U.S. patent application number 11/974994 was filed with the patent office on 2008-02-21 for reduced computation in joint detection.
This patent application is currently assigned to InterDigital Technology Corporation. Invention is credited to Raj Mani Misra, Ariela Zeira.
Application Number | 20080043875 11/974994 |
Document ID | / |
Family ID | 33492662 |
Filed Date | 2008-02-21 |
United States Patent
Application |
20080043875 |
Kind Code |
A1 |
Misra; Raj Mani ; et
al. |
February 21, 2008 |
Reduced computation in joint detection
Abstract
A user equipment receives a plurality of transmitted data
signals in a communication system. The transmitted data are
received. A channel response is determined for each received data
signal. A system response is determined based on in part the
channel signals. The system response is expanded to be piecewise
orthogonal. Data from the received data signals is retrieved based
on in part the expanded system response.
Inventors: |
Misra; Raj Mani; (Fremont,
CA) ; Zeira; Ariela; (Huntington, NY) |
Correspondence
Address: |
VOLPE AND KOENIG, P.C.;DEPT. ICC
UNITED PLAZA, SUITE 1600
30 SOUTH 17TH STREET
PHILADELPHIA
PA
19103
US
|
Assignee: |
InterDigital Technology
Corporation
Wilmington
DE
|
Family ID: |
33492662 |
Appl. No.: |
11/974994 |
Filed: |
October 17, 2007 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
11010703 |
Dec 13, 2004 |
7295596 |
|
|
11974994 |
Oct 17, 2007 |
|
|
|
09662404 |
Sep 14, 2000 |
6831944 |
|
|
11010703 |
Dec 13, 2004 |
|
|
|
60153801 |
Sep 14, 1999 |
|
|
|
Current U.S.
Class: |
375/267 ;
375/E1.027 |
Current CPC
Class: |
H04B 1/71052 20130101;
H04B 2201/70707 20130101; H04L 25/0204 20130101; H04B 1/71055
20130101; G06F 17/16 20130101; H04L 25/0242 20130101 |
Class at
Publication: |
375/267 |
International
Class: |
H04B 1/69 20060101
H04B001/69 |
Claims
1. A circuit for receiving a plurality of transmitted data signals,
the circuit comprising: an antenna for receiving the transmitted
data signals; a channel estimation device for determining a channel
response for each received data signal; and a joint detection
device having an input configured to receive the channel responses
and the received data signals for determining a system response
based on in part the channel signals, expanding the system response
to be piecewise orthogonal, and retrieving data from the received
data signals based on in part the expanded system response.
2. The circuit of claim 1 for use in a time division duplex using
code division multiple access communication system.
3. The circuit of claim 2 wherein each of the transmitted data
signals has an associate code and is transmitted in a shared
frequency spectrum and the system response is determined by
convolving associated chip codes with the channel response.
4. The circuit of claim 2 wherein the channel estimation device
measures the channel response using a received training sequence
associated with the data signals.
5. The circuit of claim 1 wherein the system response is a system
response matrix, further comprising dividing the system response
matrix into blocks of columns prior to the expanding.
6. The circuit of claim 5 wherein the expanding is by padding zeros
in the column blocks such that each column block is orthogonal.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation of U.S. patent
application Ser. No. 11/010/703, filed Dec. 13, 2004, which is a
continuation of U.S. patent application Ser. No. 09/662,404 filed
Sep. 14, 2000, now U.S. Pat. No. 6,831,944, issued Dec. 14, 2004,
which in turn claims priority to U.S. Provisional Patent
Application No. 60/153,801, filed on Sep. 14, 1999.
BACKGROUND
[0002] The invention generally relates to wireless communication
systems. In particular, the invention relates to joint detection of
multiple user signals in a wireless communication system.
[0003] FIG. 1 is an illustration of a wireless communication system
10. The communication system 10 has base stations 12.sub.1 to
12.sub.5 which communicate with user equipments (UEs) 14.sub.1 to
14.sub.3. Each base station 12.sub.1 has an associated operational
area where it communicates with UEs 14.sub.1 to 14.sub.3 in its
operational area.
[0004] In some communication systems, such as code division
multiple access (CDMA) and time division duplex using code division
multiple access (TDD/CDMA), multiple communications are sent over
the same frequency spectrum. These communications are typically
differentiated by their chip code sequences. To more efficiently
use the frequency spectrum, TDD/CDMA communication systems use
repeating frames divided into time slots for communication. A
communication sent in such a system will have one or multiple
associated chip codes and time slots assigned to it based on the
communication's bandwidth.
[0005] Since multiple communications may be sent in the same
frequency spectrum and at the same time, a receiver in such a
system must distinguish between the multiple communications. One
approach to detecting such signals is single user detection. In
single user detection, a receiver detects only the communication
from a desired transmitter using a code associated with the desired
transmitter, and treats signals of other transmitters as
interference.
[0006] In some situations, it is desirable to be able to detect
multiple communications simultaneously in order to improve
performance. Detecting multiple communications simultaneously is
referred to as joint detection. Some joint detectors use Cholesky
decomposition to perform a minimum mean square error (MMSE)
detection and zero-forcing block equalizers (ZF-BLEs). These
detectors have a high complexity requiring extensive receiver
resources.
[0007] Accordingly, it is desirable to have alternate approaches to
joint detection.
SUMMARY
[0008] A user equipment receives a plurality of transmitted data
signals in a communication system. The transmitted data signals are
received. A channel response is determined for each received data
signal. A system response is determined based on in part the
channel signals. The system response is expanded to be piecewise
orthogonal. Data from the received data signals is retrieved based
on in part the expanded system response.
BRIEF DESCRIPTION OF THE DRAWING(S)
[0009] FIG. 1 is a wireless communication system.
[0010] FIG. 2 is a simplified transmitter and a receiver using
joint detection.
[0011] FIG. 3 is an illustration of a communication burst.
[0012] FIG. 4 is an illustration of reduced computation joint
detection.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT(S)
[0013] FIG. 2 illustrates a simplified transmitter 26 and receiver
28 using joint detection in a TDD/CDMA communication system. In a
typical system, a transmitter 26 is in each UE 14.sub.1 to 14.sub.3
and multiple transmitting circuits 26 sending multiple
communications are in each base station 12.sub.1 to 12.sub.5. A
base station 12.sub.1 will typically require at least one
transmitting circuit 26 for each actively communicating UE 14.sub.1
to 14.sub.3. The joint detection receiver 28 may be at a base
station 12.sub.1, UEs 14.sub.1 to 14.sub.3 or both. The joint
detection receiver 28 receives communications from multiple
transmitters 26 or transmitting circuits 26.
[0014] Each transmitter 26 sends data over a wireless communication
channel 30. A data generator 32 in the transmitter 26 generates
data to be communicated over a reference channel to a receiver 28.
Reference data is assigned to one or multiple codes and/or time
slots based on the communications bandwidth requirements. A
spreading and training sequence insertion device 34 spreads the
reference channel data and makes the spread reference data
time-multiplexed with a training sequence in the appropriate
assigned time slots and codes. The resulting sequence is referred
to as a communication burst. The communication burst is modulated
by a modulator 36 to radio frequency. An antenna 38 radiates the RF
signal through the wireless radio channel 30 to an antenna 40 of
the receiver 28. The type of modulation used for the transmitted
communication can be any of those known to those skilled in the
art, such as direct phase shift keying (DPSK) or quadrature phase
shift keying (QPSK).
[0015] A typical communication burst 16 has a midamble 20, a guard
period 18 and two data bursts 22, 24, as shown in FIG. 3. The
midamble 20 separates the two data bursts 22, 24 and the guard
period 18 separates the communication bursts to allow for the
difference in arrival times of bursts transmitted from different
transmitters. The two data bursts 22, 24 contain the communication
burst's data and are typically the same symbol length.
[0016] The antenna 40 of the receiver 28 receives various radio
frequency signals. The received signals are demodulated by a
demodulator 42 to produce a baseband signal. The baseband signal is
processed, such as by a channel estimation device 44 and a joint
detection device 46, in the time slots and with the appropriate
codes assigned to the communication bursts of the corresponding
transmitters 26. The channel estimation device 44 uses the training
sequence component in the baseband signal to provide channel
information, such as channel impulse responses. The channel
information is used by the joint detection device 46 to estimate
the transmitted data of the received communication bursts as soft
symbols.
[0017] The joint detection device 46 uses the channel information
provided by the channel estimation device 44 and the known
spreading codes used by the transmitters 26 to estimate the data of
the various received communication bursts. Although joint detection
is described in conjunction with a TDD/CDMA communication system,
the same approach is applicable to other communication systems,
such as CDMA.
[0018] One approach to joint detection in a particular time slot in
a TDD/CDMA communication system is illustrated in FIG. 4. A number
of communication bursts are superimposed on each other in the
particular time slot, such as K communication bursts. The K bursts
may be from K different transmitters. If certain transmitters are
using multiple codes in the particular time slot, the K bursts may
be from less than K transmitters.
[0019] Each data burst 22, 24 of the communication burst 16 has a
predefined number of transmitted symbols, such as Ns. Each symbol
is transmitted using a predetermined number of chips of the
spreading code, which is the spreading factor (SF). In a typical
TDD communication system, each base station 12.sub.1 to 12.sub.5
has an associated scrambling code mixed with its communicated data.
The scrambling code distinguishes the base stations from one
another. Typically, the scrambling code does not affect the
spreading factor. Although the terms spreading code and factor are
used hereafter, for systems using scrambling codes, the spreading
code for the following is the combined scrambling and spreading
codes. Each data burst 22, 24 has Ns.times.SF chips.
[0020] The joint detection device 46 estimates the value that each
data burst symbol was originally transmitted. Equation 1 is used to
determine the unknown transmitted symbols. r=Ad+n Equation 1
[0021] In Equation 1, the known received combined chips, r, is a
product of the system response, A, and the unknown transmitted
symbols, d. The term, n represents the noise in the wireless radio
channel.
[0022] For K data bursts, the number of data burst symbols to be
recovered is Ns.times.K. For analysis purposes, the unknown data
burst symbols are arranged into a column matrix, d. The d matrix
has column blocks, d.sub.1 to d.sub.Ns, of unknown data symbols.
Each data symbol block, d.sub.i, has the i.sup.th unknown
transmitted data symbol in each of the K data bursts. As a result,
each column block, d.sub.i, has K unknown transmitted symbols
stacked on top of each other. The blocks are also stacked in a
column on top of each other, such that d.sub.1 is on top of d.sub.2
and so on.
[0023] The joint detection device 46 receives a value for each chip
as received. Each received chip is a composite of all K
communication bursts. For analysis purposes, the composite chips
are arranged into a column matrix, r. The matrix r has a value of
each composite chip, totaling Ns*SF chips.
[0024] A is the system response matrix. The system response matrix,
A, is formed by convolving the impulse responses with each
communication burst chip code. The convolved result is rearranged
to form the system response matrix, A (step 48).
[0025] The joint detection device 46 receives the channel impulse
response, h.sub.i, for each i.sup.th one of the K communication
bursts from the channel estimation device 44. Each h.sub.i has a
chip length of W. The joint detection device convolves the channel
impulse responses with the known spreading codes of the K
communication bursts to determine the symbol responses, s.sub.1 to
s.sub.K, of the K communication bursts. A common support sub-block,
S, which is common to all of the symbol responses is of length
K.times.(SF+W-1).
[0026] The A matrix is arranged to have Ns blocks, B.sub.1 to
B.sub.Ns. Each block has all of the symbol responses, s.sub.1 to
s.sub.K arranged to be multiplied with the corresponding unknown
data block in the d matrix, d.sub.1 to d.sub.Ns. For example,
d.sub.1 is multiplied with B. The symbol responses, S.sub.1 to
S.sub.K, form a column in each block matrix, B.sub.i, with the rest
of the block being padded with zeros. In the first block, B.sub.1,
the symbol response row starts at the first row. In the second
block, the symbol response row is SF rows lower in the block and so
on. As a result, each block has a width of K and a height of
Ns.times.SF. Equation 2 illustrates an A block matrix A = [ s 1 s 2
s K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 s 1 s 2 s K 0 0 0 0 0 0 0 0 s 1 s
2 s K 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] = [ B .times. 1 .times.
.times. B .times. 2 .times. .times. .times. .times. B .times. N
.times. s ] Equation .times. .times. 2 ##EQU1##
[0027] The n matrix has a noise value corresponding to each
received combined chip, totaling Ns.times.SF chips. For analysis
purposes, the n matrix is implicit in the received combined chip
matrix, r.
[0028] Using the block notation, Equation 1 can be rewritten as
Equation 3. r = [ B .times. 1 .times. .times. B .times. 2 .times.
.times. B 3 .times. .times. .times. .times. B .times. N .times. s ]
.times. [ d 1 d 2 d 3 d N s ] + n = i = 1 N s .times. B i .times. d
i + n Equation .times. .times. 3 ##EQU2##
[0029] Using a noisy version of the r matrix, the value for each
unknown symbol can be determined by solving the equation. However,
a brute force approach to solving Equation 1 requires extensive
processing. To reduce the processing, the system response matrix,
A, is repartitioned. Each block, B.sub.i is divided into Ns blocks
having a width of K and a height of SF. These new blocks are
referred to as A.sub.1 to A.sub.L and 0. L is the length of the
common support S, as divided by the height of the new blocks,
A.sub.1 to A.sub.L, per Equation 4. L = SF + W - 1 SF Equation
.times. .times. 4 ##EQU3##
[0030] Blocks A.sub.1 to A.sub.L are determined by the supports,
s.sub.1 to s.sub.K, and the common support, S. A 0 block having all
zeros. A repartitioned matrix for a system having a W of 57, SF of
16 and an L of 5 is shown in Equation 5. A = [ A 1 0 0 0 0 0 0 0 0
0 0 A 2 A 1 0 0 0 0 0 0 0 0 0 A 3 A 2 A 1 0 0 0 0 0 0 0 0 A 4 A 3 A
2 A 1 0 0 0 0 0 0 0 A 5 A 4 A 3 A 2 A 1 0 0 0 0 0 0 0 A 5 A 4 A 3 A
2 A 1 0 0 0 0 0 0 0 A 5 A 4 A 3 A 2 A 1 0 0 0 0 0 0 0 A 5 A 4 A 3 A
2 0 0 0 0 0 0 0 0 A 5 A 4 A 3 A 1 0 0 0 0 0 0 0 0 A 5 A 4 A 2 A 1 0
0 0 0 0 0 0 0 A 5 A 3 A 2 A 1 0 0 0 0 0 0 0 0 A 4 A 3 A 2 A 1 ]
Equation .times. .times. 5 ##EQU4##
[0031] To reduce the complexity of the matrix, a piecewise
orthogonalization approach is used. Any of the blocks B.sub.i for i
being L or greater is non-orthogonal to any of the preceding L
blocks and orthogonal to any blocks preceding by more than L. Each
0 in the repartitioned A matrix is an all zero block. As a result
to use a piecewise orthogonalization, the A matrix is expanded
(step 50).
[0032] The A matrix is expanded by padding L-1 zero blocks to the
right of each block of the A matrix and shifting each row in the A
matrix by its row number less one. To illustrate for the A1 block
in row 2 of FIG. 2, four (L-1) zeros are inserted between A2 and A1
in row 2. Additionally, block A1 (as well as A2) is shifted to the
right by one column (row 2-1). As a result, Equation 5 after
expansion would become Equation 6. A exp = [ .times. A 1 0 0 0 0 0
0 0 0 0 0 0 A 2 0 0 0 A 1 0 0 0 0 0 0 0 A 3 0 0 0 A 2 0 0 0 A 1 0 0
0 A 4 0 0 0 A 3 0 0 0 0 0 0 0 A 5 0 0 0 A 4 0 0 0 0 0 0 0 0 0 0 0 A
5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
.times. ] Equation .times. .times. 6 ##EQU5##
[0033] To accommodate the expanded A matrix, the d matrix must also
be expanded, d.sub.exp. Each block, d.sub.1 to d.sub.Ns, is
expanded to a new block, d.sub.exp1 to d.sub.expNs. Each expanded
block, d.sub.exp1 to d.sub.expNs, is formed by repeating the
original block L times. For example for d.sub.exp1 a first block
row would be created having L versions of d1, stacked one below the
other.
[0034] As a result, Equation 1 can be rewritten as Equation 7. r =
A exp d exp + n = [ B exp .times. .times. 1 .times. .times. B exp
.times. .times. 2 .times. .times. B exp .times. .times. 3 .times.
.times. .times. .times. B exp .times. .times. N s ] .times. [ d exp
.times. .times. 1 d exp .times. .times. 2 d exp .times. .times. 3 d
exp .times. .times. N s ] + n = i = 1 N s .times. B exp .times.
.times. i .times. d exp .times. .times. i + n , Equation .times.
.times. 7 ##EQU6## Equation 7 can be rewritten to partition each
B.sub.expi orthogonally in L partitions, U.sub.j.sup.(I), j=1 to L,
as in Equation 8. r = A exp d exp + n = i = 1 N s .times. [ U 1 ( i
) .times. .times. U 2 ( i ) .times. .times. .times. .times. U L ( i
) ] .times. [ d i d i d i d i ] + n = i = 1 N s .times. j = 1 L
.times. U j ( i ) .times. d i = i = 1 N s .times. B i .times. d i +
n Equation .times. .times. 8 ##EQU7##
[0035] To reduce computational complexity, a QR decomposition of
the A.sub.exp matrix is performed (step 52). Equation 9 illustrates
the QR decomposition of A.sub.exp. A.sub.exp=Q.sub.expR.sub.exp
Equation 9 Due to the orthogonal partitioning of A.sub.exp, the QR
decomposition of A.sub.exp is less complex. The resulting Q.sub.exp
and R.sub.exp matrices are periodic with an initial transient
extending over L blocks. Accordingly, Q.sub.exp and R.sub.exp can
be determined by calculating the initial transient and one period
of the periodic portion. Furthermore, the periodic portion of the
matrices is effectively determined by orthogonalizing A.sub.1 to
A.sub.L. One approach to QR decomposition is a Gramm-Schmidt
orthogonalization.
[0036] To orthogonalize A.sub.exp as in Equation 6, B.sub.exp1 is
orthogonalized by independently orthogonalizing each of its
orthogonal partitions, {U.sub.j.sup.(i)}, j=1 . . . L. Each
{A.sub.j}, j=1 . . . L is independently orthogonalized, and the set
is zero-padded appropriately. {Q.sub.j} are the orthonormal sets
obtained by orthogonalizing {U.sub.j.sup.(i)}. To determine
B.sub.exp2, its U.sub.1.sup.(2) needs to be orthogonalized with
respect to only Q.sub.2 of B.sub.exp1 formed previously.
U.sub.2.sup.(2), U.sub.2.sup.(2) and U.sub.4.sup.(2) only need to
be orthogonalized with respect to only Q.sub.3, Q.sub.4 and
Q.sub.5, respectively. U.sub.5.sup.(2) needs to be orthogonalized
to all previous Qs and its orthogonalized result is simply a
shifted version of Q.sub.5 obtained from orthogonalizing
B.sub.exp1.
[0037] As the orthogonalizing continues, beyond the initial
transient, there emerges a periodicity which can be summarized as
follows. The result of orthogonalizing B.sub.expi, i.gtoreq.6 can
be obtained simply by a periodic extension of the result of
orthogonalizing B.sub.exp5.
[0038] The orthogonalization of B.sub.exp5, is accomplished as
follows. Its Q.sub.5 is obtained by orthogonalizing A.sub.5, and
then zero padding. Its Q.sub.4 is obtained by orthogonalizing the
support of Q.sub.5 and A.sub.4, [sup(Q.sub.5) A.sub.4], and then
zero padding. Since sup(Q.sub.5) is already an orthogonal set, only
A.sub.4 needs to be orthogonalized with respect to sup(Q.sub.5) and
itself. Its Q.sub.3 is obtained by orthogonalizing [sup(Q.sub.5)
sup(Q.sub.4) A.sub.3] and then zero padding. Its Q.sub.2 is
obtained by orthogonalizing [sup(Q.sub.5) sup(Q.sub.4) sup(Q.sub.3)
A.sub.2] and then zero padding. Its Q. is obtained by
orthogonalizing [sup(Q.sub.5) sup(Q.sub.4) sup(Q.sub.3)
sup(Q.sub.2) A.sub.1] and then zero padding. Apart from the initial
transient, the entire A.sub.exp can be efficiently orthogonalized,
by just orthogonalizing A.sub.p per Equation 10.
A.sub.p=[A.sub.5A.sub.4A.sub.3A.sub.2A.sub.1] Equation 10 By
effectively orthogonalizing the periodic portion of A.sub.exp by
using only A.sub.p, computational efficiency is achieved. Using a
more compact notation, Q.sub.i.sup.s,for sup (Q.sub.i), this
orthogonalization of A.sub.p results in the orthonormal matrix,
Q.sub.p, of Equation 11.
Q.sub.p=[Q.sub.5.sup.SQ.sub.4.sup.SQ.sub.3.sup.SQ.sub.2.sup.SQ.sub.1.sup.-
S] Equation 11 The periodic part of Q.sub.exp is per Equation 12.
PeriodicPartofQ exp = [ .times. 0 0 0 0 0 0 0 0 0 0 Q 1 S 0 0 0 0 0
0 0 0 0 Q 2 S 0 0 0 Q 1 S 0 0 0 0 0 0 Q 3 S 0 0 0 Q 2 S 0 0 0 0 0 0
Q 4 S 0 0 0 Q 3 S 0 0 0 0 0 0 Q 5 S 0 0 0 Q 4 S 0 0 0 0 0 0 0 0 0 0
Q 5 S 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .times. ] Equation
.times. .times. 12 ##EQU8##
[0039] To constructing the upper triangular matrix R.sub.exp,
A.sub.i.sub.j is a block of size K.times.K representing the
projections of each column of A.sub.i onto all the columns of
Q.sub.j.sup.s. For example, the first column of A.sub.4.sub.5
represents the projections of the first column of A.sub.4 on each
of the K columns of Q.sub.5.sup.s. Similarly, A.sub.4.sub.4
represents the projections of the first column of A.sub.4 on each
of the K columns of Q.sub.4.sup.S. However, this block will be
upper triangular, because the k.sup.th column of A.sub.4 belongs to
the space spanned by the orthonormal vectors of Q.sub.5.sup.S and
the first k vectors of Q.sub.4.sup.S. This block is also orthogonal
to subsequent vectors in Q.sub.4.sup.S, leading to an upper
triangular A.sub.4.sub.4. Any A.sub.i.sub.j with i=j will be upper
triangular. To orthogonalize other blocks, the following
results.
[0040] The first block of B.sub.exp5, viz., U.sub.1.sup.(5) is
formed by a linear combination of {Q.sub.j.sup.S}, j=1 . . . 5,
with coefficients given by A.sub.1.sub.j, j=1 . . . 5. The second
block, U.sub.2.sup.(5), is formed by a linear combination of
{Q.sub.j.sup.S}, j=2 . . . 5, with coefficients given by
A.sub.2.sub.j, j=2 . . . 5. The third block, U.sub.3.sup.(5), is
formed by a linear combination of {Q.sub.j.sup.S}, j=3 . . . 5,
with coefficients given by A.sub.2.sub.j, j=3 . . . 5. The fourth
block, U.sub.4.sup.(5),is formed by a linear combination of
{Q.sub.j.sup.S}, j=4, 5, with coefficients given by A.sub.2.sub.j,
j=4, 5. The fifth block, U.sub.5.sup.(5), is formed by
Q.sub.5.sup.S.times.A.sub.5.sub.5.
[0041] Accordingly, the coefficients in the expansion of subsequent
B.sub.expi, i.gtoreq.6 are simply periodic extensions of the above.
Since the R.sub.exp entries are computed during the
orthogonalization of A.sub.exp, no additional computations are
needed to construct R.sub.exp. Disregarding the initial transient,
the remainder of R.sub.exp is periodic, and two periods of it are
shown in Equation 13. R exp = [ 0 0 0 0 0 0 0 0 0 0 .times. 0
.times. .times. A .times. 1 5 .times. 0 .times. 0 .times. 0 .times.
A .times. 1 4 .times. 0 0 0 0 0 0 0 0 0 .times. A .times. 2 5
.times. 0 0 0 .times. A .times. 1 5 .times. 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 .times. A .times. 1 3 .times. 0 0 0 0 0 0 0 0 0
.times. A .times. 2 4 .times. 0 0 0 .times. A .times. 1 4 .times. 0
0 0 0 0 .times. A .times. 3 5 .times. 0 0 0 .times. A .times. 2 5
.times. 0 0 0 0 0 0 0 0 0 0 0 .times. A .times. 1 2 .times. 0 0 0 0
0 0 0 0 0 0 .times. A .times. 2 3 .times. 0 0 0 .times. A .times. 1
3 .times. 0 0 0 0 0 .times. A .times. 3 4 .times. 0 0 0 .times. A
.times. 2 4 .times. 0 0 0 0 0 .times. A .times. 4 5 .times. 0 0 0
.times. A .times. 3 5 .times. 0 .times. .times. A .times. 1 1
.times. 0 0 0 0 0 0 0 0 0 .times. A .times. 2 2 .times. 0 0 0
.times. A .times. 1 2 .times. 0 0 0 0 0 .times. A .times. 3 3
.times. 0 0 0 .times. A .times. 2 3 .times. 0 0 0 0 0 .times. A
.times. 4 4 .times. 0 0 0 .times. A .times. 3 4 .times. 0 0 0 0 0
.times. A .times. 5 5 .times. 0 0 0 .times. A .times. 4 5 .times. 0
0 0 0 0 .times. A .times. 1 1 .times. 0 0 0 0 0 0 0 0 0 .times. A
.times. 2 2 .times. 0 0 0 0 0 0 0 0 0 .times. A .times. 3 3 .times.
0 0 0 0 0 0 0 0 0 .times. A .times. 4 4 .times. 0 0 0 0 0 0 0 0 0
.times. A .times. 5 5 .times. 0 0 0 0 0 0 0 0 0 0 .times. A .times.
1 1 ] Equation .times. .times. 16 ##EQU9## The least squares
approach to solving Q.sub.exp and R.sub.exp is shown in Equation
14. Q.sub.expR.sub.expd.sub.exp=r Equation 14 By pre-multiplying
both sides of Equation 14 by the transpose of Q.sub.exp,
Q.sub.exp.sup.T, and using
Q.sub.exp.sup.TQ.sub.exp=I.sub.LKN.sub.s, Equation 14 becomes
Equation 15. R.sub.expd.sub.exp=Q.sub.exp.sup.Tr Equation 15
Equation 15 represents a triangular system whose solution also
solves the LS problem of Equation 14.
[0042] Due to the expansion, the number of unknowns is increased by
a factor of L. Since the unknowns are repeated by a factor of L, to
reduce the complexity, the repeated unknowns can be collected to
collapse the system. R.sub.exp is collapsed using L coefficient
blocks, CF.sub.1 to CF.sub.L, each having a width and a height of
K. For a system having an L of 5, CF.sub.1 to CF.sub.5 can be
determined as in Equation 16.
CF.sub.1=A.sub.1.sub.1+A.sub.2.sub.2+A.sub.3.sub.3+A.sub.4.sub.4+A.sub.5.-
sub.5
CF.sub.2=A.sub.1.sub.2+A.sub.2.sub.3+A.sub.3.sub.4+A.sub.4.sub.5
CF.sub.3=A.sub.1.sub.3+A.sub.2.sub.4+A.sub.3.sub.5
CF.sub.4=A.sub.1.sub.4+A.sub.2.sub.5 CF.sub.5=A.sub.1.sub.5
Equation 16 Collapsing R.sub.exp using the coefficient blocks
produces a Cholesky like factor, G (step 54). By performing
analogous operations on the right hand side of Equation 15 results
in a banded upper triangular system of height and width of
K.times.Ns as in Equation 17. [ .times. Tr 1 Tr 2 Tr 3 Tr 4 CF 5 0
0 0 0 0 0 Tr 1 Tr 2 CF 3 CF 4 CF 5 0 0 0 0 0 0 CF 1 CF 2 CF 3 CF 4
CF 5 0 0 0 0 0 0 CF 1 CF 2 CF 3 CF 4 CF 5 0 0 0 0 0 0 CF 1 CF 2 CF
3 CF 4 CF 5 0 0 0 ] .times. [ .times. d 1 d 2 d 3 d N s ] = .times.
r ^ Equation .times. .times. 17 ##EQU10## Tr.sub.1 to Tr.sub.4 are
the transient terms and {circumflex over (r)}. By solving the upper
triangle via back substitution, Equation 17 can be solved to
determine d (step 56). As a result, the transmitted data symbols of
the K data bursts is determined.
[0043] Using the piecewise orthogonalization and QR decomposition,
the complexity of solving the least square problem when compared
with a banded Cholesky decomposition is reduced by a factor of
6.5.
* * * * *