.31)
[0733] Therefore, since the inputs X.sub.1 and Y.sub.1 to the fine
stage satisfy (7.8), no additional tables are needed for the fine
stage. Henceforth we choose m=N/3+1, n=N/3+2 and k=N/3+1.
[0734] 7.2.2 Hardware Reduction in the Fine Computation Stage
7124
[0735] Since (7.8) is satisfied, we can obtain the fine angle
.phi..sub.2 by computing the quotient Y.sub.1/X.sub.1. From (7.24),
we have X.sub.0.ltoreq.X.sub.1.ltoreq.X.sub.0+Y.sub.0, hence
1.ltoreq.X.sub.1.ltoreq.4. In order to use the same reciprocal
table as in the coarse stage, X.sub.1 should be scaled such
that
1.ltoreq.X.sub.1<2. (7.32)
[0736] This can be satisfied by shifting X.sub.1 to the right if
X.sub.1.gtoreq.2. Of course Y.sub.1 should also be shifted
accordingly so that Y.sub.1/X.sub.1 remains unchanged.
[0737] As in the coarse stage, the reciprocal table accepts N/3+1
MSBs of X.sub.1 and returns Z.sub.1. We define the reciprocal error
.delta..sub.1=1/X.sub.1-Z.sub.1. Since the same reciprocal table is
used as in the coarse stage, .delta..sub.1 and .delta..sub.0 must
have the same bound. Since
.delta..sub.1=.delta..sub.0=1/X.sub.0-Z.sub.0=1/X.sub.0-1/[X.sub.0]+.DELTA-
..sub.Z.sub..sub.0 (7.33)
[0738] we can use (7.20) and (7.22) to obtain
-0.75.times.2.sup.-N/3-1<.delta..sub.1<0.75.times.2.sup.-N/3-1.
(7.34)
[0739] The bound on Y.sub.1 can be found using (7.31) and
(7.32):
.vertline.Y.sub.1.vertline.<0.75.times.2.sup.-N/3+1. (7.35)
[0740] Now we can obtain the final error bound in approximating
X.sub.1/Y.sub.1 by Y.sub.1/Z.sub.1, according to (7.34) and (7.35),
as
.vertline.Y.sub.1/X.sub.1-Y.sub.1X.sub.1.vertline.=.vertline.Y.sub.1.delta-
..sub.1.vertline.<(0.75).sup.2.times.2.sup.-2N/3. (7.36)
[0741] Clearly, this approximation error is too large. To reduce
the maximum error below 2.sup.-N, the bound on
.vertline..delta..sub.1.vertli- ne. should be approximately
2.sup.-2N/3, which would require the reciprocal table to accept
2N/3 bits as input. That is, the table needed for such a
high-resolution input would be significantly larger than the one
already employed by the coarse stage.
[0742] To overcome this difficulty, we can apply the Newton-Raphson
iteration method (Koren, I., Computer Arithmetic Algorithms,
Prentice Hall, Englewood Cliffs, N.J. (1993)) to reduce the initial
approximation error .delta..sub.1. We now briefly explain how this
method works. First, let us define the following function:
.function.(Z)=1/Z-X.sub.1. (7.37)
[0743] Obviously, we can obtain Z=1/X.sub.1 by solving
.function.(Z)=0. In other words, we can find 1/X.sub.1 by searching
for the Z value such that .function.(Z) intersects the Z-axis, as
shown in FIG. 70A.
[0744] Shifting the Z-axis down by X.sub.1, we obtain a new
function 121 f 1 ( Z ) = 1 Z ,
[0745] shown in FIG. 70B. At Z.sub.1=1/[X.sub.1], Z.sub.1 being the
initial guess, the slope of .function..sub.1(Z)=1/Z is 122 f 1 ' (
Z 1 ) = - 1 Z 1 2 . ( 7.38 )
[0746] The tangent, shown as the dashed line 7102, intersects the
.function..sub.1(Z)=X.sub.1 line at a new point Z.sub.2. From FIG.
70B, Z.sub.2 is much closer to the desired value 1/X.sub.1 than the
initial guess Z.sub.1. Let us now find Z.sub.2. According to FIG.
7-3, we must have 123 X 1 - 1 / Z 1 Z 2 - Z 1 = - 1 Z 1 2 . ( 7.39
)
[0747] Expressing Z.sub.2 in terms of Z.sub.1 and X.sub.1 we
have
Z.sub.2=Z.sub.1(2-X.sub.1Z.sub.1) (7.40)
[0748] Thus, we can obtain Z.sub.2, a more accurate approximation
of 1/X.sub.1 from Z.sub.1. One may wonder how accurate Z.sub.2 is
in approximating 1/X.sub.1. Let us now examine the approximation
error bound.
[0749] Substituting Z.sub.1=1/X.sub.1-.delta..sub.1 into (7.40), we
have 124 Z 2 = ( 1 / X 1 - 1 ) ( 2 - X 1 ( 1 / X 1 - 1 ) ) = 1 / X
1 - X 1 1 2 . ( 7.41 )
[0750] According to (7.32), (7.34) and (7.35), after one
Newton-Raphson iteration, the error in Y.sub.1Z.sub.2 is reduced
to
.vertline.Y.sub.1/X.sub.1-Y.sub.1Z.sub.2.vertline.=.vertline.Y.sub.1X.sub.-
1.delta..sub.1.sup.2.vertline.<(0.75).sup.3.times.2.sup.-N.
(7.42)
[0751] Thus, a rather accurate result is obtained with just one
iteration.
[0752] Finally, the fine angle can be computed by multiplying
Z.sub.2 by Y.sub.1:
.phi..sub.2.apprxeq.Y.sub.1Z.sub.2=Y.sub.1Z.sub.1(2-X.sub.1Z.sub.1).
(7.43)
[0753] Although there are three multipliers involved in (7.43), the
size of these multipliers can be reduced with just a slight
accuracy loss by truncating the data before feeding them to the
multipliers. The computational procedure of (7.43) is as
follows:
[0754] 1) The inputs to the fine stage, X.sub.1 and Y.sub.1, are
truncated to 2N/3+2 and N+3 bits to the right of their radix
points, respectively. Since the N/3-1 MSBs in Y.sub.1 are just sign
bits, as indicated by (7.35), they do not influence the complexity
of the multiplier that produces Y.sub.1Z.sub.1. The corresponding
quantization errors are bounded by
0.ltoreq..DELTA..sub.X.sub..sub.1<2.sup.-2N/3-2 (7.44)
0.ltoreq..DELTA..sub.Y.sub..sub.1<2.sup.-N-3 (7.45)
[0755] 2) Both quantized X.sub.1 and Y.sub.1 are multiplied by
Z.sub.1.
[0756] 3) To form 2-X.sub.1Z.sub.1, instead of generating the two's
complement of X.sub.1Z.sub.1, we can use the ones complement with
only an insignificant error. Since this error is much smaller, in
comparison to the truncation error in the next step, we can neglect
it.
[0757] 4) The product Y.sub.1Z.sub.1 is truncated to N+3 bits. We
would also truncate the ones complement of X.sub.1Z.sub.1. But
since the inverted LSBs of X.sub.1Z.sub.1 will be discarded, we can
truncate X.sub.1Z.sub.1 to 2N/3+2 bits and then take its ones
vcomplement. The corresponding quantization errors, as discussed
above, are:
0.ltoreq..DELTA..sub.X.sub..sub.1.sub.Z.sub..sub.1<2.sup.-2N/3-2
(7.46)
0.ltoreq..DELTA..sub.Y.sub..sub.1.sub.Z.sub..sub.1<2.sup.-N-3
(7.47)
[0758] After including all the error sources due to simplification,
we now analyze the effects of these errors on the final result
.phi..sub.2. Taking the errors into account, we can rewrite (7.43)
as:
.phi..sub.2.apprxeq.((Y.sub.1-.DELTA..sub.Y.sub..sub.1)(1/X.sub.1-.delta..-
sub.1)+.DELTA..sub.Y.sub..sub.1.sub.Z.sub..sub.1)(2-(X.sub.1-.DELTA..sub.X-
.sub..sub.1)(1/X.sub.1-.delta..sub.1)+.DELTA..sub.X.sub..sub.1.sub.Z.sub..-
sub.1) (7.48)
[0759] Expanding this product and neglecting terms whose magnitudes
are insignificant, we have
.phi..sub.2.apprxeq.Y.sub.1/X.sub.1-Y.sub.1X.sub.1.delta..sub.1.sup.2+(Y.s-
ub.1/X.sub.1.sup.2).DELTA..sub.X.sub..sub.1-(1/X.sub.1).DELTA..sub.Y.sub..-
sub.1+(Y.sub.1/X.sub.1)(.DELTA..sub.X.sub..sub.1.sub.Z.sub..sub.1)+.DELTA.-
.sub.Y.sub..sub.1.sub.Z.sub..sub.1. (7.49)
[0760] As mentioned in Section 7.1, Y.sub.1/X.sub.1 is an
approximation of tan.sup.-1(Y.sub.1/X.sub.1). Its approximation
error, defined in (7.8), is bounded by
.vertline..DELTA.
.sub.tan.vertline.=.vertline.(Y.sub.1/X.sub.1).sup.3/3.v-
ertline.<(0.75).sup.3.times.2.sup.-N/3. (7.50)
[0761] Replacing Y.sub.1/X.sub.1 by
tan.sup.-1(Y.sub.1/X.sub.1)+(Y.sub.1/X- .sub.1).sup.3/3 in (7.49),
we have
.phi..sub.2.apprxeq.tan.sup.-1(Y.sub.1/X.sub.1)+(Y.sub.1/X.sub.1).sup.3/3--
Y.sub.1X.sub.1.delta..sub.1.sup.2+(Y.sub.1/X.sub.1.sup.2).DELTA..sub.X.sub-
..sub.1-(1/X.sub.1).DELTA..sub.Y.sub..sub.1+(Y.sub.1/X.sub.1).DELTA..sub.X-
.sub..sub.1.sub.Z.sub..sub.1+.DELTA..sub.Y.sub..sub.1.sub.Z.sub..sub.1.
(7.51)
[0762] The total error, .di-elect
cons.=.phi..sub.2-tan.sup.-1(Y.sub.1/X.s- ub.1), is
.epsilon.=(Y.sub.1/X.sub.1){(Y.sub.1/X.sub.1).sup.2/3-(X.sub.1.delta..sub.-
1).sup.2+.DELTA..sub.X.sub..sub.1/X.sub.1+.DELTA..sub.X.sub..sub.1.sub.Z.s-
ub..sub.1}-(1/X.sub.1).DELTA..sub.Y.sub..sub.1+.DELTA..sub.Y.sub..sub.1.su-
b.Z.sub..sub.1. (7.52)
[0763] All terms in the subtotal
(Y.sub.1/X.sub.1).sup.2/3-(X.sub.1.delta.-
.sub.1).sup.2+.DELTA..sub.X.sub..sub.1/X.sub.1+.DELTA..sub.X.sub..sub.1.su-
b.Z.sub..sub.1 are non-negative. Thus, the lower bound of this
subtotal is the minimum value of -(X.sub.1.delta..sub.1).sup.2,
which is -0.75.sup.2.times.2.sup.-2/N3=-0.56.times.2.sup.-2N/3,
according to (7.34). Correspondingly, its upper bound is the sum of
the maximum values of the other three terms:
(0.75.sup.2/3+2.sup.-2+2.sup.-2).times.2.sup.-2-
N/3=0.68.times.2.sup.-2N/3.
[0764] Finally, we can obtain the total error bound as: 125 <
0.75 .times. 2 - N / 3 .times. 0.68 .times. 2 - 2 N / 3 + 2 - N - 3
+ 2 - N - 3 = 0.76 .times. 2 - N . ( 7.53 )
[0765] 7.3 Magnitude Calculation
[0766] Once the angle of the vector (X.sub.0,Y.sub.0) is known, its
magnitude can be obtained by multiplying X.sub.0 by 1/cos .phi.,
whose values can be pre-calculated and stored in a ROM, thereby
requiring only a single multiplication. However, if we use all the
available bits to index the ROM table, it is likely that a very
large ROM will be needed.
[0767] As we know from the preceding discussion, the coarse angle
.phi..sub.1 is an approximation of .phi.. Similarly 1/cos
.phi..sub.1 approximates 1/cos .phi.. Therefore, we can expand the
coarse-stage ROM 7110 to include also the 1/cos .phi..sub.1 values.
That is, for each input [Y.sub.0Z.sub.0], the coarse-stage ROM
would output both .phi..sub.1=tan.sup.-1([Y.sub.0Z.sub.0]) and
1/cos .phi..sub.1. Since X.sub.0 and Y.sub.0 satisfy (7.2) and
(7.3), the 1/cos .phi. value is within the interval [1,{square
root}{square root over (2)}].
[0768] For many applications, the magnitude value is used only to
adjust the scaling of some signal level, and high precision is not
necessary. For applications where a higher precision is desired, we
propose the following approach:
[0769] First, instead of using the above-mentioned table of 1/cos
.phi..sub.1 values, we pre-calculate and store in ROM the 1/cos
.phi..sub.M values, where .phi..sub.M contains only the m MSBs of
.phi.. Obviously a small table, one of comparable size to the 1/cos
.phi..sub.1 table, is needed. Then, we can look up the table
entries for the two nearest values to .phi., namely .phi..sub.M and
.phi.'.sub.M=.phi..sub.M+- 2.sup.-m. Then a better approximation of
1/cos .phi. can be obtained by interpolating between the table
values 1/cos .phi..sub.M and 1/cos .phi.'.sub.M as 126 1 / cos 1
cos M + 1 / cos M ' - 1 / cos M M ' - M .times. ( - M ) . ( 7.54
)
[0770] Let .phi..sub.L=.phi.-.phi..sub.M. Obviously, .phi..sub.L
simply contains the LSBs of .phi.. We can now rewrite (7.54) as
1/cos .phi..apprxeq.1/cos .phi..sub.M+(1/cos .phi.'.sub.M-1/cos
.phi..sub.M).times..phi..sub.L.times.2.sup.m (7.55)
[0771] which involves only a multiplication and a shift operation,
in addition to two adders.
[0772] 7.4 Converting Arbitrary Inputs
[0773] In previous sections we have restricted the input values to
lie within the bounds of (7.2) and (7.3). However, if the
coordinates of (X.sub.0,Y.sub.0) do not satisfy that condition, we
must map the given point to one whose coordinates do. Of course,
the resulting angle must be modified accordingly. To do that, we
replace X.sub.0 and Y.sub.0 by their absolute values. This maps
(X.sub.0,Y.sub.0) into the first quadrant. Next, the larger of
.vertline.X.sub.0.vertline. and .vertline.Y.sub.0.vertline. is used
as the denominator in (7.1) and the other as the numerator. This
places the corresponding angle in the interval [0, .pi./4]. We can
now use the procedure discussed previously to obtain .phi.. Once we
get .phi., we can find the angle .phi. that corresponds to the
original coordinates from .phi.. First, if originally
.vertline.X.sub.0.vertline.<.vertline.Y.sub.0.vertline. we
should map .phi. to [.pi./4, .pi./2] using .phi.'=.pi./2-.phi..
Otherwise .phi.'=.phi.. We then map this result to the original
quadrant according to Table 7.1.
[0774] Next, let us examine the affect of the above-mentioned
mapping on the magnitude calculation. Since the negation and
exchange of the original X.sub.0 and Y.sub.0 values do not change
the magnitude, whose value is (X.sub.0.sup.2
+Y.sub.0.sup.2).sup.1/2, the result obtained using the X.sub.0 and
Y.sub.0 values after the mapping needs no correction. However, if
the input values were scaled to satisfy (7.3), we then need to
scale the computed magnitude to the original scale of X.sub.0 and
Y.sub.0.
9TABLE 7.1 Converting Arbitrary Inputs Modification Original
coordinates to the angle X.sub.0 < 0, Y.sub.0 > 0 .phi. =
.pi. - .phi.' X.sub.0 < 0, Y.sub.0 < 0 .phi. = .pi. + .phi.'
X.sub.0 > 0, Y.sub.0 < 0 .phi. = 2.pi. - .phi.'
[0775] 7.5 Test Result
[0776] We have verified our error bound estimation by a bit-level
simulation of the rectangular-to-polar converter 7100. To test the
core algorithm described in Section 7.2, we generated the pair of
inputs X.sub.0 and Y.sub.0 randomly within the range described by
(7.2) and (7.3). This test was run repeatedly over many different
values of X.sub.0 and Y.sub.0, and the maximum error value was
recorded. Choosing N=9 for this simulation, the error bound
estimate according to (7.53) is 0.0015. Our test results yielded
the error bounds [-0.00014, 0.00051], well within the calculated
bound.
[0777] 7.6 Conclusion
[0778] An efficient rectangular-to-polar converter is described.
The angle computation of a complex number is partitioned into
coarse and fine computational stages. Very small arctangent and
reciprocal tables are used to obtain a coarse angle. These tables
should provide just enough precision such that the remaining fine
angle is small enough to approximately equal its tangent value.
Therefore the fine angle can be obtained without a look-up table.
The computations are consolidated into a few small multipliers,
given a precision requirement. While a low-precision magnitude can
be obtained quite simply, a high-precision result can be achieved
by combining the angle computation with the angle rotation
processor 3900 of Section 5.
[0779] The applications of this converter include the
implementation of the converter 6106 in the symbol synchronizer
6100 and the synchronizer 6400. However, the converter is not
limited to symbol synchronization. It also provides efficient
implementation of computational tasks for many communication
systems, such as constant-amplitude FSK and PSK modems (Chen, A,
and Yang, S., "Reduced complexity CORDIC demodulator implementation
for D-AMPS and digital IF-sampled receiver," in Proc. Globecom
1998, vol. 3 (1998), pp. 1491-1496; Boutin, N., IEEE Trans.
Consumer Electron. 38:5-9 (1992)), DMT modems (Arivoli, T., et al.,
"A single chip DMT modem for high-speed WLANs," in Proc. 1998
Custom Integrated Circuits Conf. (May 1998), pp. 9-11), as well as
carrier synchronization (Andronico, M., et al., "A new algorithm
for fast synchronization in a burst mode PSK demodulator," in Proc.
1995 IEEE Int. Conf. Comm., vol. 3 (June 1995), pp. 1641-1646;
Fitz, M. P., and Lindsey, W. C., IEEE Trans. Comm. 40:1644-1653
(1992) where the computation of phase and magnitude from the
rectangular coordinates is essential.
8. Exemplary Computer System
[0780] Embodiments of invention may be implemented using hardware,
software or a combination thereof and may be implemented in a
computer system or other processing system. In fact, in one
embodiment, the invention is directed toward a software and/or
hardware embodiment in a computer system. An example computer
system 7702 is shown in FIG. 77. The computer system 7702 includes
one or more processors, such as processor 7704. The processor 7704
is connected to a communication bus 7706. The invention can be
implemented in various software embodiments that can operate in
this example computer system. After reading this description, it
will become apparent to a person skilled in the relevant art how to
implement the invention using other computer systems and/or
computer architectures.
[0781] Computer system 7702 also includes a main memory 7708,
preferably a random access memory (RAM), and can also include a
secondary memory or secondary storage 7710. The secondary memory
7710 can include, for example, a hard disk drive 7712 and a
removable storage drive 7714, representing a floppy disk drive, a
magnetic tape drive, an optical disk drive, etc. The removable
storage drive 7714 reads from and/or writes to a removable storage
unit 7716 in a well known manner. Removable storage unit 7716,
represents a floppy disk, magnetic tape, optical disk, etc. which
is read by and written to by removable storage drive 7714. As will
be appreciated, the removable storage unit 7716 includes a computer
usable storage medium having stored therein computer software
and/or data.
[0782] In alternative embodiments, secondary memory 7710 may
include other similar means for allowing computer software and data
to be loaded into computer system 7702. Such means can include, for
example, a removable storage unit 7720 and an storage interface
7718. Examples of such can include a program cartridge and
cartridge interface (such as that found in video game devices), a
removable memory chip (such as an EPROM, or PROM) and associated
socket, and other removable storage units 7720 and interfaces 7718
which allow software and data to be transferred from the removable
storage unit 7720 to the computer system 7702.
[0783] Computer system 7702 can also include a communications
interface 7722. Communications interface 7722 allows software and
data to be transferred between computer system 7702 and external
devices 7726. Examples of communications interface 7722 can include
a modem, a network interface (such as an Ethernet card), a
communications port, a PCMCIA slot and card, etc. Software and data
transferred via communications interface 7722 are in the form of
signals, which can be electronic, electromagnetic, optical or other
signals capable of being received by the communications interface
7722. These signals are provided to the communications interface
7722 via a channel 7724. This channel 7724 can be implemented using
wire or cable, fiber optics, a phone line, a cellular phone link,
an RF link and other communications channels.
[0784] Computer system 7702 may also include well known peripherals
7703 including a display monitor, a keyboard, a printers and
facsimile, and a pointing device such a computer mouse, track ball,
etc.
[0785] In this document, the terms "computer program medium" and
"computer usable medium" are used to generally refer to media such
as the removable storage devices 7716 and 7718, a hard disk
installed in hard disk drive 7712, semiconductor memory devices
including RAM and ROM, and associated signals. These computer
program products are means for providing software (including
computer programs that embody the invention) and/or data to
computer system 7702.
[0786] Computer programs (also called computer control logic or
computer program logic) are generally stored in main memory 7708
and/or secondary memory 7710 and executed therefrom. Computer
programs can also be received via communications interface 7722.
Such computer programs, when executed, enable the computer system
7702 to perform the features of the present invention as discussed
herein. In particular, the computer programs, when executed, enable
the processor 7704 to perform the features of the present
invention. Accordingly, such computer programs represent
controllers of the computer system 7702.
[0787] In an embodiment where the invention is implement using
software, the software may be stored in a computer program product
and loaded into computer system 7702 using removable storage drive
7714, hard drive 7712 or communications interface 7722. The control
logic (software), when executed by the processor 7704, causes the
processor 7704 to perform the functions of the invention as
described herein.
[0788] In another embodiment, the invention is implemented
primarily in hardware using, for example, hardware components such
as application specific integrated circuits (ASICs), stand alone
processors, and/or digital signal processors (DSPs). Implementation
of the hardware state machine so as to perform the functions
described herein will be apparent to persons skilled in the
relevant art(s). In embodiments, the invention can exist as
software operating on these hardware platforms.
[0789] In yet another embodiment, the invention is implemented
using a combination of both hardware and software.
9. Appendices
[0790] The following Appendices are included.
[0791] 9.1 Appendix A: Proof of the Zero ISI Condition.
[0792] In Section 3.4 we have examined the condition for a
real-valued function .function.(t) to have zero crossings at
integer multiples of the sampling period, i.e., for satisfying
(3.16). We have stated that .function.(t) satisfies (3.16) if and
only if {circumflex over (F)}(k), defined as samples of F(.OMEGA.)
in (3.8) (F is the frequency response of .function.), satisfy
(3.18). Here, -.infin.<k<.infin. is an integer. We now
provide the proof.
[0793] Proof. First let us define a periodic, of period N,
extension of .function.(t) as 127 f c ( t ) = n = - .infin. .infin.
f ( t - Nn ) . ( A .1 )
[0794] Its Fourier transform consists of a sequence of impulses 128
F c ( ) = k = - .infin. .infin. F ^ ( k ) ( - 2 N k ) . ( A .2
)
[0795] Next, consider an impulse chain. 129 c ( t ) = n = - .infin.
.infin. ( t - n ) . ( A .3 )
[0796] whose Fourier transform is 130 C ( ) = k = - .infin. .infin.
( - 2 k ) . ( A .4 )
[0797] The convolution F.sub.c{circle over (x)}C can be expressed
as 131 F c ( ) C ( ) = F c ( ) m = - .infin. .infin. ( - 2 m ) = m
= - .infin. .infin. F c ( - 2 m ) ( A .5 )
[0798] Substituting (A.2) into (A.5) yields: 132 F c ( ) C ( ) = m
= - .infin. .infin. ( k = - .infin. .infin. F ^ ( k ) ( - 2 m - 2 N
k ) ) = k = - .infin. .infin. m = - .infin. .infin. F ^ ( k ) ( - 2
N ( k + Nm ) ) = k = - .infin. .infin. ( m = - .infin. .infin. F ^
( k - Nm ) ) ( - 2 N k ) . ( A .6 )
[0799] Therefore, we have the following relationships 133 ( 3.16 )
f c ( t ) = { 1 t = Nm , m an integer 0 all other integers . f c (
t ) c ( t ) = m = - .infin. .infin. ( t - Nm ) F c ( ) C ( ) = k =
- .infin. .infin. ( - 2 N k ) ( A .6 ) k = - .infin. .infin. ( m =
- .infin. .infin. F ^ ( k - Nm ) ) ( - 2 N k ) = k = - .infin.
.infin. ( - 2 N k ) m = - .infin. .infin. F ^ ( k - Nm ) = 1 ( 3.18
)
[0800] This concludes the proof.
[0801] 9.2 Appendix B: Impulse Response of the Simplified
Interpolators
[0802] In Section 2, after introducing a preliminary interpolation
method, we have shown that we can trade one angle rotator for a
multiplier by conceptually modifying the input samples, then by
"correcting" the interpolated value obtained from the "modified"
samples. A simpler implementation structure as well as better
performance in interpolating most practical signals have been
facilitated. We now derive the impulse response of this
interpolation filter. As discussed in Section 2, the interpolated
sample is computed as 134 y ( ) = 1 N k = - N / 2 + 1 N / 2 - 1 c k
W N - k - K ( B .1 )
[0803] where K is defined in (2.30), and 135 c k = m = - N / 2 + 1
N / 2 y ~ ( m ) W m km = m = - N / 2 + 1 N / 2 ( y ( m ) + mK ) W N
km ( B .2 )
[0804] where k=0, . . . , N/2-1.
[0805] Substituting (2.30) into (3.2), we have 136 c k = m = - N /
2 + 1 N / 2 y ( m ) W N km - m = - N / 2 + 1 N / 2 ( m 2 N n = - N
/ 2 + 1 N / 2 ( - 1 ) n y ( n ) ) W N km = m = - N / 2 + 1 N / 2 y
( m ) W N km - n = - N / 2 + 1 N / 2 ( m = - N / 2 + 1 N / 2 m 2 N
W N km ) ( - 1 ) n y ( n ) = m = - N / 2 + 1 N / 2 y ( m ) W N km -
m = - N / 2 + 1 N / 2 ( n = - N / 2 + 1 N / 2 n 2 N W N km ) ( - 1
) m y ( m ) ( B .3 )
[0806] Replacing K and c.sub.k in (B.1) by (2.30) and (B.3),
respectively, we have 137 y ( ) = 1 N k = - N / 2 + 1 N / 2 - 1 m =
- N / 2 + 1 N / 2 ( W N km - n = - N / 2 + 1 N / 2 n 2 N W N kn ( -
1 ) m ) y ( m ) W N - k + 2 N m = - N / 2 + 1 N / 2 ( - 1 ) m y ( m
) = m = - N / 2 + 1 N / 2 y ( m ) ( 1 N k = - N / 2 + 1 N / 2 - 1 W
N k ( m - ) - 2 N ( - 1 ) m ( n = - N / 2 + 1 N / 2 n N k = - N / 2
+ 1 N / 2 - 1 W N k ( n - ) - ) ) = m = - N / 2 + 1 N / 2 y ( m ) f
( - m ) . ( B .4 )
[0807] where .function.(t), the impulse response of the simplified
interpolation filter discussed in Section 2.6.1, is now defined as
138 f ( t ) = 1 N k = - N / 2 + 1 N / 2 - 1 W N - kt - 2 N ( - 1 )
m ( n = - N / 2 + 1 N / 2 - 1 n N k = - N / 2 + 1 N / 2 - 1 W N k (
n - t - m ) - ( t + m ) ( B .5 )
[0808] for -m.ltoreq.t.ltoreq.1-m, m=-N/2+1, . . . , N/2.
Otherwise, .function.(t)=0.
[0809] The frequency response, of course, can be obtained by taking
the Fourier transform of .function.(t). To modify the frequency
response of .function.(t), we can multiply the c.sub.k coefficient
in (B.3) by a value denoted by {circumflex over (F)}.mu.(k). In
designing an optimal interpolation filter as discussed in Section
4, we search for the {circumflex over (F)}.mu.(k) value that
minimizes (4.4), i.e., F.sub..mu.(.omega.) most accurately
approximates the desired response (4.3).
[0810] 9.3 Appendix C: Fourier Transform of g(nT.sub.s-.mu.)
[0811] Since, in Section 6, the g(nT.sub.s) are samples of the
continuous-time pulse g(t), assuming, without loss of generality,
that T.sub.s=1, it is well-known (Freeman, H., Discrete-Time
Systems, Wiley, New York, N.Y. (1965)) that the Fourier transforms
are related as 139 G ( j ) = k = - .infin. .infin. G ^ ( + 2 K ) (
C .1 )
[0812] where G(e.sup.j.omega.) and (.omega.) are the Fourier
transforms of g(n) and g(t), respectively. Since g(t) is
bandlimited, i.e., .vertline.(.omega.).vertline.=0 for
.vertline..omega..vertline.>.pi., we have
G(e.sup.j.omega.)=(.omega.), -.pi..ltoreq..omega..ltoreq..pi..
(C.2)
[0813] Using the Fourier transform's time-shifting property, the
Fourier transform of g(t-.mu.) is
e.sup.j.omega..mu.(.omega.). (C.3)
[0814] Since the g(n-.mu.) are samples of g(t-.mu.), for the same
reason as the above, their Fourier transforms are the same in the
interval -.pi..ltoreq..omega..ltoreq..pi., as in (C.2). Thus,
according to (C.2) and (C.3) we have
FT(g(n-.mu.))=e.sup.j.omega..mu.G(e.sup.j.omega.),
-.pi..ltoreq..omega..lt- oreq..pi.. (C.4)
[0815] 9.4 Appendix D: Interpolation on Non-Center Intervals
[0816] When we first discussed the interpolation problem in Section
2, we focused on interpolating between the two samples in the
middle of the set of samples used to generate a synchronized
sample. What is the impact on interpolation performance when we
interpolate in an interval not at the center of the samples being
used? FIG. 74 shows such an example for N=4, where the
interpolation is performed between y(0) and y(1) using y(-2),
y(-1), y(0) and y(1) (as opposed to using y(-1), y(0), y(1) and
y(2), as seen in FIG. 2-1).
[0817] Using the procedure described in Section 2, given N samples
y(n), n=-N+2, . . . , 1, we first compute the Fourier coefficients
as 140 c k = n = - N + 2 1 y ( i ) W N kn , k = - N 2 + 1 , , N 2 .
( D .1 )
[0818] Comparing (D.1) to (2.9), their only difference is the range
of the summation. As in Section 2, for a given offset
0.ltoreq..mu.<1, the synchronized sample y(.mu.) can be computed
as: 141 y ( ) = 1 N Re ( c 0 + 2 k = 1 N / 2 - 1 c k W N - k + c N
/ 2 j ) . ( D .2 )
[0819] We can express y(.mu.) in terms of y(n) by substituting
(D.2) into (D.1), as 142 y ( ) = 1 N n = - N + 2 1 y ( n ) ( 1 + 2
k = 1 N / 2 - 1 cos 2 k N ( - n ) + cos ( - n ) ) = 1 N n = - N + 2
1 y ( n ) f ( - n ) where ( D .3 ) f ( t ) = ( 1 + 2 k = 1 N / 2 -
1 cos 2 k N t + cos t - 1 t - N + 2 0 otherwise ( D .4 )
[0820] is the impulse response of the corresponding interpolation
filter. For N=4,.function.(t) is plotted in FIG. 75A. Taking the
Fourier transform of .function.(t), we obtain the corresponding
frequency response, which is shown in FIG. 76A.
[0821] Comparing the FIG. 76A frequency response to FIG. 7A, both
for N=4, we can see that the interpolation performance degraded
significantly, as shown by the ripples in the passband and large
sidelobes in the stopband in FIG. 76A, when the interpolation is
not done in the center interval.
[0822] However, using the optimization method discussed in Section
4, we can "reshape" the impulse response .function.(t) such that
the corresponding frequency response is a better approximation to
the ideal interpolation frequency response.
[0823] The impulse response of an optimized interpolation filter
for a non-center interval is illustrated in FIG. 75B. The
corresponding frequency response is shown in FIG. 76B, which is
clearly better than FIG. 76A, since it has less ripple in the
passband and more attenuation in the stopband.
[0824] Using samples y(-N+2), . . . , y(-1), and y(0), to
interpolate obviously reduces the latency in generating
synchronized samples, as compared to using samples y(-N/2+1), . . .
, y(0), . . . , y(N/2), since the interpolator does not have to
wait until samples y(1), . . . , y(N/2), become available. In
applications where low latency takes a higher priority than
interpolation accuracy, this approach will be useful.
[0825] 9.5 Appendix E
[0826] The following documents are incorporated by reference in
their entireties:
[0827] 1. Buchanan, K., et al., IEEE Pers. Comm. 4:8-13 (1997);
[0828] 2. Reimers, U., IEEE Comm. Magazine 36:104-110 (1998);
[0829] 3. Cho, K., "A frequency-agile single-chip QAM modulator
with beamforming diversity," Ph.D. dissertation, University of
California, Los Angeles (1999);
[0830] 4. Oerder M., and Meyr, H., IEEE Trans. Comm. 36:605-612
(1988);
[0831] 5. Pollet T., and Peeters, M., IEEE Comm. Magazine 37:80-86
(1999);
[0832] 6. Gardner, F. M., IEEE Trans. Comm. 41:502-508 (1993);
[0833] 7. Erup, L., et al., IEEE Trans. Comm. 41:998-1008
(1993);
[0834] 8. Farrow, C., "A continuously variable digital delay
element," in Proc. IEEE Int. Symp. Circuits Syst. (June 1988), pp.
2641-2645;
[0835] 9. Proakis, J. G., Digital Communications, McGraw-Hill, New
York, N.Y. (1993);
[0836] 10. Proakis, J. G., Digital Signal Processing, Macmillan,
New York, N.Y. (1992);
[0837] 11. Vesma, J., and Saramki, T., "Interpolation filters with
arbitrary frequency response for all-digital receivers," in Proc.
1996 IEEE Int. Symp. Circuits Syst. (May 1996), pp. 568-571;
[0838] 12. Ahn, Y., et al., "VLSI design of a CORDIC-based
derotator," in Proc. 1998 IEEE Int. Symp. Circuits Syst., Vol. II
(May 1998), pp. 449-452;
[0839] 13. Wang, S., et al., IEEE Trans. Comp. 46:1202-1207
(1997);
[0840] 14. Madisetti, A., et al., IEEE J. Solid-State Circuits
34:1034-1043 (1999);
[0841] 15. Tan, L. and Samueli, H., IEEE J. Solid-State Circuits
30:193-200 (1995;
[0842] 16. Madisetti, A., "VLSI architectures and IC
implementations for bandwidth efficient communications," Ph.D.
dissertation, University of California, Los Angeles (1996);
[0843] 17. Vankka, J., IEEE Trans. Ultrasonics, Ferroelectronics
and Freq. Control 44:526-534 (1997);
[0844] 18. Daneshrad, B., "System design of a 1.6 Mbps all-digital
QAM transceiver for digital subscriber line applications," Ph.D.
dissertation, University of California, Los Angeles (1993);
[0845] 19. Tan, L., "High performance architectures and circuits
for QAM transceivers," Ph.D. dissertation, University of
California, Los Angeles (1995);
[0846] 20. Gardner, S., Electron. Eng. 71:85-92 (1999);
[0847] 21. Gardner, S., Electron. Eng. 71:75-83 (1999);
[0848] 22. Sabel, L., and Cowley, W., "A recursive algorithm for
the estimation of symbol timing in PSK burst modems," in Proc.
Globecom 1992, vol. 1 (1992), pp. 360-364;
[0849] 23. Chen, A., et al., "Modified CORDIC demodulator
implementation for digital IF-sampled receiver," in Proc. Globecom
1995, vol. 2 (1995), pp. 1450-1454;
[0850] 24. Boutin, N., IEEE Trans. Consumer Electron. 38:5-9
(1992);
[0851] 25. Meyr, H., et al., Digital Communication Receivers:
Synchronization, Channel Estimation and Signal Processing, Wiley,
New York, N.Y. (1998);
[0852] 26. Chen, A., and Yang, S., "Reduced complexity CORDIC
demodulator implementation for D-AMPS and digital IF-sampled
receiver," in Proc. Globecom 1998, vol. 3 (1998), pp.
1491-1496;
[0853] 27. Andronico, M., et al., "A new algorithm for fast
synchronization in a burst mode PSK demodulator," in Proc. 1995
IEEE Int. Conf Comm., vol. 3 (June 1995), pp. 1641-1646;
[0854] 28. Critchlow, D., "The design and simulation of a
modulatable direct digital synthesizer with non-iterative
coordinate transformation and noise shaping filter," M.S. thesis,
University of California, San Diego (1989);
[0855] 29. Arivoli, T., et al, "A single chip DMT modem for
high-speed WLANs," in Proc. 1998 Custom Integrated Circuits Conf.
(May 1998), pp. 9-11;
[0856] 30. Fitz, M. P., and Lindsey, W. C., IEEE Trans. Comm.
40:1644-1653 (1992);
[0857] 31. Fowler, D. L., and Smith, J. E., "An accurate high speed
implementation of division by reciprocal approximation," in Proc.
9th Symp. on Computer Arithmetic (1989), pp. 60-67;
[0858] 32. Koren, I., Computer Arithmetic Algorithms, Prentice
Hall, Englewood Cliffs, N.J. (1993);
[0859] 33. Fu, D., and Willson, Jr., A. N., "Interpolation in
timing recovery using a trigonometric polynomial and its
implementation," in Globecom 1998 Comm. Theory Mini Conf. Record
(November 1998), pp. 173-178;
[0860] 34. Lang, T., and Antelo, E., IEEE Trans. Comp. 47:736-749
(1998);
[0861] 35. Moeneclaey, M., IEEE Trans. Comm. 31:1029-1032
(1983);
[0862] 36. Vuori, J., IEEE Trans. Consum. Electron. 45:1098-1103
(1999); and
[0863] 37. Freeman, H., Discrete-Time Systems, Wiley, New York,
N.Y. (1965).
10. Conclusion
[0864] Example implementations of the methods, systems and
components of the invention have been described herein. As noted
elsewhere, these example implementations have been described for
illustrative purposes only, and are not limiting. Other
implementation embodiments are possible and covered by the
invention, such as but not limited to software and
software/hardware implementations of the systems and components of
the invention. Such implementation embodiments will be apparent to
persons skilled in the relevant art(s) based on the teachings
contained herein.
[0865] While various application embodiments of the present
invention have been described above, it should be understood that
they have been presented by way of example only, and not
limitation. Thus, the breadth and scope of the present invention
should not be limited by any of the above-described exemplary
embodiments.
* * * * *