U.S. patent application number 13/374524 was filed with the patent office on 2012-07-19 for spectrally constrained impulse shortening filter for a discrete multi-tone receiver.
This patent application is currently assigned to Tellabs Operations, Inc.. Invention is credited to Gopal Harikumar, Daniel Marchok.
Application Number | 20120183034 13/374524 |
Document ID | / |
Family ID | 21991290 |
Filed Date | 2012-07-19 |
United States Patent
Application |
20120183034 |
Kind Code |
A1 |
Harikumar; Gopal ; et
al. |
July 19, 2012 |
Spectrally constrained impulse shortening filter for a discrete
multi-tone receiver
Abstract
A channel in a multiple carrier communication system is
equalized by computing a target spectral response, shortening the
impulse response of the channel so that a significant part of an
energy of the impulse response is confined to a region that is
shorter than a target length and filtering the signal based on the
target spectral response.
Inventors: |
Harikumar; Gopal; (South
Bend, IN) ; Marchok; Daniel; (Buchanan, MI) |
Assignee: |
Tellabs Operations, Inc.
Naperville
IL
|
Family ID: |
21991290 |
Appl. No.: |
13/374524 |
Filed: |
December 30, 2011 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
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12238140 |
Sep 25, 2008 |
8102928 |
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13374524 |
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11759789 |
Jun 7, 2007 |
7430242 |
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12238140 |
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10608329 |
Jun 27, 2003 |
7254178 |
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11759789 |
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09054468 |
Apr 3, 1998 |
6631175 |
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10608329 |
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Current U.S.
Class: |
375/224 |
Current CPC
Class: |
H04L 25/03012 20130101;
H04L 2025/03414 20130101 |
Class at
Publication: |
375/224 |
International
Class: |
H04B 17/00 20060101
H04B017/00 |
Claims
1. A method for equalizing a channel in a multiple carrier
communication system, the channel being configured to receive a
signal and including a spectrally constrained impulse shortening
filter, the method comprising: measuring received noise power
spectral density; computing a target spectral response based on the
measured noise power; selecting a frequency response of the
spectrally constrained impulse shortening filter based on the
target spectral response; and filtering the communication signal
with the spectrally constrained impulse shortening filter.
Description
RELATED APPLICATION(S)
[0001] This application is a continuation of U.S. application Ser.
No. 12/238,140, filed Sep. 25, 2008, which is a continuation U.S.
application Ser. No. 11/759,789, filed Jun. 7, 2007, which is a
continuation of Ser. No. 10/608,329, filed Jun. 27, 2003, now U.S.
Pat. No. 7,254,178, which is a continuation U.S. application Ser.
No. 09/054,468, filed Apr. 3, 1998, now U.S. Pat. No.
6,631,175.
[0002] The entire teachings of the above application(s) are
incorporated herein by reference.
BACKGROUND
[0003] The invention relates to time-domain equalization in a
discrete multi-tone (DMT) receiver.
[0004] Conventional single carrier modulation techniques translate
data bits for transmission through a communication channel by
varying the amplitude and/or phase of a single sinusoidal carrier.
By contrast, DMT, which is also referred to as Orthogonal Frequency
Division Multiplexing (OFDM) or Multicarrier Modulation (MCM),
employs a large number Of sinusoidal subcarriers, e.g., 128 or 256
subcarriers. The available bandwidth of the communication channel
is divided into subchannels and each subchannel communicates a part
of the data. A DMT system may employ quadrature amplitude
modulation (QAM) for each of the subcarriers.
[0005] OFDM-based systems transmit blocks of information bits. The
time required to transmit one such block is called the symbol
period. The time domain waveform that corresponds to one such block
of bits is called a symbol.
[0006] Intersymbol interference (ISI) arises from the
characteristics of practical communication channels and limits the
rate at which information can be transmitted through them.
Specifically, communication channels typically have an Effective
Discrete-Time Impulse Response (EDIR) that is greater than one
sample time in length, which causes ISI. ISI is a well-known
phenomenon in single-carrier communication systems and there are
many techniques for reducing it. The process of such ISI reduction
is called equalization. ISI is discussed, for example, in Proakis,
Digital Communications, McGraw Hill, 2nd Edition, 1989.
[0007] Equalization in OFDM-based systems is achieved by a two
stage process. First, at the transmitter, an end-portion of each
symbol is affixed to the beginning of the symbol to form what is
called a Cyclic Prefix (CP). A cyclic prefix that is greater than
the EDIR of the channel prevents one symbol from interfering with
another. Furthermore, it also facilitates a simple method of
neutralizing the time-domain spread of each symbol forced by the
channel. This is achieved by the use of a simple frequency domain
process in the receiver which requires one multiplication operation
for each used subcarrier of the OFDM system.
[0008] The use of a Cyclic Prefix to reduce ISI is discussed, for
example, in: Cimini, "Analysis and Simulation of a Digital Mobile
Channel using Orthogonal Frequency Division Multiplexing," IEEE
Transactions on communications, pp 665-675 July 1985; Chow, "A
Discrete Multi-Tone Transceiver System for HDSL applications," IEEE
Journal on Selected Areas of Communications, 9(6):895-908, August
1991; "DMT Group VDSL PMD Draft Standard Proposal," Technical
Report, T1E1.4/96-329R2, ANSI 1997.
[0009] Another problem arising in conventional DMT systems is noise
bleeding, which is the phenomenon of noise in one frequency band
interfering with a signal whose subcarrier is in another frequency
band. Noise bleeding is caused, in general, by the Discrete Fourier
Transform (DFT) operation at the receiver. Noise bleeding is
discussed in, for example, Worthen et. al., "Simulation of VDSL
Test Loops," Technical Report T1E1.4/97-288, ANSI 1997.
[0010] In a perfectly synchronized DMT system, the signal in one
frequency band does not interfere with a signal whose subcarrier is
in another frequency band. However, noise from one band may
interfere with other less noisy bands and render them unusable.
Techniques for dealing with noise-bleeding include wavelet-based
solutions. However, wavelet-based solutions are, in general,
computationally intensive.
SUMMARY
[0011] The invention provides a spectrally constrained impulse
shortening filter (SCISF) for use, for example, in DMT systems. The
SCISF serves two primary functions.
[0012] First, the SCISF reduces intersymbol interference (ISI) by
reducing the length of the effective discrete-time impulse response
(EDIR) of the communication channel. Conventional impulse
shortening filters may have deep nulls in their frequency response.
By contrast, the SCISF has a filter characteristic that is
essentially free from undesired nulls that may attenuate or
completely eliminate certain subcarriers.
[0013] Second, the SCISF reduces noise bleeding between subchannels
by attenuating noisy channels in a manner that does not reduce the
signal to noise ratio (SNR) in these channels, but reduces the
noise power that may appear in the sidelobes of adjacent
subchannels. The SCISF accomplishes these functions by applying a
frequency constraint to the signal based on a target spectral
response.
[0014] In one general aspect, the invention features equalizing a
channel in a multiple carrier communication system. The system
includes a spectrally constrained impulse shortening filter.
Received noise power spectral density is measured. A target
spectral response is computed based on the measured noise power. A
frequency response for the spectrally constrained impulse
shortening filter is selected based on the target spectral
response. The communication signal is filtered with the spectrally
constrained impulse shortening filter.
[0015] Embodiments may include one or more of the following
features. The noise power spectral density may be measured at the
output of the discrete Fourier transform. The spectrally
constrained impulse shortening filter may be a time domain digital
filter.
[0016] In another aspect, the invention features equalizing a
channel in a multiple carrier communication system. The channel has
an impulse response and is configured to receive a signal having a
cyclic prefix. A target spectral response is computed. The impulse
response of the channel is shortened so that a significant part of
the energy of the impulse response is confined to a region that is
shorter than a target length, and the signal is filtered based on
the target spectral response.
[0017] The target length may be a length of the cyclic prefix. The
target spectral response may be computed from measured noise power
density, which may be measured at the output of a discrete Fourier
transform. For example, the target spectral response may be the
inverse of the measured noise power spectral density. The filtering
step may be performed with a filter having a frequency response
selected to match the target spectral response. The shortening of
the impulse response and/or the filtering may be performed by a
time domain digital filter.
[0018] In another general aspect, the invention features selecting
an impulse response for a spectrally constrained impulse shortening
filter in a multiple carrier communication system. Received noise
power spectral density is measured. A cost function is computed
using the noise power. The cost function is dependent on the
impulse response. The dimensionality of a space over which the cost
function is defined is reduced and the cost function is minimized.
The noise power spectral density may be measured at the output of a
discrete Fourier transform. The cost function may be used to
compute coefficients for the spectrally constrained impulse
shortening filter.
[0019] In another general aspect, the invention features a
spectrally constrained impulse shortening filter for a multiple
carrier communication system. The system includes a channel that
has an impulse response. A digital filter structure receives the
signal and apply a frequency characteristic to the signal. The
frequency characteristic is determined by filter coefficients.
Filter coefficients are selected to shorten the impulse response of
the channel so that a significant part of the energy of the impulse
response is confined to a region that is shorter than a target
length and to apply a frequency characteristic to the signal based
on a target spectral response. The selected filter coefficients are
input to the taps of the filter.
[0020] In another general aspect, the invention features a receiver
for receiving a multiple carrier signal from a communication
channel having an impulse response. An analog-to-digital converter
receives the signal from the communication channel. A spectrally
constrained impulse shortening filter receives the signal from the
analog-to-digital converter and shortens the impulse response of
the channel so that a significant part of an energy of the impulse
response is confined to a region that is shorter Than a target
length. The filter also applies a frequency characteristic to the
signal based on a target spectral response. A discrete Fourier
transform receives the output of the spectrally constrained impulse
shortening filter and a decoder receives outputs of the discrete
Fourier transform.
[0021] In another aspect, the invention features a modem in which
an encoder receives digital data and outputs a constellation of
complex values. An inverse discrete Fourier transform receives the
constellation from the encoder. A digital-to-analog converter is
connected to the inverse discrete Fourier transform and outputs a
signal to a communication channel. An analog-to-digital converter
receives the signal from the communication channel. A spectrally
constrained impulse shortening filter shortens an impulse response
of the channel so that a significant part of an energy of the
impulse response is confined to a region that is shorter than a
target length. The signal is filtered based on a target spectral
response. A discrete Fourier transform connected to the filter and
a decoder is connected to the discrete Fourier transform and
outputs digital data.
[0022] In another aspect, the invention features software for
causing a processor in a communication system to perform the
following operations: measure received noise power spectral
density; and compute a target spectral response based on the
measured noise power. The software may also include instructions
for causing a processor to compute filter coefficients based on the
target spectral response.
[0023] In another aspect, the invention features software for
causing a processor in a communication system to measure received
noise power spectral density. A cost function is computed using the
noise power. The cost function is dependent on an impulse response
of a spectrally constrained impulse shortening filter. The
dimensionality of a space over which the cost function is defined
is reduced, and the cost function is minimized.
[0024] The techniques described here are not limited to any
particular hardware or software configuration. They may find
applicability in any computing or processing environment that may
be used for a communication system. The techniques may be
implemented in hardware or software, or a combination of the two.
Preferably, the techniques are implemented in computer programs
executing on a digital signal processor that includes a processor
and a storage medium readable by the processor (including volatile
and non-volatile memory).
[0025] Other features and advantages will be apparent from the
following description, including the drawings and the claims.
BRIEF DESCRIPTION OF THE DRAWINGS
[0026] FIG. 1 is a block diagram of a discrete multi-tone
communication system.
[0027] FIG. 2 is a plot of effective discrete-time impulse response
(EDIR) of a communication channel including transmit and receive
filters.
[0028] FIG. 3 is a plot of the shortened EDIR due to a spectrally
constrained impulse shortening filter (SCISF).
[0029] FIG. 4 is a block diagram of a SCISF.
[0030] FIG. 5 is a plot of transmit signal power, signal power at
SCISF input and noise power at SCISF input.
[0031] FIG. 6 is a plot of signal and noise power at the output of
the SCISF.
[0032] FIG. 7 shows the filter response of a discrete Fourier
transform for one frequency band.
[0033] FIG. 8 is a plot of the target spectral response of the
SCISF, G.sub.d(.omega.), versus the actual frequency response,
G(.omega.).
[0034] FIG. 9 is a plot of signal-to-noise ratio at the output of
the SCISF, the output of the DFT with the SCISF and the output of
the DFT without the SCISF.
DESCRIPTION
[0035] As shown in FIG. 1, a discrete multi-tone (DMT)
communication system 10 has a transmitter 12 and a receiver 14. The
transmitter 12 accepts an input data bit stream which passes
through a constellation encoder 20. The encoder 20 divides the
serial input bit stream into blocks of data. These blocks of data
are further subdivided into smaller blocks corresponding to
subchannels. Each of these smaller blocks are used to compute a
complex value representing a constellation point. Each
Constellation point corresponds to a subsymbol. The subsymbols are
then output by the encoder. Taken together, the subsymbols
constitute a symbol.
[0036] The subsymbols are supplied to an inverse discrete Fourier
transform (IDFT) 30, which may be implemented, for example, in a
digital signal processor. The IDFT 30 outputs N time samples of a
symbol. The time samples are processed by a parallel to serial
converter 40 to form a single stream of time samples.
[0037] Following the parallel to serial converter 40, a prefixer 50
adds a cyclic prefix to the beginning of each symbol to reduce
intersymbol interference (ISI). Alternatively, the cyclic prefix
may be added in the parallel to serial converter. After the cyclic
prefix is added, the resulting signal passes through a
digital-to-analog (D/A) converter 60 for transmission to the
receiver 14 through a communication channel 70. An analog transmit
filter 65 may be included following the D/A converter to band limit
the transmitted signal.
[0038] At the receiver 14, the signal passes through an
analog-to-digital (A/D) converter 75 and then through an impulse
shortening filter 90. An analog receive filter 85 may be included
prior to the A/D converter in order to band limit the received
signal. A prefix stripper 100 strips the cyclic prefixes from the
resulting symbols and a serial to parallel converter 110 divides
the stream of time samples into parallel signal paths that form the
inputs to a discrete Fourier transform (DFT) 120. The DFT 120
converts the time samples into subsymbols. A decoder 130 converts
the subsymbols into a data bits and outputs the resulting data.
[0039] As discussed above, a cyclic prefix is added to each symbol
prior to transmission through the communication channel to reduce
the effects of ISI. The cyclic prefix is formed by copying the last
.nu. time samples from the end of a symbol and placing them at the
beginning of the symbol. To eliminate ISI, the length of the cyclic
prefix, .nu., is chosen to be longer than the effective
discrete-time impulse response (EDIR) of the channel. However,
because the cyclic prefix constitutes redundant data, increasing
the length of the cyclic prefix reduces the efficiency of the
Communication system. For example, in a system having N time
samples per symbol and a cyclic prefix of .nu. time samples, the
efficiency of the system will be reduced by N/(N+.nu.). To maximize
efficiency it is necessary either to minimize .nu. or to maximize
N. However, increasing N increases the complexity, latency and
computational requirements of the system and at some point becomes
impractical. Accordingly, it is desirable to minimize .nu..
[0040] An impulse shortening filter having an impulse response,
g(n), may be employed in the receiver to minimize the length of the
cyclic prefix by decreasing the EDIR, g(n)*h(n), of the effective
communication channel, which includes the transmit and receive
filters, the impulse shortening filter and the physical
transmission channel. The use of an impulse shortening filter is
referred to as time domain equalization. Decreasing the EDIR allows
a shorter cyclic prefix to be used without increasing ISI.
[0041] FIG. 2 is a plot of the EDIR for a DMT test configuration
having a communication channel that is 4500 feet in length and
operates at a sampling frequency of 11.04 MHz (test loop 4, as
described in "Very-high Speed Digital Subscriber Lines: System
Requirements," Technical Report T1E1.4197-131R1, ANSI 1998). The
EDIR includes the effects of a transmit filter, the communication
channel and a receive filter. FIG. 3 shows the impulse response as
shortened by the addition of an impulse shortening filter.
[0042] The impulse shortening filter is selected so that a
significant part of the energy of the joint impulse response of the
filter and the effective communication channel, g(n)*h(n), is
confined to a region that is shorter in length than the length of
the cyclic prefix. Some prior algorithms for computing g(n)
considered only shortening the EDIR and did not consider the
spectral characteristics of the resulting impulse shortening
filter. Such filters often had deep nulls in some frequency bands,
which rendered some of the corresponding subchannels useless.
[0043] Since increasing the length of the cyclic prefix reduces
system efficiency, the receiver may dynamically compute an optimal
length for the cyclic prefix and may send that information to the
transmitter. For example, the receiver may compute a set of impulse
responses for the impulse shortening filter based on a set of
predetermined cyclic prefix lengths. The receiver then computes the
system throughput for each particular cyclic prefix length. The
length that maximizes system throughput is selected and the result
of that selection is communicated to the transmitter. The
transmitter then operates using the selected cyclic prefix
length.
[0044] To avoid the possible attenuation of frequency bands, the
spectral response of the impulse shortening filter is further
required to have a spectral response that |G(.omega.)|, meets a
specified spectral constraint. A spectral constraint of the form
|G(.omega.)H(.omega.)|>.tau., where .tau. is a threshold, is
sufficient to avoid nulls in the frequency response of the impulse
shortening filter. However, it is possible to compute a spectral
constraint or target spectral response, |G.sub.d(.omega.)|, that
provides additional performance improvements, such as reducing
noise bleeding between subchannels. A spectrally constrained
impulse filter is configured to have a spectral response that
approximates the target spectral response.
[0045] As shown in FIG. 4, the spectrally constrained impulse
shortening filter (SCISF) 90 may be implemented as a time domain
digital filter, which has a digital filter structure 210 with a
number of taps 220 or filter coefficient inputs for adjusting the
filter response. The coefficients may be computed and supplied to
the taps by a digital signal processor (DSP). Alternatively, the
SCISF may be implemented entirely in software, i.e., within a
DSP.
[0046] A desired spectral response may be applied to a received
signal using a filter that is separate from the impulse shortening
or time domain equalization (TEQ) filter. For example, an analog
filter may be placed prior to the A/D converter. However, the
adjustability of such a filter would be limited. As a further
example, a digital filter could be added prior to the TEQ filter.
Both of these configurations suffer the disadvantage that the TEQ
filter may distort the desired spectral characteristics of the
added filter. A filter also might be positioned after the TEQ
filter, which would reduce noise bleeding, but might reduce the
impulse shortening provided by the TEQ filter. Accordingly, the
SCISF integrates the TEQ (i.e., impulse shortening) function with
the desired spectral response in a single filter.
[0047] In summary, the filter characteristic, g(n), of the SCISF
satisfies two conditions. First, the effective length of the
convolution of the filter characteristic with the impulse response
of the communication channel, g(n)*h(n), is less than a target
length. Second, the cost function (error function) between the
target spectral response, G.sub.d(.omega.), and the actual filter
spectral response, G(.omega.), is minimized.
[0048] The target spectral response is an ideal filter
characteristic that is selected to maximize the data bit throughput
in the subchannels of a DMT system by reducing the impact of noise.
There are many sources for noise in a DMT communication system,
such as near-end cross-talk (NEXT), radio frequency interference
(RFI) and noise generated in the communication channel (white
noise). As shown in FIGS. 5 and 6, the noise spectral density is
generally not uniform across the frequency band of the
communication system. This non-uniformity contributes to the
problem of noise bleeding, in which noise in one frequency band
interfering with a signal in another frequency band.
[0049] In general, noise bleeding is caused by sidelobes of filters
in a DFT. FIG. 7 shows the filter response of a DFT for one
frequency band or bin (i.e., bin 128). The first sidelobes (250)
are only 13 dB below the main lobe (260). Therefore, noise located
outside of bin 128, but within the first sidelobe of bin 128, i.e.,
approximately mid-way between bins 126 and 127, would appear in bin
128 with an attenuation of only 13 dB. Consequently, noisy
subchannels in a DMT system may degrade the performance of
non-noisy subchannels.
[0050] The target spectral response is essentially a spectral
constraint that attenuates noisy channels more than non-noisy
channels. The signal and noise in the noisy channels are reduced
equally, so the attenuation does not affect the signal-to-noise
ratio in these channels. However, because the absolute noise level
in the noisy channels is reduced, there is less noise available to
enter the sidelobes of the non-noisy channels. Hence, the noise
bleeding problem is minimized.
[0051] To determine the target spectral response, the noise power
spectral density (noise PSD) at the receiver must be known. The
noise PSD may be determined, for example, by performing a
periodigram on received data. This measurement is more complicated
if a transmitter is transmitting, since the transmitted signal must
be separated from the noise measurement. The noise PSD is
determined by: (i) slicing the received constellation of
subcarriers after the DFT to determine the nearest valid
constellation point; (ii) determining an error signal based on the
difference between the received constellation point and the valid
constellation point; (iii) performing an IDFT on the error signal;
and (iv) generating a periodigram (with windowing) from the error
signals. The noise PSD may then be determined from the
periodigram.
[0052] An example of a noise PSD characteristic for a DMT
communication system is shown in FIG. 5 (test loop 4, as described
in "Very-high Speed Digital Subscriber Lines: System Requirements,"
Technical Report T1E1.4/97-131R1, ANSI 1998). The transmit signal
power is measured at the output of the transmit filter in the
transmitter. The signal and noise PSD plots shown in FIG. 5 are
measured at the input of the A/D converter in the receiver, which
is prior to the SCISF.
[0053] Measured noise PSD is used by a digital signal processor
(DSP) to compute a target spectral response, G.sub.d(.omega.),
using the algorithm described below. Alternatively, the inverse of
the noise PSD may be used as an approximation for the target
spectral response. A spectral response, G(.omega.), is then
determined for the SCISF that minimizes the error between the
spectral response of the SCISF and the target spectral response. A
set of filter coefficients may then be generated to configure the
SCISF to the determined characteristic. These calculations may be
done periodically to adjust the performance of the communication
system. Frequency domain equalization coefficients and symbol
Synchronization may also be adjusted based on these
calculations.
[0054] FIG. 8 is a plot of the target spectral response of the
SCISF, G.sub.d(.omega.), versus the actual frequency response,
G(.omega.). The difference between the responses is only a few dB.
FIG. 6 shows the signal and noise PSD at the output of the
SCISF.
[0055] FIG. 9 shows the dramatic effect of the SCISF on system
performance. Without the SCISF (i.e., using a filter that provides
Only impulse shortening), the signal-to-noise ratio (SNR) decreases
significantly at the output of the Fourier transform (i.e., FFT or
DFT) to less than about 7 dB. This decrease is due, in large part,
to noise bleeding caused by the sidelobes of the Fourier transform.
By contrast, with the SCISF, the SNR at the output of the Fourier
transform tracks the SNR at the output of the SCISF within a few
dB. Overall, the SCISF provides an improvement in SNR.
[0056] The target spectral response of the SCISF, G.sub.d(.omega.)
and the actual frequency response, G(.omega.), are derived from an
energy constraint for the SCISF, i.e., the target spectral response
must localize the energy of the effective impulse response within a
desired frequency band. The energy constraint is combined with a
desired spectral response based on the measured noise power
spectral density. The resulting cost function (or error function)
is then minimized to obtain a practical filter characteristic for
the SCISF. This process is presented in detail below.
[0057] In the following derivation, all vectors are column vectors
by default. Vectors are denoted by bold lower case letters (e.g.,
t). The size m.sub.t of a vector t, is written t(m.sub.t). The
components of a vector are denoted by lower case letters, e.g.,
t(m.sub.t)=[t.sub.0 . . . t.sub.m.sub.t.sub.-1].sup.T. The
convolution of length m.sub.t+m.sub.h-1 of vectors t(m.sub.t) and
h(m.sub.h) is denoted t*h. The Toeplitz matrix of a vector
t(m.sub.t), t.sub.(m,n).DELTA.[t.sub.m . . . t.sub.n].sup.T, is
written T.sub.(m,n).sup.m.sup.x, and
T.sub.(m,n).sup.m.sup.xx=(t*x).sub.(m,n). The Discrete Time Fourier
Transform (DTFT) of a vector t may be represented as:
T ( .omega. ) = n = 0 m t - 1 t n - j .omega. n . ( 1 )
##EQU00001##
[0058] The EDIR of the channel and the impulse response of the
SCISF are expressed as h(m.sub.h) and g(m.sub.g), respectively. The
cyclic prefix has a length of m.sub.c per symbol. The energy of the
effective impulse response must be localized to a contiguous region
of target length m.sub.l, where m.sub.l<m.sub.c, while
satisfying a spectral constraint. The energy criterion to be
satisfied by the SCISF may be written as:
( g * h ) m : m + m 1 - 1 2 ( g * h ) 2 .gtoreq. .alpha. ( 2 )
##EQU00002##
for some 0.ltoreq.m.ltoreq.m.sub.g+m.sub.h-2 and some
0<.alpha.<1. Defining the set S.sup.m as
s m = { g .di-elect cons. R m s : ( g * h ) m : m + m 1 - 1 2 ( g *
h ) 2 .gtoreq. .alpha. } ( 3 ) ##EQU00003##
the impulse response of the SCISF must belong to S.sup.n for some
m.
[0059] Let .omega..sub.1, . . . , .omega..sub.N be the location of
the sub-carriers in the frequency domain. The spectral constraint
may be applied by selecting g.epsilon.S.sup.m, so that the cost
function
J ( g ) = .DELTA. ? ( G ( .omega. i ) - G d ( .omega. i ) ) 2 ?
indicates text missing or illegible when filed ( 4 )
##EQU00004##
is minimized for the desired target spectral response
G.sub.d(.omega.).
[0060] Typically, this optimization would have to be repeated for
all possible m to select the filter impulse response, g, that
achieves the lowest possible value of J. However, as discussed
below, optimization may be limited to a few well chosen values of
m.
[0061] The determination of the filter impulse response may be done
in two stages. First, the desired magnitude frequency response
G.sub.d(.omega.) of the impulse shortening filter g over the bins
used by the DMT system is obtained. Second, J is optimized over
S.sup.m for a specific value of m.
[0062] To determine the target spectral response G.sub.d(.omega.),
it is necessary to use expressions for the signal to noise ratios
observed in the various frequency bins at the output of the DFT in
the receiver. The DMT system has M tones (subcarriers), N of which
(those from M.sub.1 through M.sub.2) are used, and the
communication channel has an analog frequency response H.sub.c(f).
The analog noise power spectral density observed at the input of
the receiver A/D is S.sub.n(f). Prior to conversion in the A/D
converter, the received analog signal may be filtered by an
anti-aliasing filter with transfer function H.sub.a(f). The EDIR in
the absence of the impulse shortening filter is h(n). After the A/D
converter, the signal is fed into the impulse shortening filter
with an impulse response of g(n). The impulse shortening filter
ensures that (h(n)*g(n)) is shorter in length than the cyclic
prefix. G(.omega.) is the discrete time Fourier transform of g(n).
Under these conditions, the expected signal energy .mu.(k) observed
in bin k at the output of the length-2M receiver DFT is given
by:
.mu. ( k ) = C 1 D k H ( .pi. k M ) 2 G ( .pi. k M ) 2 ; H (
.omega. ) = H c ( .omega. 2 .pi. T ) H a ( .omega. 2 .pi. T ) ( 5 )
##EQU00005##
Where C.sub.1 is a constant, 1/T the sampling frequency and D.sub.k
the transmitted power in bin k. The noise power .eta.(k) in bin k
is:
.eta. ( k ) = C 2 [ S .eta. ( .omega. 2 .pi. T ) G ( .omega. ) 2 H
a ( .omega. 2 .pi. T ) 2 ] * [ sin 2 ( M .omega. ) sin 2 ( .omega.
2 ) ] | .omega. = .pi. k M ( 6 ) ##EQU00006##
where C.sub.2 is another constant and * denotes the convolution.
Assuming that the noise in the bands corresponding to unused tones
is sufficiently attenuated by the anti-alias filter, .eta.(k) is
approximately equal to:
.eta. ( k ) .apprxeq. C 3 l = M 1 M 2 s .eta. ( l 2 MT ) G ( .pi. l
M ) 2 H a ( l 2 MT ) 2 ( .alpha. ( k - l ) + .alpha. ( 2 M - k - l
) ) , ( 7 ) ##EQU00007##
where .alpha.(n) is defined as:
.alpha. ( n ) = .intg. - .pi. 2 M .pi. 2 M [ sin 2 ( M ( ? M -
.lamda. ) ) sin 2 ( 1 2 ( ? M - .lamda. ) ) ] .lamda. , ? indicates
text missing or illegible when filed ( 8 ) ##EQU00008##
M.sub.1 . . . M.sub.2 are the used tones and C.sub.3 another
constant. Defining x to be the vector of frequency magnitudes to be
solved for as:
x = .DELTA. [ G ( .pi. M 1 M ) 2 G ( .pi. M 2 M ) 2 ] , ( 9 )
##EQU00009##
the SNR in bin k can be seen to be of the form
b k x k a k T x , ##EQU00010##
in which b.sub.k are scalars and a.sub.k are vectors.
[0063] To determine the target spectral response, x is chosen to
maximize the bit throughput. Approximating the capacity of bin k by
{square root over (SNR(k))}, the optimal spectral profile is
obtained by minimizing the cost function F, where:
F ( x ) = k = M 1 M 2 b k x k a k T x . ( 10 ) ##EQU00011##
This minimization is performed over:
X={x.epsilon.R.sup.N:.parallel.x.parallel.=1, x.sub.i.gtoreq.0,
1.ltoreq.i.ltoreq.N}
and can be accomplished by any one of a variety of constrained
optimization strategies, as discussed in Bertsekas, Nonlinear
Programming, Athena Scientific, Belmont, Mass., 1995. A median
filter may be applied to the output of the optimization algorithm
to smooth the resulting target spectral response.
[0064] Let A.DELTA.H.sub.(0:m.sub.k.sub.+m.sub.g.sub.-2) and
B.DELTA.H.sub.(m:m+m.sub.l.sub.-1). The energy constraint in
equation (3) can be written as:
S m = { g .di-elect cons. R m g : g T B T Bg g T A T Ag .gtoreq.
.alpha. } . ( 12 ) ##EQU00012##
[0065] Matrix A has full column rank, since it corresponds to a
full convolution, so R.sub.A.DELTA.A.sup.TA is invertible. Let
R.sub.B.DELTA.B.sup.TB. Defining q=R.sub.A.sup.0.5h, where
R.sub.A.sup.0.5 is the square root of the positive definite matrix
R.sub.A, the energy constraint set can be written in terms of q
as:
Q = .DELTA. { q .di-elect cons. R m g : q T R A - 0.5 R B R A - 0.5
q q T q .gtoreq. .alpha. } . ( 13 ) ##EQU00013##
[0066] The next step is to reduce the dimensionality, i.e., the
number of variables to search over in the optimization process. For
example, in a video digital subscriber line(VDSL) application, an
impulse shortening filter having a few hundred taps may be
required. Searching over a variable space so large is difficult and
impractical. Instead, the optimization is performed by searching
over a cleverly chosen lower-dimensional subset of the variable
space. This simplification in the optimization process may be done
without significant reduction in the performance of the
communication system.
[0067] The reduction in dimensionality is accomplished by a
transformation of variables. Let
C.DELTA.R.sub.A.sup.-0.5R.sub.BR.sub.A.sup.-0.5. Let
S=U.SIGMA.U.sup.T be the singular-value decomposition of C, where
.SIGMA. is a diagonal matrix whose (i, i)th element is
.sigma..sub.i and where the .sigma..sub.i are arranged in
descending order. If .sigma..sub.i<.alpha., there is no feasible
solution corresponding to delay m. If .sigma..sub.i.gtoreq..alpha.,
let
U = [ U 1 U 2 ] .SIGMA. = [ .SIGMA. 1 0 0 .SIGMA. 2 ] , ( 14 )
##EQU00014##
where U.sub.1 has size (m.sub.g,m.sub.d) and .SIGMA..sub.1 has size
(m.sub.d,m.sub.d) for some m.sub.d. These equations define the
dimension of the range space of the matrix U.sub.1, over which the
search is confined. The dimension m.sub.d may be chosen either to
include all .sigma..sub.i greater than some threshold .beta.,
.beta.<.alpha., or m.sub.d may be a fixed number. Simulations
indicate that m.sub.d can be less than m.sub.g by more than an
order of magnitude without significantly affecting communication
system performance.
[0068] The dimensionality reduction is achieved by a further
transformation of variables: q=U.sub.1v. The energy constraint set
now becomes:
V = .DELTA. { v .di-elect cons. R m d : v T .SIGMA. 1 v v T v
.gtoreq. .alpha. } = { v .di-elect cons. R m d : v .noteq. 0 ,
.alpha. v 2 - v T .SIGMA. 1 v .ltoreq. 0 } ( 15 ) ##EQU00015##
using the identity U.sub.1.sup.TCU.sub.1=.SIGMA..sub.1.
[0069] Next, the cost function is expressed in terms of v. For a
particular v, the corresponding g is given by
g=R.sub.A.sup.-0.5U.sub.1v, which leads to:
[ G ( .omega. 1 ) G ( .omega. N ) ] = Fg ; F = [ 1 - j .omega. 1 -
j 2 .omega. 1 - j ( m g - 1 ) .omega. 1 1 - j .omega. 1 - j 2
.omega. N - j ( m g - 1 ) .omega. N ] ( 16 ) ##EQU00016##
Let D.DELTA.FAR.sub.A.sup.-0.5U.sub.1. Let D.sup.R.DELTA.real(D)
and D.sup.I.DELTA.imag(D). D.sup.R and D.sup.I are real matrices of
size (N, m.sub.d). Let d.sub.R,n.sup.T, 1.ltoreq.n.ltoreq.N and
d.sub.I,n.sup.T, 1.ltoreq.n.ltoreq.N be the rows of D.sup.R and
D.sup.I, respectively. Then:
G ( .omega. n ) = ( d R , n T v ) 2 + ( d I , n T v ) 2 = v T ( d R
, n d R , n T + d I , n d I , n T v ) v = v T T n v
##EQU00017##
where
.GAMMA..sub.n.DELTA.d.sub.R,nd.sub.R,n.sup.T+d.sub.I,nd.sub.I,n.sup-
.T. These definitions result in:
J o ( v ) = .DELTA. j ( g ( v ) ) = n = 1 N ( v T .GAMMA. n v - G d
( .omega. n ) ) 2 ( 17 ) ##EQU00018##
The gradient and Hessian of J.sub.o are:
.gradient. J o ( v ) = n = 1 N 2 ( 1 - G d ( .omega. n ) v T
.GAMMA. n v ) .GAMMA. n v ( 18 ) H J o ( v ) = n = 1 N 2 [ ( 1 - G
d ( .omega. n ) v T .GAMMA. n v ) .GAMMA. n + G d ( .omega. n ) ( v
T .GAMMA. n v ) 3 .GAMMA. n vv T .GAMMA. n ] ( 19 )
##EQU00019##
[0070] The projection P.sub..nu.(y) of any y.epsilon.R.sup.md on to
V is defined to be
P V ( y ) = .DELTA. arg min v .di-elect cons. V y - v 2 ( 20 )
##EQU00020##
[0071] To develop an algorithm to optimize the cost function
J.sub.o over V, an expression must be derived for P.sub..nu.(y).
There is no closed-form expression for this projection operator,
however there is a very efficient algorithm to compute it. If
y.epsilon..nu., P.sub..nu.(y)=y. If not, the projection onto V is
the same as the projection on to its boundary, defined by:
boundary (V).DELTA.{v.epsilon.R.sup.m.sup.d:v.noteq.0,
.alpha..parallel.v.parallel..sup.2-v.sup.T.SIGMA..sub.1v=0}
(21)
The latter may be accomplished using LaGrange multipliers as
follows. Defining a modified cost function S as:
s(v).DELTA..parallel.y-v.parallel..sup.2+.mu.(.alpha..parallel.v.paralle-
l..sup.2-v.sup.T.SIGMA..sub.1v) (22)
The equation .gradient..sub.S(v)=0 must be solved on the boundary
of V. This reduces to the following simultaneous equations:
[(1+.mu..alpha.)I-.mu..SIGMA..sub.1]v=0 (23)
.alpha..parallel.v.parallel..sup.2-v.sup.T.SIGMA..sub.1v=0
(24).
Solving for v in terms of .mu. from (23) and substituting in (24),
the following equation is obtained for .mu.:
y T diag ( .alpha. - .sigma. i [ 1 + .mu. ( .alpha. - .sigma. i ) ]
2 ) y = 0 ( 25 ) ##EQU00021##
Equation (25) may be written as a polynomial equation of order
2m.sub.d in .mu.. The polynomial equation must be solved for the
real and positive roots, which may be done using one of the variety
of efficient root-finding algorithms in the literature. One of the
real, positive roots must be chosen by enumeration so that, when
substituted into the expression for v, it leads to the smallest
.parallel.y-v.parallel..sup.2.
[0072] Since the gradient and Hessian of the cost function J.sub.0
and the projection operator onto V are available, the optimization
can be done very efficiently. For example, the penalty method may
be used, as described in Bertsekas, Nonlinear Programming, Athena
Scientific, Belmont, MA, 1995, or an iterative strategy consisting
of gradient descent followed by projection onto V. Both techniques
have been tested and perform well.
[0073] Other embodiments are within the scope of the following
claims.
* * * * *