Method and apparatus for receiving digital wireless transmissions using multiple-antenna communication schemes

A signal detection technique for multiple-input multiple-output (MIMO) communications systems embodied in a method and apparatus for detecting a plurality of transmitted signals with use of a plurality of receiving antennas. An iterative procedure decodes one of a plurality of transmitted signals at each iteration using an intermediate matrix at each iteration to determine the transmitted signal to be decoded. The intermediate matrix for each successive iteration is advantageously computed in a recursive manner with use of a Schur complement operation performed based on the inverse of a modified version of the intermediate matrix used in the previous iteration.

FIELD OF THE INVENTION

The present invention relates generally to the field of wireless radio-frequency communication systems, and more particularly to a method and apparatus for detecting transmitted signals in a digital wireless communication system employing multiple-antenna communications.

BACKGROUND OF THE INVENTION

Multiple-antenna communications systems, also known as Multiple-Input Multiple-Output (MIMO) systems, are known to be able to achieve very high spectral efficiencies in scattering environments, with no increase in bandwidth or transmitted power. In particular, it is known that such a multipath wireless channel is capable of huge capacities, provided that the multipath scattering is sufficiently rich and is properly exploited through the use of an appropriate processing architecture and multiple antennas (both at transmission and reception).

One such MIMO system is described, for example, in U.S. Pat. No. 6,097,771, issued on Aug. 1, 2000 to G. Foschini, entitled “Wireless Communications System Having A Layered Space-Time Architecture Employing Multi-Element Antennas.” U.S. Pat. No. 6,097,771, which is commonly assigned to the assignee of the present invention, is hereby incorporated by reference as if fully set forth herein. The architecture described in U.S. Pat. No. 6,097,771 has been shown to be theoretically capable of approaching the Shannon capacity for multiple transmitters and receivers. (As is well-known to those of ordinary skill in the art, the Shannon capacity of a system refers to the information-theoretic capacity limit of the system.)

Another such MIMO system is described, for example, in U.S. Pat. No. 6,317,466, issued on Nov. 13, 2001 to G. Foschini et al., entitled “Wireless Communications System Having A Space-Time Architecture Employing Multi-Element Antennas At Both The Transmitter And Receiver” (hereinafter “Foschini et al.”). U.S. Pat. No. 6,317,466, which is also commonly assigned to the assignee of the present invention, is also hereby incorporated by reference as if fully set forth herein. The architecture described in Foschini et al, provides for a technique having a significantly lower computational complexity than that of U.S. Pat. No. 6,097,771, but which nonetheless can still achieve a substantial portion of the Shannon capacity.

Specifically, in the system of Foschini et al, a data stream is split into M uncorrelated sub-streams of symbols, each of which is transmitted by one of M transmitting antennas. The M sub-streams are picked up by N receiving antennas after having been perturbed by a channel matrix H. (The channel matrix H represents the signal interference or signal loss which naturally occurs as a result of the transmission channel.) The sub-stream signal with the highest signal-to-noise ratio is advantageously detected first and this involves the calculation of the pseudo-inverse of H or the calculation of a minimum mean-square error filter. The effect of the detected symbol as well as the effect of the corresponding transmission channel is then advantageously removed (mathematically) from the N received signals. This process repeats with the next strongest sub-stream signal among the remaining undetected signals. Thus, this approach detects M symbols (one from each of the M sub-streams) in M iterations. Moreover, it has been proven that this decoding order is optimal from a performance point of view. However, the computational complexity of the Foschini et al, technique is still reasonably high (albeit lower than that of U.S. Pat. No. 6,097,771).

This complexity problem was addressed, for example, in U.S. Pat. No. 6,600,796, issued on Jul. 29, 2003 to B. Hassibi, entitled “Method And Apparatus For Receiving Wireless Transmissions Using Multiple-Antenna Arrays” (hereinafter “Hassibi”). U.S. Pat. No. 6,600,796 is commonly assigned to the assignee of the present invention and is hereby incorporated by reference as if fully set forth herein. In Hassibi, it was recognized that mathematical matrix inversion operations are inherently costly (in computational complexity), and, making use of that recognition, an improved technique for detecting the M transmitted signals was disclosed. In particular, in the technique of Hassibi, as each transmitted symbol is detected the effect of the detected symbol and of the corresponding channel is advantageously subtracted from the N received signals without performing any mathematical matrix inversion operations.

Although the prior art techniques such as that of Foschini et al, and especially that of Hassibi have considerably reduced the computational complexity of the signal detection process for MIMO systems over the earlier techniques, their complexity nonetheless rises significantly as the number of antennas grow. That is, while reasonably efficient when used with a modest number of antennas, these techniques become more cumbersome particularly when the number of transmitting antennas becomes large (e.g., greater than 10). Therefore, an improved signal detection technique for MIMO systems, whose computational complexity does not increase as quickly with increasing numbers of antennas, would be highly desirable.

SUMMARY OF THE INVENTION

In accordance with the present invention, an improved signal detection technique for MIMO systems is embodied in a method and apparatus for detecting a plurality of transmitted signals with use of a plurality of receiving antennas. In particular, and in accordance with one illustrative embodiment of the present invention, an iterative procedure decodes one of a plurality of transmitted signals at each iteration using an intermediate matrix at each iteration to determine the transmitted signal to be decoded. The intermediate matrix for each successive iteration is advantageously computed in a recursive manner with use of a Schur complement operation performed based on the inverse of a modified version of the intermediate matrix used in the previous iteration. (A Schur complement is a well-known matrix operation fully familiar to those skilled in the art.)

More specifically, a method and apparatus for detecting a plurality of transmitted signals transmitted across a channel by respective transmitting antenna elements in a multiple-input multiple-output communications system is provided. The method, for example, comprises the steps of (a) collecting a plurality of received signals from respective receiving antenna elements in said communications system; (b) determining a channel matrix H of estimated channel coefficients based on said plurality of received signals; (c) computing an estimate of a selected one of said transmitted signals, said estimate based on said plurality of received signals and on an intermediate matrix Q, thereby resulting in detection of the selected one of said transmitted signals, wherein said intermediate matrix Q is a function of the channel matrix H; and (d) repeating at least step (c) one or more times to detect an additional one or more of said transmitted signals, wherein said intermediate matrix Q as used in step (c) for each such repeated execution thereof is re-computed based on a function of an inverse of a Schur complement of an element in the inverse of a modified version of the intermediate matrix Q used in the previous execution of step (c).

DETAILED DESCRIPTION

An Overview of an Illustrative MIMO System Architecture

FIG. 1shows an illustrative MIMO system architecture in which an illustrative embodiment of the present invention may be advantageously employed. As shown in the figure, a single data stream is transmitted across a wireless channel with use of a communications link comprising M transmitting antennas and N receiving antennas. The channel is advantageously presumed to be a Rayleigh channel, familiar to those of ordinary skill in the art. In particular, the channel is flat-fading (meaning that the signals are narrow-band), and it is presumed that there is no Inter-Symbol Interference (ISI) between adjacent symbols.

The figure shows transmitter10, channel14and receiver15. Specifically, demultiplexer11of transmitter10separates the data stream into M uncorrelated data sequences which are then modulated by modulators12-1through12-M respectively into M substreams, s1(k), . . , sM(k), each of which is then sent through corresponding transmitting antennas13-1through13-M. After traversing channel14(whose behavior is represented by the matrix H—see discussion ofFIG. 2, below), the transmitted signals are captured by receiver15, specifically comprising receiving antennas16-1through16-N and corresponding demodulators17-1through17-N, which produce intermediate signals x1(k) through xN(k), respectively. Then, in accordance with the principles of the present invention, signal detector18generates the recovered data stream from intermediate signals x1(k) through xN(k) in accordance, for example, with an illustrative embodiment of the present invention.

FIG. 2shows an illustration of a how the transmission channel for a MIMO system may be represented as a channel matrix in accordance with an illustrative embodiment of the present invention. In particular, the figure shows transmitting antennas13-1through13-M, used to transmit signals s1(k), s2(k), . . . , sM(k), respectively; channel matrix H representative of transmission channel14; and the portion of receiver15comprising receiving antennas16-1through16-N and corresponding demodulators17-1through17-N, which produce intermediate signals x1(k) through xN(k), respectively. Also shown are summation elements21-1through21-N which represent the (unavoidable) inclusion of noise signals w1(k) through wN(k), respectively, into the signals received by receiving antennas16-1through16-N.

Specifically, channel matrix H comprises channel vectors h:1through h:M, representing the channel's effect on each of the M transmitted signals, respectively. More specifically, channel vector h:1comprises channel matrix entries h11through hN1, representing the channel's effect on transmitted signal s1(k) at each of receiving antennas16-1through16-N, respectively; channel vector h:2comprises channel matrix entries h12through hN2, representing the channel's effect on transmitted signal s2(k) at each of receiving antennas16-1through16-N, respectively, . . . , and channel vector h:Mcomprises channel matrix entries h1Mthrough hNM, representing the channel's effect on transmitted signal sM(k) at each of receiving antennas16-1through16-N, respectively.

The operation of the illustrative MIMO architecture is described more formally, below. First, at the receivers, at sample time k, we have:

x⁡(k)=∑m=1M⁢⁢h:m⁢sm⁡(k)+w⁡(k)⁢=H⁢⁢s⁡(k)+w⁡(k),⁢k=1,2,…⁢⁢K,wherex⁡(k)=[x1⁡(k)x2⁡(k)LxN⁡(k)]T⁢=[h1:H⁢s⁡(k)+w1⁡(k)h2:H⁢s⁡(k)+w2⁡(k)LhN:H⁢s⁡(k)+wN⁡(k)]T(1)
is the N-dimensional received vector,

H=[h11h12Lh1⁢⁢Mh21h22Lh2⁢MMMOMhN1hN2LhNM]=[h:1h:2Lh.M]=[h1·Hh2:HMhN:H]
is an N×M complex matrix assumed to be constant for K symbol periods, vectors hn:and hmare respectively of length M and N,
s(k)=[s1(k)s2(k)L sM(k)]T
is the M-dimensional transmitted vector,
w(k)=[w1(k)w2(k)L wN(k)]T
is a zero-mean complex additive white Gaussian noise (AWGN) vector with covariance

Rww=E⁢{w⁡(k)⁢wH⁡(k)}=σw2⁢IN×N,(2)
andTandHdenote respectively the transpose and the conjugate transpose of a matrix or a vector, and where IN×Nrepresents the N×N identity matrix. (It will be assumed herein that the additive noise w(k) is independent both in time and space.)

The transmitted vector s(k) has a total power PT. This power is advantageously held constant regardless of the number of transmitting antennas M and corresponds to the trace of the covariance matrix of the transmitted vector:

PT=tr⁡[Rss]=Constant=∑m=1M⁢⁢σsm2.(3)
It will be assumed herein that all of the antennas transmit with the same power,
σs12=σs22=KσsM2=σs2,
such that
PT=Mσs2.  (4)

Now define a parameter ρ that relates PTand σw2as follows:

ρ=PTσw2(5)
This parameter advantageously corresponds to the average receive signal-to-noise ratio (SNR) per antenna.

Therefore, in accordance with the principles of the present invention, an original information sequence for wireless transmission is demultiplexed into M data sequences sm(k), m=1, . . . , M (called substreams), and each one of them is sent through a transmitting antenna. These M substreams are assumed to be uncorrelated, which implies that the covariance matrix of the transmitted vector s(k) is diagonal:

Rss=E⁢{s⁡(k)⁢sH⁡(k)}=σs2⁢IM×M(6)
Also assume that N≧M and that H has full column rank, i.e., that rank[H]=M.

Assume that the transmitter has no knowledge of the channel. In this case, the Shannon capacity of the (M, N) flat-faded channel is given by the following well-known formula, familiar to those skilled in the art:

C=log2⁡[det⁡(IN×N+ρM⁢HHH)]⁡[bps⁢/⁢Hz]=log2⁡[det⁡(IM×M+ρM⁢HH⁢H)].(7)
One important observation that can be made from Equation (7) is that, for rich scattering channels (meaning that the elements of the channel matrix are independent of one another), the MIMO channel capacity grows roughly proportionally to M.
An Illustrative MIMO Signal Detection Technique

In a MIMO system such as the illustrative system shown inFIG. 1, the detection of the transmitted symbols at the receivers typically comprises first determining the value of the complex channel matrix H, or, more precisely, an estimate thereof. Most typically, and as is well known to those skilled in the art, H is determined by having the transmitter send a training sequence (comprising a sequence of symbols which is known in advance by the receiver) at the beginning of each burst. (As is also well known, most communications systems separate the sequence of symbols to be transmitted into individual portions referred to as bursts.) The length of a burst is advantageously equal to K=K1+K2symbols where the K1symbols are used for training and the K2symbols are used for the transmission of the actual data information. The propagation coefficients (i.e., the values of the elements of the matrix H) may then be assumed to be constant during an entire burst (since it occurs over a reasonably short period of time), after which they may change to new independent random values which it is assumed they maintain for another K symbols, and so on. Note that no distinction will be made between H and its estimate herein.

Thus, given the channel matrix H and the set of received signals x(k) as described above, the goal of a MIMO system is to determine (or, more precisely, to estimate) the transmitted signals s(k). Note in particular that the transmitted signals s(k) have been coded (illustratively by modulators12-1through12-M ofFIG. 1) with use of a predetermined symbol constellation, and therefore, the estimates of the transmitted signals (which will be referred to herein as ŝ(k)=[ŝ1ŝ2LŝM]T) should fall into that constellation.

As is well know to those skilled in the art, one general approach to the problem of signal detection in MIMO systems comprises a Sequential Nulling and Cancellation (SNC) technique, such as is described, for example, in Foschini et al. An SNC technique typically consists of performing the following steps:

1. The M transmitted signals are first estimated by filtering the N received signals using, for example, either the well-known minimum mean-square-error (MMSE) filtering technique or the well-known zero forcing (ZF) technique. In either case, an estimation matrix (which we will identify herein as G) is the mathematical result.

2. Among the M estimated source signals, the substream with the smallest estimation variance or the strongest SNR is chosen for detection.

3. The interference contributed by the detected substream is cancelled (removed or “subtracted out”) from the N received signals.

4. Return to step 1 and repeat with the number of substreams advantageously reduced by one, until all substreams have been detected.

Thus, it can be seen that with M iterations, each of the substreams are advantageously detected.

FIG. 3shows an illustration of how an estimation matrix may be advantageously used to decode the plurality of transmitted signals in accordance with an illustrative embodiment of the present invention. Illustratively, the figure shows a MIMO environment with three transmitting antennas and four receiving antennas. (That is, in accordance with the description provided herein, M=3 and N=4.) Note that the use of an estimation matrix as shown inFIG. 3is a characteristic of both certain prior art MIMO signal detection schemes and certain MIMO signal detection schemes in accordance with various illustrative embodiments of the present invention.

As shown in the figure, estimation matrix31(G), which may be advantageously determined with use of either the MMSE technique or the ZF technique, is used to compute the estimates of the transmitted signals. In particular the signal vector y(k) is computed as y(k)=GHx(k) at each iteration of the SNC process. Then, as each substream, yi(k), for some i (which represents the substream with the smallest estimation variance or the strongest SNR) is detected in turn, the corresponding estimate of the transmitted signal may be advantageously determined by computing ŝi(k)=Q{yi(k)}, where Q[·] represents the quantization procedure according to the constellation being used. As further shown in the figure, substream2is illustratively detected first, followed by substream1and finally by substream3.

A Prior Art MIMO Signal Decoding Technique

FIG. 4shows a high-level flowchart of a sequential nulling and cancellation scheme for use in decoding the plurality of transmitted signals in accordance with a prior art technique. In this prior art technique, the estimation matrix (e.g., estimation matrix31, G, as shown inFIG. 3) is computed with use of either the pseudo-inverse of the channel matrix H or, preferably, the MMSE filter G. In particular, define an error vector signal at time k between the input s(k) and its estimate:

J=E⁢{eH⁡(k)⁢e⁡(k)}=tr⁡[E⁢{e⁡(k)⁢eH⁡(k)}].(9)
The minimization of Equation (9) leads to the Wiener-Hopf equation, familiar to those skilled in the art—namely:
GHRxx=Rsx,  (10)
where
Rxx=E{x(k)xH(k)}  (11)
is the output signal covariance matrix, and
Rsx=E{s(k)xH(k)}  (12)
is the cross-correlation matrix between the input and output signals.

From Equation (10), it can be seen that the MMSE filter is:
G=[HHH+αIN×N]−1H,(13)
where

α=σw2σs2.(14)
It can easily be seen that Equation (13) is equivalent to:

G=H⁡[HH⁢H+α⁢⁢IM×M]-1=HQ.(15)
The second form—that is, Equation (15)—is more useful and more efficient in practice since M≦N and the size of the matrix to be inverted in Equation (15) is either smaller than or equal in size to the matrix to be inverted in Equation (13).

Instead of the MMSE filter, a prior art SNC technique can alternatively directly use the pseudo-inverse (familiar to those skilled in the art) of H, which is:
GPIH=[HHH]−1HH(16)
As can easily be seen, the only difference between the matrices G and GPIis that the G is “regularized” by a diagonal matrix αIM×Mwhile GPIis not. This regularization introduces a bias but Equation (15) actually gives a much more reliable result than Equation (16) when the matrix HHH is ill-conditioned and the estimation of the channel is noisy. In practice, depending on the condition number of the matrix HHH, a different value may be used for α than the one given in Equation (14). For example, if this condition number is very high and the SNR is also high, it will be better to take a higher value for α. Thus, the MMSE filter can be seen as a biased pseudo-inverse of H.

More specifically, in the prior art SNC algorithm being described, the detection of the symbols sm(k) is performed over M iterations. Note that the order in which the components of s(k) are detected is important to the overall performance of the system. Let the ordered set
S={p1,p2,L,pM}  (17)
be a permutation of the integers 1, 2, . . . , M specifying the order in which components of the transmitted symbol vector s(k) are extracted.

Thus, returning toFIG. 4, the illustrative prior art SNC technique comprises the following steps.

Initialization Step (as Shown in Block40of the Figure):

Use the training sequence to determine the matrix H and set the initial matrix HM=H for the first iteration. Also, determine the received signals x(k) and set x1(k)=x(k) for the first iteration.

Step1(as Shown in Block41of the Figure):

Using the MMSE filter or the pseudo-inverse, compute:
y(k)=GHx(k).  (18)
(Note that y(k) represents the estimates of the transmitted signals.) In particular, if using the MMSE filter approach, matrix G is computed by Equation (13) above; is using the zero-forcing approach, matrix G is computed by Equation (16) above. In addition, x(k) and H are determined in the initialization sequence for the first iteration, and as described below (with reference to block44of the figure) for all subsequent iterations.
Step2(as Shown in Block42of the Figure):

The element of y(k) with the highest SNR is detected. This element is associated with the smallest diagonal entry of Q for the MMSE filter (as will be more clearly explained below), or the column of G having the smallest norm for the pseudo-inverse (in the case where zero-forcing has been used). For example, if such a column determined in the m'th iteration is pm, then the estimate of the chosen transmitted signal is given by:
ŝpm(k)=Q[ypm(k)],  (19)
with Q[·] indicating the slicing or quantization procedure in accordance with the given symbol constellation in use.
Step3(as Shown in Block43of the Figure):

Assuming that ŝpm(k)=spm(k), then spm(k) is cancelled from the received vector x(k), resulting in a modified received vector, namely:

x2⁡(k)=x⁡(k)-sp1⁡(k)⁢h:p1=∑m≠p1⁢h:m⁢sm⁡(k)+w⁡(k)=HM-1⁢sM-1⁡(k)+w⁡(k),(20)
where HM−1is an N×(M−1) matrix derived from H by removing its p1'th column and sM−1(k) is a vector of length M−1 obtained from s(k) by removing its p1'th component.
Step4(as Shown in Block44of the Figure):

Unless all M transmitted signals have already been decoded, steps1–3are repeated for components p2,L,pMby operating in turn on the progression of modified received vectors x2(k),L,xM(k). Note that at the m'th iteration, the N×(M−m) matrix HM−mmay be derived from H by removing m of its columns—namely, columns p1,L,pm. It is well known that this ordering (i.e., choosing the transmitted signal having the highest SNR at each iteration in the detection process) is optimal among all possible orderings.

The following more formally summarizes the illustrative prior art SNC algorithm (using the MMSE filter):

yp1⁡(k)=qM,:l1H⁢HMH⁢x1⁡(k)Move the l1'th entry of vector f(k) to the end
ŝp1(k)=Q[yp1(k)]
Recursion (i.e., Subsequent Iterations), for m=1,2,K,M−1:(a) xm+1(k)=xm(k)−ŝpm(k)hM,:lm(b) Determine HM−mby removing the lm'th column of HM−m+1

(e)⁢⁢ypm+1⁡(k)=qM-m,lm+1H⁢HM-mH⁢xm+1⁡(k)(f) Move the l1'th entry of vector f(k) to the position behind the (M−m)'th entry(g) ŝpm+1(k)=Q[ypm+1(k)]
Solutions:

The estimates of the transmitted signals: [ŝp1(k) ŝp2(k)LŝpM(k)]T

An Illustrative MIMO Decoding Technique According to the Present Invention

FIG. 5shows a flow chart of a sequential nulling and cancellation scheme for use in decoding the plurality of transmitted signals in accordance with an illustrative embodiment of the present invention. In accordance with the illustrative embodiment of the present invention, the matrix G is advantageously computed indirectly (rather than directly). Specifically, recall that:
G=HR−1(21)
where
R=HHH+αIM×M(22)

The covariance matrix of the error signal, e(k)=s(k)−y(k), is:

Rcc=E⁢{e⁡(k)⁢eH⁡(k)}=σw2⁢R-1=σ2w⁢Q.(23)
Clearly, the element of y(k) with the highest SNR is the one with the smallest error variance, so that:

p1=arg⁢⁢minm⁢⁢qmm,(24)
where qmmare the diagonal elements of the matrix Q=R−1.

The matrix R can be rewritten as follows:

R=∑n=1N⁢hn⁢hn:H+α⁢⁢IM×M,(25)
which means that R can be advantageously computed recursively in N iterations, as follows:

R[l]=∑n=1l⁢hn*⁢hn:H+α⁢⁢IM×M=R[l-1]+hl:⁢hl:H(26)
and
R[N]=R, R[0]=αIM×M(27)
Using the Sherman-Morrison formula (a well-known mathematical transformation fully familiar to those skilled in the art and also known as the “second lemma inversion”), Q can also be computed recursively, as follows:

Q[l]=Q[l-1]-Q[l-1]⁢hl:⁢hl:H⁢Q[l-1]1+hl:H⁢Q[l-1]⁢hl:.(28)
With the initialization

Q[0]=1α⁢IM×M,
we obtain
Q[N]=[HHH+αIM×M]−1.
Note that if the process begins at iteration M+1 with the initialization

Note that it is well known that the computation of any recursion introduces potential numerical instabilities because of the finite precision of processor units. This instability however occurs only after a very large number of iterations. Since, in this case, the number of iterations to compute Q is limited by the number of receiving antennas N, such numerical instabilities are unlikely to occur. In any event, the numerical stability can be advantageously improved by increasing the value of α at the time of initialization.

Note also that in accordance with the illustrative embodiment of the present invention as described herein, Equation (28) is advantageously computed only one time at the first iteration. Once Q[N]is computed, p1may be easily determined based on Equation (24) above.

Now, continuing the illustrative process for the first iteration, the input estimate may be computed as follows:

yp1⁡(k)=∑m=1M⁢qp1,m⁢h.⁢mH⁢x⁡(k)(29)
and
ŝp1(k)=Q[yp1(k)].  (30)
Note that the last step of the illustrative decoding procedure in accordance with the present invention is the same as the last step (“Step3”) of the prior art approach described above.

For each of the following iterations (after the first), the process is as follows. First, note that the matrix Q can advantageously be deflated recursively. Specifically, note that:

Pp1⁢M=[10LL00OMM10L1M10OM100L010L00↑↑p1M]M×M.
Since
(HPp1M)H(HPp1M)+αIM×M=Pp1M(HHH+αIM×M)Pp1M,  (32)
it follows that the rows and columns p1and M of the matrix R may be advantageously permuted. Equivalently, in accordance with an alternative illustrative embodiment of the present invention, the rows and columns p1and M of the matrix Q may be permuted. This can be easily seen from the fact that
(Pp1MRPp1M)−1=Pp1MR−1Pp1M=Pp1MQPp1M.  (33)
Note that these permutations advantageously allow for the removal of the effect of the channel h:p1quite easily. Specifically:

It can easily be shown that:

A=[EFGH],
then the Schur complement, S, of (partition) H in (matrix) A is S=E−FH−1G.

Furthermore, from Equation (36), it may be deduced that:

RM-1-1=QM-1=[TM-1+vM-1⁢vM-1H/βp1]-1(37)
and using the (well-known) Sherman-Morrison formula, we obtain

QM-1=TM-1-1-TM-1-1⁢vM-1⁢vM-1H⁢TM-1-1βp1+vM-1H⁢TM-1-1⁢vM-1.(38)
Clearly, Equation (38) shows that the matrix Q can be advantageously deflated recursively. Specifically, in the general case:

QM-m=TM-m-1-TM-m-1⁢vM-m⁢vM-mH⁢TM-m-1βpm+vM-mH⁢TM-m-1⁢vM-m(39)RM-m=HM-mH⁢HH-m+α⁢⁢I(M-m)×(M-m)(40)
Note that RM−mmay advantageously be easily determined without direct computation. In particular, and in accordance with the illustrative embodiment of the present invention, RM−mis determined from RM+1−mby removing the last line and column thereof—only RM=R is calculated directly, during the first iteration of the illustrative procedure.

Thus, returning toFIG. 5, the illustrative MIMO decoding technique in accordance with an illustrative embodiment of the present invention comprises the following steps.

Initialization Step (as Shown in Block50of the Figure):

Use the training sequence to determine the initial matrix H; determine the received signals x(k) and set x1(k)=x(k) for the first iteration.

Step1(as shown in Block51of the Figure):

Step2(as Shown in Block52of the Figure):

The element of y(k) with the highest SNR is detected. This element is associated with the smallest diagonal entry of Q for the MMSE filter. If such a column is pm, then the estimate of the chosen transmitted signal is given by:
ŝpm(k)=Q[ypm(k)].
Step3(as Shown in Block53of the Figure):

Assuming that ŝpm(k)=spm(k), then spm(k) is cancelled from the received vector xm(k), resulting in a modified received vector, xm+1(k).

Step4(as Shown in Block54of the Figure):

Assuming that lmis the index of the chosen transmitted signal, the rows and columns lmand M−m+1 (the last) are permuted in both RM+1−mand QM+1−m.

Step5(as Shown in Block55of the Figure):

Partition matrix RM+1−mto determine RM−m, vM−m, and βpmand remove the last row and column of QM+1−mto determine TM−m−1.

Step6(as Shown in Block56of the Figure):

Step7(as Shown in Block57of the Figure):

Unless all M transmitted signals have already been decoded, repeat steps2–6for components p2,L,pMby operating in turn on the progression of modified received vectors x2(k),L,xM(k).

The following more formally summarizes the illustrative MIMO decoding technique in accordance with the illustrative embodiment of the present invention as presented herein:

Initialization and First Iteration:

yp1⁡(k)=∑i=1M⁢⁢qM,l1⁢t⁢h:tH⁢x1⁡(k)Interchange the entries l1and M of the vector f(k)ŝp1(k)=Q[yp1(k)]
Recursion (Subsequent Iterations), for m=1, 2, K, M−1:(a) xm+1(k)=xm(k)−ŝpm(k)hpm(b) Permute the rows and columns lmand M−m+1 of RM+1−m(c) Permute the rows and columns lmand M−m+1 of QM+1−m(d) Determine RM−m, vM−m, and βpmfrom RM+1−mas shown above(e) Determine

(h)⁢⁢ypm+1⁡(k)=∑i=1M-m⁢⁢qM-m,lm+1l⁢h:fi⁡(k)H⁢xm+1⁡(k)(i) Interchange the entries lm+1and M−m of the vector f(k)(j) ŝpm+1(k)=Q [ypm+1(k)]
Solutions:The estimates of the transmitted signals: [ŝp1(k)ŝp2(k)LŝpM(k)]TThe decoding order: f(k)=[pMpM−1Lp1]T
Addendum to the Detailed Description

Thus, for example, it will be appreciated by those skilled in the art that the block diagrams herein represent conceptual views of illustrative circuitry embodying the principles of the invention. Similarly, it will be appreciated that any flow charts, flow diagrams, state transition diagrams, pseudocode, and the like represent various processes which may be substantially represented in computer readable medium and so executed by a computer or processor, whether or not such computer or processor is explicitly shown. Thus, the blocks shown, for example, in such flowcharts may be understood as potentially representing physical elements, which may, for example, be expressed in the instant claims as means for specifying particular functions such as are described in the flowchart blocks. Moreover, such flowchart blocks may also be understood as representing physical signals or stored physical data, which may, for example, be comprised in such aforementioned computer readable medium such as disc or semiconductor storage devices.