00502331

I184 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 5 , MAY 1996

Blind Separation of Synchronous CO-Channel Digital Signals Using an Antenna Array-Part I: Algorithms

Shilpa Talwar, Student Member, IEEE, Mats Viberg, Member, IEEE, and Arogyaswami Paulraj, Fellow, IEEE

Abstruct- We propose a maximum-likelihood (ML) approach for separating and estimating multiple synchronous digital signals arriving at an antenna array. The spatial response of the array is assumed to be known imprecisely or unknown. We exploit the finite alphabet property of digital signals to simultaneously estimate the array response and the symbol sequence for each signal. Uniqueness of the estimates is established for signals with linear modulation formats. We introduce a signal detection technique based on the finite alphabet property that is different from a standard linear combiner. Computationally efficient algorithms for both block and recursive estimation of the signals are presented. This new approach is applicable to an unknown array geometry and propagation environment, which is particularly useful in wireless communication systems. Simulation results demonstrate its promising performance.

I. INTRODUCTION

IRELESS communication systems are witnessing rapid advances in volume and range of services. A major

challenge for these systems today is the limited radio frequency spectrum available. Approaches that increase spectrum efficiency are therefore of great interest. One promising approach is to use antenna arrays at cell sites. Array processing techniques can then be used to receive and transmit multiple signals that are separated in space. Hence, multiple co-channel users can be supported per cell to increase capacity. In this paper, we study the problem of separating multiple synchronous digital signals received at an antenna array [l] . The goal is to reliably demodulate each signal in the presence of other co-channel signals and noise. The complementary problem of transmitting to multiple receivers with minimum interference at each receiver has been studied in [2]-[4].

Several algorithms have been proposed in the array processing literature for separating co-channel signals based on availability of prior spatial or temporal information. The traditional spatial algorithms combine high resolution direction- finding techniques such as MUSIC and ESPRIT [5], [6] with optimum beamforming to estimate the signal waveforms [7], [8]. However, these algorithms require that the number of

Manuscript received December 7, 1994; revised October 16, 1995. The work of S. Talwar was suppoited by the Computational Science Graduate Fellowship Program of the Office of Scientific Computing, U.S. Department of Energy. The associate editor coordinating the review of this paper and approving it for publication was Prof. Michael D. Zoltowski.

S. Talwar is with the Scientific Computing and Computational Mathematics Program, Stanford Univerity, Stanford, CA 94305 USA.

M. Viberg is with the Department of Applied Electronics, Chalmers University of Technology, S-41296 Gothenberg, Sweden.

A. Paulraj is with the Information Systems Laboratory, Stanford University. Stanford, CA 94305 USA.

Publisher Item Identifier S 1053-587X(96)03072-3.

signal wavefronts including multipath reflections be less than the number of sensors, which restricts their applicability in a wireless setting. In the recent past, several property-restoral techniques have been developed that exploit the temporal structure of communication signals while assuming no prior spatial knowledge. These techniques take advantage of signal properties such as constant modulus (CM) [9], discrete alphabet [lo], [ l l ] , self-coherence [12], and high-order statistical properties [13], [14]. In this paper, we propose a new property- restoral approach that takes advantage of the finite alphabet (FA) property of digital signals.

Our approach is termed blind since it does not require any training signals for signal demodulation. This is particularly useful in situations where training signals are not available. For example, in communications intelligence, training signals are not accessible. In cellular applications, blind algorithms can be used to reject interference from adjacent cells. In IS- 54, for example, adjacent cell interference appears only over a partial burst and training signals do not help. Blind algorithms are also bandwidth-efficient due to the elimination of training sets. Moreover, the study of blind algorithms can be used to complement existing non-blind techniques. For example, as a result of our investigation of uniqueness for the blind problem, we propose a minimal set of training signals that can be used in a non-blind multi-user scenario. An important advantage of our approach is that in non-blind scenarios, training sets can easily be incorporated to initialize our algorithms.

The algorithms presented herein can be used to demodulate multiple synchronous digital signals in a coherent multipath environment. To guarantee unique signal estimates, we assume that the number of signals does not exceed the number of sensors, and that the channel is constant over a sufficient number of snapshots. The synchronous assumption is reasonable in microcell-air interfaces where symbol timing can be effec- tively controlled. Extension of our approach to asynchronous transmission and delay spread channels is relatively straight- forward, the main differences in these scenarios being (i) instead of matched filtering, the array output is oversampled; and (ii) the signal estimation step is replaced by maximum- likelihood sequence estimation (MLSE) since the channel is no longer memoryless. Recently, some non-ML techniques have also been proposed that use subspace information to first synchronize the signals and remove intersymbol interference, and then use one of the synchronous FA algorithms presented herein to separate the signals [15]-[17].

The outline of this paper is as follows. We introduce the data model in Section 11. In Section 111, we consider the problem

1053-587)3/96$05.00 0 1996 IEEE

TALWAR et ul : SYNCHRONOUS CO-CHANNEL DIGITAL SIGNALS-PART I 1185

of uniquely identifying the signals when the array response structure is unknown. In Section IV, the ML estimator for the array responses and symbol sequences is discussed. Two efficient block algorithms are presented in Section V, and their convergence is analyzed. Recursive extensions of these algorithms are discussed in Section VI. In Section VII, we present simulation results to demonstrate the performance of these algorithms. Finally, we conclude with directions for future work in Section VIII.

11. PROBLEM FORMULATION

Consider d narrowband signals impinging at an array of m, sensors with arbitrary characteristics. The signal waveform received at each sensor is demodulated with respect to the carrier frequency (assuming perfect carrier phase lock recovery). The m x 1 vector of sensor outputs, x(t), in the absence of multipath, is given by

d

x(t) = X P k a ( B k ) s k ( t ) + v(t ) (1) k = l

where l ) k is the amplitude of the kth signal, .(e,) is the array response vector to a signal from direction Oh. s k ( . ) is the kth signal waveform, and V( .) is additive white noise with covariance n21.

In a realistic communication scenario, however, there are multiple reflected and diffracted paths from the source to the array. These paths arrive from different angles and with different attenuations and time delays. The array output becomes

k = l 1 = 1

where q k is number of subpaths for the kth signal, and a k l and q~ are, respectively, the attenuation and time delay corresponding to lth subpath. We assume that the propagation delays associated with these paths is much smaller than the inverse bandwidth of the signals. The delays can thus be modeled as phase-shifts under the narrowband assumption. The new data model becomes

d

x ( t ) = Cpkaksk ( t ) + ~ ( t ) (2)

where a k is now the total array response vector a k = cvk le - JwrT~’a (Ok , ) , and w, is the carrier frequency. The

spatial structure of the array response vector a k cannot be exploited if the number of paths is larger than the number of sensors. However, we can exploit the temporal structure of digital signals with memoryless linear modulation formats

k = l

N

where N is the number of symbols in a data batch (burst), {bk(.)) is the symbol sequence of the kth user, T is the symbol period, and 9(.) is the symbol waveform. We let 9 ( . ) be a square-root raised cosine waveform so that matched filtering yields a pulse that satisfies the Nyquist criterion.

For simplicity, we assume that the symbols belong to the alphabet R = (k1, &3, . . . , f ( L - 1)) for real signals, and fl = {&1,&3, . . . ,&(L-I)j~B{ztj,*,j3.. . . . & j ( L - 1 ) } for complex signals. These correspond to the important cases of PAM and QAM modulation formats.

Now, assuming that the signals are symbol-synchronous, we perform matched filtering on (2), and sample the filtered array output at symbol rate. This yields the following equivalent discrete representation of the data:

d

X(.) = C P k a k b k ( n ) + v(n). ( 3 )

The noise term ~ ( n ) remains white, and it is easily seen that the output of the matched filter is a sufficient statistic for determining the transmitted symbols [18]. We can rewrite (3) in matrix form

(4)

k=l

~ ( n ) = As(n) + ~ ( n )

where x(n) is the filtered data, s(n) = [bl(n) . . . b d ( n ) l T , v(n) is additive white noise, and A is an m x d matrix of array responses scaled by the signal amplitudes A = hlal . . .pdad].

Assuming that the channel is constant over the available N symbol periods, we obtain the following block formulation of the data

X(N) = AS(N) + V ( N ) ( 5 )

where X(N) = [x ( l ) . . . x (N) ] ,S (N) = [ s ( l ) . . . s (N) ] , and V ( N ) = [v ( l ) . . .v(N)]. The matrix A represents the spatial structure of the data, and the matrix S represents its temporal structure. The problem addressed in this paper is the combined estimation of the array response matrix A and the symbol matrix S ( N ) , given the array output X ( N ) . We assume that the number of signals is known or has been estimated [19]. For notational convenience, we denote X G X ( N ) and S S ( N ) , from here on.

111. IDENTIFIABILITY

Before discussing the estimation problem, we consider the problem of uniqueness of signal estimates in the absence of noise. This problem can be viewed as a nonlinear factorization of the data matrix X m x ~ into factors A m x d and S d x ~ , such that X = AS. In the case that columns of matrix A lie on the array manifold (defined as the set {.(e): 0 E [ 0 , 2 ~ ] } ) and S is an an arbitrary full-rank matrix, it is well known that this factorization is “unique” provided (i) any set of m vectors from the array manifold is linearly independent; and (ii) d < m [20]. There is an ordering ambiguity in the signal estimates, since

x AS = A P ~ P S = A s where A and S is also a valid solution pair for any permutation matrix P.

Our problem is the opposite of the standard problem. We assume that A is an arbitrary full-rank matrix, but the elements of S belong to a finite alphabet R. We first consider the

1186 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 5, MAY 1996

case of binary antipodal signals, i.e., R = {*1}, and then generalize to larger alphabets. As before, the solution to the nonlinear system of equations X = AS is not unique under our assumptions. For example, A = AT and S = T-lS is a solution for any non-singular matrix Tdxd that is a diagonal with f 1 entries or a permutation matrix or a product of the two. In this case, there is an additional ambiguity in the sign

Dropping the superscripts in (11), we have the following equivalent system of equations:

of the estimated signals, beyond the usual ordering ambiguity. However, the sign ambiguity can be easily removed once the signals have been estimated by appropriate decoding of each symbol sequence. Hence, the sign and ordering ambiguities do not present a serious problem as the correct signals are still received. We define the system X = AS to be identifiable if all simultaneous solutions can be written as AT and T-lS, where T is a non-singular matrix with exactly one non-zero element (+ 1 or - 1) in each row and column. For convenience, we call such a matrix an admissible transform matrix (ATM). The following theorem gives a sufJicient condition for the identifiability of binary signals.

Theorem 3.1: Let X = AS where Amxd is an arbitrary full-rank matrix with d 5 m, and S d x ~ is a full-rank matrix with i 1 elements. If the columns of S include all the 2d-1 possible distinct (up to a sign) d-vectors with f l elements, then A and S can be uniquely identified up to a matrix T with exactly one non-zero element { + 1 - l} in each row and column.

Pro03 It is easily seen that the condition on S also implies that it is full rank. Suppose there exists another pair, A and S, both full rank, such that X = AS. Then

X = AS = AS. (6)

Solving for A, we get

A = ASSt (7)

where ( .) t denotes the pseudo-inverse. We multiply (7) by S to obtain

AS = ASStS

and together with (6), this implies A(S - SStS) = 0. Since A is full rank, we must have

s = SStS. (8)

Note StS is a projection matrix that projects the rows of S onto the row space of S. Thus, from (8), we see that S and S must share a common row space. That is

S = TS. (9)

Now, it remains to show that T is an ATM. We normalize the first element of each column of S to +l

by postmultiplying (9) by a diagonal matrix D with diagonal elements D,, = SI, ( j = 1 . . . N )

(10) SD = TSD j S(l) = TS(l).

Next, we postmultiply the resulting system (10) by a N x N permutation matrix that reorders the columns of S(l) to make

where S is an arbitrary full-rank matrix of il elements, and S is a normalized matrix with the first n = 2d-1 columns distinct. We can partition S = [S, S,], where S, is the submatrix of n distinct columns and S, is the submatrix of remaining columns. Note that the columns in S , are repeated from the first n, and thus provide no extra information in determining T. Therefore, we consider only the equations defined by S,. Each row t of T must satisfy t.TS, = S z , where S: is a subvector of the corresponding row of and can be of any one of the 2" possible n-vectors. More explicitly, for tT = [ t d . . tzt l] , we have a system of equations of the type

t d + ( i d - 1 + . . . + t 2 + t l ) = i 1 t d - (td-1 + ' ' ' + t 2 + t l ) =

t d + ( t d - 1 + . ' ' + t 2 - t l ) = It1 t d - (td-1 + . . . + t 2 - t l ) = fl.

In the Appendix, we show by induction that the solution of this system has the form t; = kl for some i and tj = 0 for j = 1: . . . d ; j # i. This implies that each row of S is a multiple of a row of S. Since S has rank d , all rows of S, up to a sign, are included in S. Therefore, the rows of T are distinct, and it is an ATM. 0

The identifiability of larger alphabets follows easily from the binary case. Consider the alphabet R of length L with symbols {fl, 1 3 ; . . . f(L - I)}. We assume that the columns of S include all the 1 = Ld/2 possible distinct (up to a sign) d- vectors. In particular, they include n = 2d-1 distinct vectors of il elements and n = 2d-1 distinct vectors of k ( L - I) elements. As before (without loss of generality), we can normalize each column of S such that the first symbol is positive, and permute the columns so that we can partition S = [S, (L - l)S, S,], where S, is a submatrix of n distinct vectors with fl elements and S, is the remaining submatrix. Each row t of T again satisfies

tTS = tT[S, (L - 1)s" S,] = ST (12)

where ST is the corresponding row of S, and is partitioned likewise as ST = [Sr, Szn ST]. Although there are 1 distinct equations defined by the columns of S, the subset of equations determined by the first 2n columns alone results in a trivial solution for t, i.e., only one non-zero element. This follows from (12)

tT s, = s;, (13) tT ( L - l)S, = ST" (14)

;he first 2d-1 columns distinct which implies (L - l)ST,, = "'. Since the elements of both SI,, and %2,% belong to the alphabet Ci, the only possibilty for S(l)p = TS(1)p j S(2) = TS(2). (11)

TALWAR et al.: SYNCHRONOUS CO-CHANNEL DIGITAL SIGNALS-PART I 1 I87

the entries of SI.^ are f l s . Thus, from (13), we see that the problem is reduced to one of identifiability of binary signals. For this problem, we have shown previously that t is a trivial solution, and consequently, T is an ATM.

Finally, the generalization to complex signals is straightfor-

are independent, and the probability of receiving each symbol in R is equal, p is bounded by

(16)

ward. In this case, we see that the-signals can be identified uniquely up to a factor { + 1 ~ - 1, +j , - j } . As before, we have the system of equations S = T S , where S and S are now complex matrices with elements in R = {fl , 5 3 , . . . f ( L - 1)) 69 {ij, f j 3 , . . . , f j ( L - 1)). We are interested in finding a condition on the columns of S such that T is an ATM. For complex signals, we extend the definition of an ATM to a non- singular matrix with one non-zero element{ & 1, kj ), in each row and column.

We begin by noting that multiplication of complex matrices is isomorphic to multiplication of real matrices with twice the dimensions [21]. In particular, we have

1 Rc{S) -Im{S} R.e{T} -Im{T) [Im{S} Rc{S}] = [Im{T} Re{T}

Re{S} -Im{S} [Im{S} Re{S}

and by denoting each of the real block matrices above with a subscript ( .)n, we get S R = TRSR. The matrix of received signals Sn now has dimensions d x 2N, for n! = 2d. If- the first N columns of S R include all the 1 = Ld/2 possible distinct vectors, then there is exactly one non-zero element, a + 1 or a - 1, in each of the d rows of the submatrix [Re{T} - Im{T}]. Furthermore, each row of this submatrix must be distinct since S is full rank, and thus, T = Rc(T} +j Im{T} is an ATM. It is important to note here that the identifiability of d complex signals is equivalent to that of d = 2d real signals. We can summarize the above discussion by the following theorem.

Theorem 3.2: Let X = AS where A,n,xd is an arbitrary full-rank matrix with d 5 m, and S ~ ~ x l ~ ~ is a full-rank matrix with elements in R.

1) Real Case: If the columns of S include all the L d / 2 possible distinct (up to a sign) d-vectors with elements in fi = {*I, f 3 , . . . f ( L - I)}, then A and S can be uniquely identified up to a matrix T with exactly one non-zero element, { + l ) -l}, in each row and column.

2) Complex Case: If the columns of S include all the L2d /2 possible distinct (up to a sign) d-vectors with elements in

I)}, then A and S can be uniquely identified up to a matrix T with exactly one non-zero element, {+l. -1, + j , - , j } , in each row and column.

In Theorem 3.2, we give a sufficient condition for identifiability of signals that belong to a finite alphabet 0. Now, we show that if N is sufficiently large, the probability of achieving identifiability approaches I. We consider the case with real signals, noting that the following result also holds for complex signals, with d replaced by d.

Theorem 3.3: Let p denote the probability of receiving all the Ld/2 distinct (up to a sign) columns of S in N independent snapshots, N >> Ld/2 . Assuming that the signals

, i ( L - 1)) {f,j; f , j 3 , . . . , k j ( L -

Proof See the Appendix A.2. U We see from (16) that for a fixed value of d, as N increases,

p approaches 1. The probability q = 1 - p that one of the Ld/2 distinct vectors is not picked can then be bounded by

Hence, the probability of missing one of the distinct vectors approaches zero exponentially fast, the rate of decay depending on the ratio

Nevertheless, for large values of d, the number of snapshots required for identifiability seems quite large. But the condition requiring Ld/2 distinct vectors is only sufficient, and far from being necessary. In fact, we see in Appendix A.3 that if S contains a specific set of d + 1 columns for R = {fl}, it can be determined uniquely up to an ATM. Although this result remains to be generalized, it shows that in practice identifiability can be achieved with far fewer snapshots.

( L d / 2 ) '

IV. MAXIMUM-LIKELIHOOD ESTIMATION

In this section, we consider the problem of estimating the digital signals in the presence of noise. From Section 11, we see that the signals can be modeled as unknown deterministic sequences corrupted by white Gaussian noise: ~ ( n ) = As(n) + ~ ( n ) where €[v(n)v(k)*] = 0 2 1 S n k . The maximum-likelihood (ML) estimator yields the following separable least-squares minimization problem:

in the variables A and S ( N ) , which are, respectively, continuous and discrete. We assume N is large enough to ensure unique signal and array response estimates.

It is proved in [22] that the minimization can be carried out in two steps. First, we minimize (17) with respect to A since it is unconstrained

A = X ( N ) S ( N ) t = x(N)s(N)*(s(N)s(N)*)-l

Then, substituting A back into (17), we obtain a new criterion, which is a function of S ( N ) only as follows:

where P&Ar)- = IN - S(N)*(S(N)S(N)*)-'S(N). The global minimum of (1 8) can be obtained by enumerating over all possible choices of S ( N ) . However, this search has an exponential complexity in the number of symbols N and the number of signals d (2 for complex signals), and can be computationally prohibitive, even for modest-size problems. In the next section, we consider two iterative block algorithms that have a lower computational complexity.


V. BLOCK ALGORITHMS

The block algorithms ILSP and ILSE were introduced in [1]. These algorithms take advantage of the ML estimator in (17) being separable in the variables A and S. The ML criterion is minimized with respect to the two variables using an alternating minimizations procedure. The idea is to visit the received data iteratively until a best fit with the channel (array response) and signal model is obtained. The ILSP algorithm performs well with no prior estimate of the array responses, and can be used to initialize ILSE. For a sufficiently good initialization, the ILSE algorithm converges rapidly to the ML estimate of the array responses and signal symbol sequences. Apart from their efficiency, the algorithms are naturally paral- lelizable, and can be easily extended for recursive estimation (see Section VI).

A. ILSP Algorithm

We begin by assuming that f ( A , S; X) = IIX - AS/I; in (17) is a function of unstructured continuous matrix variables A and S. Given an initial estimate A of A, the minimization of f (A , S; X) with respect to continuous S is a least-squares problem. Each element of the solution S is projected to its closest discrete values, say S.AThen a better estimate of A is obtained by minimizing f ( A , S; X ) with respect to A, keeping S fixed. This isAagain a*least-squares problem. We continue this process until A and S converge.

Iterative Least-Squares with Projection (ILSP): 1) Given Ao, k = 0 2) k = k + l

e S k = ( A ; - ~ A ~ - ~ ) - ' A ; - ~ X * S k = proj[Sk]

Ak = XS;(SkS;)-' 3) Repeat 2 until (Ak, S k ) = (Ak-1; Sk-1). In the above description of the algorithm, proj[.] implies

projection onto a discrete alphabet. However, this definition may be extended to projection onto a constant modulus or any other signal characteristic. Moreover, in scenarios where the array manifold structure is applicable, A k can be projected onto the manifold at each iteration. Hence, the ILSP algorithm outlines a general approach for imposing known structure on variables A and S in a minimization criterion of the form / / X - ASll$. The main advantage of the algorithm is its low computational complexity. At each iteration, two least- square problems are solved, each requiring U ( N m d ) flops for N >> m. In particular, ~ m d + 2 d 2 ( ~ - $1 + rnd2 flops are required to solve for A, and Nmd+2d2(m- $)+Nd2 flops to solve for S. Thus, the algorithm's complexity is polynomial in N and d.

Real Signals: If the signals belong to a real alphabet, we take advantage of this fact by constraining the imaginary part of S to be zero in each step, and thereby reducing the number of unknowns by half. Equivalently, we can minimize a slightly modified criterion f ( A n , S; X,) by noting that

IIX - ASI/$ = II[Rc{X) - Re{A}S] +j[Im{X} - Im{A)S]/l;

= IIRe{X} - Re{A}SII$ + llIm{X} - Im{A}SII$

= IIXR - ARSII$

where XR and AR are real augmented matrices as shown above. Thus, the algorithm proceeds as before, the only difference being that for real signals, we consider augmented matrices and replace (.)* by (.)'.

Initialization: A common initialization strategy in optimization for nonlinear problems with mixed discrete and continuous variables is to use the solution of the continuous problem as an estimate for the mixed problem [23]. The continuous solution for the ML criterion in (18) is Vd, the right singular vectors corresponding to the largest d singular values of of X . Hence, we can initialize ILSP with So = proj[Vd]. It is easily seen that this is equivalent to initializing with A0 = Ud, the largest d left singular vectors of X . Since it is expensive to compute the singular value decomposition (SVD), we initialize instead with Xd, the first d rows of X (equivalently A0 = Imx& which have the same rowspan as Vd in the absence of noise. This approximation works well if all signals have nearly equal powers. If some signals are much stronger than the others, then So = proj(Xd) may be rank deficient. In this case, we add a small diagonal perturbation matrix to So in order to restore its rank. In general, if S k or Ak becomes rank deficient at any iteration, we have found this simple strategy of perturbing the diagonal to be quite effective in extracting all the signals in difficult scenarios. One may also use the sofi-orthogonalization strategy proposed in [9] for the multiple-source CM problem.

The availability of prior information to initialize the ILSP algorithm can improve the possibility of global convergence, and help reduce the number of iterations. For example, in scenarios where imprecise knowledge of the spatial structure of the array is available, estimates of the array responses obtained from traditional techniques, such as MUSIC or ESPRIT, can be used to construct Ao. Alternatively, if the users initially transmit a short set of training signals, such as the sequence of d symbols that define the matrix Sd given in (56) (see the Appendix), then A0 can be estimated from the received data by Ao = XS;'. The advantage of using this particular training set is that Si', given in (59), is easily computed and is sparse. Although the algorithm is no longer blind, revisiting the data iteratively can significantly improve channel and signal estimates in situations with low SNR's.

Comparison with Relevant Algorithms: The ILSP algorithm is similar to the LS-CMA and multitarget LS-CMA algorithms proposed in the CM literature [9], [24]-[26]. There are two key differences, however. First, the FA property is stronger than CM for digital signals (with PSK modulation format) since the signals are restricted to lie on discrete points on a disk. Second, these algorithms use a MMSE beamformer to estimate the signal waveforms. They minimize the following performance criterion using the alternating projections technique:

niin //w*x - sll$. (19) W , S E C M

TALWAR et al.: SYNCHRONOUS CO-CHANNEL DIGITAL SIGNALS-PART I I I89

It is shown in [25] that for a single user and an unknown spatial noise covariance, the above criterion yields the ML signal estimate. However, when multiple signals are present, this criterion suffers from the drawback that only the strongest signals may be captured. For example, consider the case where two weight vectors have converged to the same (stronger) signal. The MMSE residual, llW*X - Sll; with S = proj[WCX], may be small in this case. Yet, the ML residual, IIX - ASllg where A = XS+, will be large since a best fit of the array response and signal model to the received data is not obtained.

Now, let us examine the ML and MMSE schemes to see precisely how they differ. We can express ILSP algorithm succinctly as

s k + 1 = proj [(xs:)+x] (20)

and a MMSE scheme such as multitarget LS-CMA [26] without soft-orthogonalization as

S k + l = proj[SkX+X]. (21)

Since the pseudo-inverse does not satisfy (XSi)+ = SkX+ in general [27], the two algorithms yield different signal estimates. Note the matrix AI, = XS; in (20) is poorly conditioned near the solution A, if the angular separation between array response vectors is small. In contrast, the data matrix X in (21) becomes poorly conditioned for high SNR’s. The MMSE algorithm is computationally less expensive. How- ever, we have found the performance of the ML algorithm to be more favorable in a blind multiple-signal scenario. In [lo], Swindlehurst et al. have proposed a decision-directed technique for digital signals using both the MMSE and ML beamformers (similar to ILSP), assuming a rough estimate of the signal of interest is available. They have shown that the asymptotic symbol error rate for the MMSE beamformer W* = Rsx.R&& is lower than the error rate for the ML beamformer W* = A+. Hence, for large N , the converged signal estimates obtained from ILSP may be improved by applying the the MMSE approach.

B. ILSE Algorithm

A limitation of the ILSP algorithm is that its performance is limited by that of the ML beamformer. This is easily seen by considering the case where A is known, and the ML criterion is to be minimized with respect to the variable S only. In ILSP, S E R is not estimated directly, but in two steps, (i) least-squares and (ii) projection. The least-squares step causes noise enhancement if the array response vectors are not well separated in angle, i.e., A is ill conditioned. The optimal approach is to enumerate over all possible S matrices with elements in R, and choose the S that minimizes I IX - AS 1 1 ’$. However, this is computationally demanding, since LdN matrices need to be considered. Fortunately, the search can be reduced to enumerating Ld vectors in R ( N times) by exploiting the following property of the Frobenius norm:

minI/X-AS//$ = min I l x ( l ) - A s ( l ) l l $ + . . . f S E R s(1)En

s (IV) E R min llx(N) - As(N)IJ$. (22)

Hence, minimization over the signal vectors s( 1) . . s ( N ) can be carried out independently. For each s(n) , the ML estimate 6(n) is obtained by enumerating over all Ld possible vectors s ( j ) E R, and choosing the one that minimizes

The ILSE algorithm proceeds in a similar fashion as ILSP. Given an initial estimate of A (possibly from ILSP), we iterate using an alternating minimization technique: minimize f (A, S; X ) with respect to S E R and then with respect to A, at each step. The residual function f (A , S; X) is decreased at each iteration, and for a reasonably good initial Ao, convergence to the global minimum is rapid. The algorithm has complexity o ( N ~ ~ L ~ ) flops per iteration: ~ m d + 2 d 2 ( ~ - $) + md2 flops to solve for A and NmLd(d + 1) flops to enumerate.

Iterative Least-Squares With Enumeration (ILSE) Given Ao, k = 0 IC = k + l

Let A = An-1 in (22), and minimize for SI , (enu-

AI, = XSi(SkSi ) - ’ Repeat 2 until (Ak) S k ) = (Ak-1, Sk-I).

meration).

C. Convergence

We have recently discovered that ILSE algorithm is very similar to the segmental K-means algorithm used for estimating parameters of hidden Markov models [28]. The K-means algorithm is an iterative scheme that alternates between two key steps: (i) segmentation performed via generalized Viterbi algorithm, and (ii) optimization. In [28], Juang and Rabiner prove fixed-point convergence of the algorithm. Their proof is based on Zangwill’ s convergence theorem, which is a general result for algorithms in nonlinear programming [29]. It is shown that the K-means algorithm satisfies the conditions of the theorem. Rather than taking the same approach, we present a simple proof that shows that ILSE algorithm converges to a fixed point in a finite number of iterations.

We first review some definitions that will be needed for the proof [29]. Let 7: V i V be a mapping from a point in space V to a point in V . An algorithm is an iterative process that generates a sequence {v~,};rP=~, given a point VO in V , by successive transformations VI, = 7(Vk-l). A point V* is a jixedpoint of T if V* = 7 ( V * ) . Let A be the set of fixed points of 7. A function f is a descent function for mapping 7 if it satisfies the conditions:

1) f : V + IR is nonnegative and continuous 2) f(V) < f (V) for V = 7 ( V ) and V 6 A 3) f (V) 5 f (V) for V = 7(V) and V E A.

Many algorithms use the objective function to be minimized as the descent function f . The conditions 1-3 require that the algorithm reduce f at each iteration until a fixed point is reached.

Within this framework, we can consider ILSE algorithm as a mapping on the cross-product space V = A x S where A = p X d and S = f l d x N . Starting with an arbitrary pair Vo =

1190 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 5 , MAY 1996

(Ao, So), the iterates VI, = (AI,, S I , ) = 7((Ak-1, Sk-1)) are determined by the following two minimization steps: (i) S I , = arg minsEn /IX - Ak-1Sll$, and (ii) AI, = arg minA I/X - ASI,~I'$. In order for the mapping 7 to be well defined, we need to ensure that for each pair (Ak-1,Sk-l) E V , the algorithm computes a unique pair (Ah, S I , ) E V . That is, the algorithm computes a unique S k for a given Ak-1, and a unique Ah for a given s k . For a given Ak-1, it may be possible that S k in (i) is not unique, such as when Ak-1 is rank deficient. In this case, we need to impose a rule that uniquely defines S k . For example, in a computer implementation of the algorithm, there is an ordering associated with enumeration, and we choose S k to be the minimizer of (i) with the lowest index. With respect to AI, in (ii), given a full rank SI , , the least-squares estimate Ah is unique. However, if S k is rank deficient, we need to compute the minimum-norm least-squares solution for Ak; that is unique.

It is natural to consider the residual function f (A: S: X) = / / X - ASll$ as a descent function of the algorithm. Clearly, f (A, S; X) is nonnegative and continuous in A and S. Con- sider

(24) (25)

AI,, S k ; X) = IIX - AlcSlcl/% = rnin IIX - ASI,~/$

5 IIX - Ak-lSkll$ (26) = niin IIX - Ak-1S/I$ (27)

5 11x - Ak-1Sk-l//$ (28) = f(Ak-1, S k - 1 : X). (29)

A

S E R

Based on our assumptions, inequality (26) is strict unless Ak = Ah-1 and inequality (28) is strict unless S I , = S k - 1 .

Hence, at each iteration the residual is strictly decreased unless (Ak-1 S k - 1 ) 1 I((Ak-1 S h - 1 ) ) is a fixed point.

The convergence proof is based on the observation that each of the iterates S k E f l d x N , belongs to a finite set. Let us assume there are P possible S I , matrices. Since there is a unique least-squares estimate Ak associated with each SI , , there are only P pairs (Ah, S I , ) that can be generated by the ILSE algorithm (for IC 2 1). Consider the sequence of pairs {(Ak, Sk)}f_+,' obtained from the algorithm. There are P + 1 iterates in the sequence taken from a set of P possible elements. Hence, at least two of the iterates must be the same. Let us assume j and j + 1 are the lowest indexes for which (Aj, S j ) and (Aj+l, Sj+l) are the same. The residual for the two iterates is equal as follows:

(30)

If I = 1, then (A,,Sj) is a fixed point of the algorithm, since (Aj+l,Sj+1) = T((Aj,SJ)) = (Aj,Sj). Let us assume (Aj, Sj) is not a fixed point, and I > 1. Then, f(Aj+l, Sj+l; X) must be strictly less than f (Aj, Sj; X); and, hence

(31)

However, since (30) and (31) are contradictory, (Aj, S j ) must be a fixed point. To summarize, we have the following convergence theorem for ILSE.

f ( A . 3+1> S - 3+l , . X ) = f (Aj ,Sj ;X) .

f(Aj+l, sj+l; X) < f (A; , s;: W .

Theorem5.1: Let (Ao,So) E C m x d x f l d x N , and the iterates (AI,; S k ) = I((Ak-1, SI,-^)), k 2 1 be determined by ILSE algorithm. Let f ( A , S ; X ) be a descent function of the algorithm. Then, there exists some j o such that for k 2 jo;(Arc.S~,) = 7((Aj,,,Sj,)) and f ( A ~ , , s k ; X ) =

Hence, a fixed point is reached in a finite number of steps, and can be detected by a lack of change in the residual. The global minimum is a fixed point of the iteration. This follows from the fact that a transition to another point would be possible only if the residual is reduced, but this cannot occur since the global minimum is the point with the lowest residual. Note that the theorem is valid for any initial guess (Ao, SO). However, the sequence { (Ah, S k ) } and the fixed point it converges to depends on the initial guess.

We have found in our simulations that for some initial iterates, the algorithm converges to fixed points that are not the global minima. This case can be detected by considering the magnitude of the residual & IIX - ASll$, where A and S are converged estimates from ILSE. If a global minimum is reached, this residual is close to the noise power, since the true residual

f ( Aj, * s,, ).

/IX - ASll$ = IlVll$ z Ntr(a21) = Nma2.

If the residual is not decreased to noise level, we have reached a non-global solution. In this case, we restart the algorithm with another initial guess. We proceed in this fashion until the global minimum is reached.

For a wide range of parameters (SNR and array response vector separation), ILSE algorithm converges to the global solution when initialized with array response estimates from ILSP. If the global solution is not reached, we re-initialize ILSP with a random guess, and then re-initialize ILSE with the estimate from ILSP. Usually one or two re-initializations are sufficient to yield the global minimum. Since both ILSP and ILSE converge very rapidly, re-initialization is not computationally expensive. Hence, the success of our approach is not hindered by the presence of additional fixed points. We cannot apply the above convergence theory to ILSP since the algorithm does not necessarily decrease the residual at each iteration. However, in our simulations, we have found that it also converges to a fixed point in a finite number of steps. Moreover, in scenarios where the array response vectors are well separated, it converges to the global minimum.

Cost Function: In Fig. 1, we symbolically depict the ML cost function /IX - AS1I2 in order to understand the path taken by ILSE algorithm. We consider a scenario with d = 2 BPSK signals at 15 dB SNR, arriving from Q = [0, lo]", and a block size of N = 3. We choose this simple case due to the complexity involved in enumerating all possible S matrices. Neverthless, this picture gives a good qualitative understanding of the path of the algorithm to a global minimum or some other fixed point.

In this figure, the x-axis corresponds to the 26 possible signal matrices S ( J ) , j = 1 . . .64. The y-axis corresponds to the least-squares estimates A(') associated with each S(')); that is, A(L) = argmina IjX - AS('))1I2 for i = 1 . . .64. The graph itself corresponds to a matrix of residuals, wherein the entry

TALWAR et al.’ SYNCHRONOUS CO-CHANNEL DIGITAL SIGNALS-PART I 1191

10 20 30 si?) 40 50 60

Fig. 1. Representation of a matrix of residuals, wherein the entry at row i and column j corresponds to the the value of the residual IIX - A(’)S(J)Il’, i ; j = 1 . . . 6 4 . Shown are paths of ILSE algorithm that lead to global minima (solid lines) and to other fixed points (dashed lines).

at row i and column j corresponds to the the value of the residual /IX - A(z)S(j) / I 2 , i , , j = 1 . . .64. In particular, the ith row in the graph corresponds to the residuals associated with matrix A(i) for all possible S ( j ) ’ s . For simplicity, we have chosen a three-color scheme to depict the residuals. A point ( i , j ) is colored gray if the value of its associated residual IIX - A(i)S(j)112 is minimum with respect to other residuals in row i . Otherwise, it is colored white. If a point is a global minimum with respect to the whole matrix, it is colored black.

For a given Ao, ILSE generates an SO, which is the starting point in our graph. From this So, the algorithm computes an A l . This step is symbolically depicted by a vertical line starting at column j such that S(J) = So, and moving up to the diagonal. Now, the algorithm generates S2 by enumerating over all possible S’s. We represent this step by a horizontal move on the line associated with A( j ) = A l . Since we seek a unique minimum, the line segment from the diagonal ends at the leftmost gray point. This follows from the fact that in our implementation, we always pick the S ( j ) with the lowest index that minimizes IIX - A S ( j ) / I 2 ; j = 1. . . LdN. Once SI is picked, the algorithm computes the corresponding Az, and this once again corresponds to a move to the diagonal. We proceed in this fashion until a global minimum or a fixed point is reached.

In this graph, fixed points are located on the diagonal. It is clear that if the leftmost minimum with respect to S for a particlular A(;) is on the diagonal, the scheme described above will not generate any more line segments, and the iteration will remain fixed at that point. Since the global minima are fixed points of the algorithm, they are also located on the diagonal. There are multiple global solutions, since the signals can be identified only up to an ATM, as described in Section 111.

Note that in rows 1, 16, 22, 27 etc., there are multiple gray points. These correspond to cases where A(’) is singular, thus, multiple S ( J ) ’ s yield the same residual. We have shown a few of the paths that may be taken by the algorithm, depending on the initial pair (Ao, SO). Paths to the global minima are indicated by solid lines, and paths to other fixed points by dashed lines. We note that for this particular scenario, 56 paths of the 64 possible paths lead to global solutions, and eight paths lead to other fixed points.

VI. RECURSIVE ALGORITHMS In this section, we consider two classes of recursive al-

gorithms for estimating the received signals. In recursive estimation, we are interested in solving the following minimization problem at symbol period n

where X(n ) = [X(n - 1) ~ ( n ) ] , and B ( n ) = diag(a”-l, an-’ i . . . ,1) is a diagonal weighting matrix for some 0 < Q < 1. Our objective is to compute A(n ) and ~ ( n ) , assuming that a good estimate of S(n - 1) (or equivalently A(. - 1)) is available. This estimate may be obtained blindly by using the block algorithms of the previous section or by a short training set. The exponential weighting is used to de-emphasize old data in a time-varying environment. The “fading memory” least-squares solution for A(n ) is given

x (n ) ] ,S (n ) = [S(n - 1)

by

A (n) = X (n)B (n) S* (n) (S ( n ) B (n ) S * (n))-’ (33)

which can be updated recursively (see [30]) as follows:

A(n ) = A(. - 1)

. s* (n )P(n - 1). (34) (4.1 - A(. - l ) s ( n ) ) + Q + s* (n )P(n - l ) s ( n )

In the above equation, P ( n ) = (S(n)B(n)S*(n) ) - l , can also be expressed recursively as

Q l ( a + s* (n )P(n - l ) s ( n ) 1 ’ P(” - l ) s (n)s*(n)P(n - 1) P ( n ) = - P ( n - 1) -

P(0) = I.

In addition, we define H ( n ) = X(n)B(n)S*(n ) , so that we can rewrite A(n ) = H ( n ) P ( n ) in (33). Using this framework, we are now ready to consider the two classes of recursive algorithms.

A. Class A

The first class includes algorithms that alternate between estimating ~ ( n ) , and then updating A(n), at each symbol period. The recursive extensions of the two block algorithms belong to this class. For each data vector ~ ( n ) , we first estimate s (n ) by minimizing

using either the least-squares with projection approach or the enumeration approach. This step requires O(md2) flops or

1192 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 5 , MAY 1996

C?(mdLd) flops, respectively. Then, A(n) is computed using (34), which requires C?(md) flops. The enumeration approach is more robust for low SNR's, and is thus recommended whenever computationally feasible (for small values of d) . The

per snapshot.

Likewise, we can partition

n - l )X(n - 1) X*(n - l)x(n) X * ( n ) X ( n ) = ["*( x*(n)X(n - 1) x* (n)x(n)

(42) recursive has flops Multiplying (40) and (42), and using properties of the trace, we can rewrite (39)

Recursive Least Squares with Enumeration (RLSE): 1) Given A(n - 1) 2) n = n + l

* Minimize (35) for s(n) 0 Update A(n) using (34)

(enumeration). (RLS).

3) Continue with the next snapshot.

B. Class B

In this class, we minimize (32) jointly over A(n) and s(n). This is achieved by substituting the weighted least- squares solution for A(n), given by (33), back into the original minimization criterion to yield a new criterion

- where PL B(n)iS(n) ' - In - ' B ( n ) t ~ ( n ) - ' and pB(il)fS(n)- is a d-dimensional projection matrix defined by the rows of S(n)B(n)' 2 as follows:

pB(nj4s(n)*

= ~ ( n ) 3 s(n)* ( ~ ( n ) ~ ( n ) ~ ( n ) * ) -l s ( ~ ) B (n) + . (37)

Note that the key difference between the criterion in (36) and the block ML criterion in (18) is that the minimization in (36) is over s (n ) only, since S ( n - 1) is assumed to be known. Hence, recursive minimization of the ML criterion is computationally tractable.

We can equivalently maximize

max tr[a2H(n - l )P(n)H*(n - 1) s ( n ) E Q

+ a H ( n - l)P(n)s(n)x*(n) + aP(n)H*(n - l)x(n)s*(n)

+ P (n ) s ( n)x* (n)x( n)s* (n)] .

H(n) = aiH(n - 1) + x(~)s(Tz)*

(43)

It is easily seen that H(n) can be updated recursively as

(44)

and thus, (43) can be simplified to

max tr [H (n) P (n) H* (n)] . s(n)ECl

Since A(n) = H(n)P(n) , we estimate s(n) by maximizing

G(n) = arg max tr[A(n)H*(n)]. s(n)Ea

(45)

Substituting H*(n) = S(n)B(n)X*(n) in (45), it follows that we choose s ( n ) to maximize the correlation between A(n)S(n)B(n)i and X(n)B(n) i .

In (43, we compute A(n) and H(n) for each of the Ld possible vectors s ( n ) E R. This is done recursively using (34) and (44), and hence the computation requires C3(mdLd) flops. Next, we compute the diagonal entries of A(n)H*(n) since we are only interested in the trace of the product. Computing the trace for all possible vectors s(n) also requires O(mdLd) flops. Hence, this recursive approach has computational complexity O(mdLd), which is the same as =SE. However, we obtain better signal estimates using this approach. The algorithm is summarized below.

(38) Recursive Projection Update (RPU): 1) Given A(" - 1) and H(n - I)

which can be expressed in terms of the trace operator 2) n = n + l Maximize (45) for s(n) Update A(n) and H(n) max t r [B (n) 4 PBCn, 3 s(n) B (n) x ( n ) *X(n)] . (39)

s ( n ) E Q \ , -

Using (37), we see that 3) Continue with the next snapshot

VII. SIMULATION RESULTS B(n) 4 PB(Ir) 4 s(n)" B(n) 3 = B(n)S(n)*P(n)S(n)B (n) We present the results of three different sets of simulations

which can be partitioned as in (40) at the bottom of the page. The above follows by noting that the weighting matrix can be expressed as

in this section. For simplicity, we assume a uniform linear array of m = 4 sensors. In the first set of simulations, we study the performance of the block algorithms for a block size of N = 100. We consider d = 3 digitally modulated BPSK signals arriving from [IO, 16,251" relative to array broadside. We assume all three signals have equal powers. Starting with

(41) 1 ' 1 B(n) =

(40) 1 - ( - l ) S * ( n - l )P(n)S(n - I)B(n - I) oB(n - l)S*(n - l)P(n)s(n) [ilZB as*(n)P(n)S(n - I)B(n - I) s* (n )P(n ) s (n )

TAI,WAR c t a/ YYNCHRONOUS CO-CHANNbL DIGITAIL SIGNALS-PART I

loo --- 14

*. - _ 12- - -?K .. , ILSP

m 10 - \

fn

c ,

, 0 K m \ .- $ 8 - %, - , c - YC 6 6 -

; 0 1 2 3 4 5 6

SNR (dB)

1 o - ~

I193

0 ‘ 2 3 4 5 6 0 1

SNR (dB)

Fig. 4. Number of iterations for ILSP and ILSE. Fig. 2. Bit error rate for ILSP.

loo

0 1 2 3 4 5 6 SNR (dB)

Bit error rnlc for ILSE

1 o - ~ I

Fig 3

A0 = I l r l x d , we first estimate A and S using ILSP. The estimate A is then used to initialize ILSE for improved array response and signal estimates. To ensure that the global minimum is achieved, we check if the residual is close to the noise power level. If this is not the case, we re-initialize ILSP with a random initial guess. This blind estimation process is repeated lo4 times, each time over a different noise realization. Hence, a total of lo6 bits are estimated for each signal.

In Fig. 2, we show the bit error rates achieved using ILSP with signal to noise ratios ranging from 0 to 6 dB. Although the signals are closely spaced in their directions of arrival (DOA’s), this algorithm is successful in separating and estimating the three signals. We observe that the bit error rates (BER’s) depend strongly on the separation between the DOA’s of the signals since sz, which is the closest to the other two signals, has the highest BER, and s3, which is well separated, has the lowest BER. This follows from the fact that using least-squares to estimate the signals (S = At X) causes

noise enhancement, especially if the columns of A are nearly dependent due to closely spaced signals. In ILSE, however, we use enumeration to estimate the signals. As seen in Fig. 3, this algorithm yields significantly lower BER’s. We achieve bit error rates lower than lop3 for SNR’s greater than 5 dB.

In Fig. 4, we present the average number of iterations required by both ILSP and ILSE. The number of iterations for ILSP decrease moderately fast with increasing SNR. ILSP requires more iterations than ILSE, but each iteration is computationally cheaper. Both algorithms converge fairly rapidly to the global solution. In this difficult scenario, re-initialization was needed in less than 4% of the runs.

Finally, in Fig. 5, we compare the use of training signals with FA property for the same scenario. In our simulations with training signals, we assume 8 out of 100 symbols in each data block are known, which represents the proportion used in IS-54. We first estimate A from the training data, and then use the RLS algorithms described in Section VI-A to estimate the signals, namely, RLS with projection (RLSP) and RLS with enumeration (RLSE). The BER’s obtained from RLSP (*) and RLSE ( 0 ) for signals s2 and s3 are shown in Fig. 5. Also shown for comparison are the BER’s obtained blindly from ILSP (dashed-dotted line) and ILSE (solid line) algorithms.

We observe that the signal estimates obtained blindly from ILSP and ILSE algorithms are slightly better than those obtained from training-based RLSP and RLSE algorithms. Note, however, that RLSE already makes strong use of the FA property. In order to make a fair comparison of FA with training signals, we must compare the performance of ILSE with RLSP. We see in Fig. 5 that ILSE performs significantly better than RLSP. Hence, FA is a powerful property for blind signal estimation. In practice, one may combine a short training set with FA algorithms to obtain good signal estimates at a low computational cost.

In the next set of simulations, we compare the performance of the blind algorithms to the case where the array response matrix is completely known. The simulation set up is the


s2 s3 i o o 1-1

SNR (dB) SNR (dB)

Fig. 5. Comparison of training-based RLSP (*) and RLSE (0) algorithms with FA-based ILSP (dashed-dotted line) and ILSE (solid line) algorithms.

same as before, except that two BPSK signals at 0 = [0.5]” are received at the array. Again, both ILSP and ILSE are used to estimate the digital signals. We see from (22) that in ILSE, we make a joint decision on all d bits in each snapshot s(n) . n = 1 . . . N . Hence, it is convenient to consider the snapshot error rate (SER), which is the probability that s(n) does not equal ~ ( T L ) . In Fig. 6, the dashed-dotted and the solid lines indicate the theoretical SER for the signal detection approach used in ILSP and ILSE, respectively, with a known A. More specifically, we detect the signals in ILSP by obtaining a least-squares estimate of S and projecting on to the discrete alphabet, and in ILSE we enumerate over all possible S matrices. For the ML minimization criterion (17), the SER curve for ILSE represents the lowest SER attainable by any algorithm. The ‘i’ and ‘0’ indicate the SER performance of the blind algorithms, initialized with A0 = I, d . Again, for less than 0.5% of the runs, we needed to re-initialize with random Ao’s. The plot shows that there is virtually no difference in the performance of the blind algorithms as compared to the non- blind approach. A more detailed analysis of these algorithms and their comparison is presented in [31] and [32].

In the final set of simulations, we study the performance of recursive algorithms. We use d training signals, given in (56), to estimate Ao. For simplicity, we choose Q = 1. As in the previous experiment, we first consider the simple scenario with two BPSK signals arriving from 0” and 5’. At 3 dB SNR, we estimate N = 250 snapshots using RLSE and RPU. We average the results over lo4 such runs, so that a total of 2.5 x lo6 bits per signal are estimated. The BER’s achieved by both the algorithms are equal: 1.45 x l o p 2 for each signal, although the number of bits in error is slightly larger for RLSE. The SER for the two algorithms is also 1.45 x 10V2. In Fig. 7, we plot the SER using RPU at each symbol period n, n = 1 . . . N . The SER plot using RLSE is virtually identical for this scenario. We observe that the SER is high initially since the estimate of A is poor. But as n increases, this estimate improves and the SER decreases. After n = 120, the SER converges to about 1.14 x lo-’, the theoretical SER for known A, as shown in Fig. 6. Finally, we make the current

i o o 1

K w cn

I 2 4 6 8

I o - ~ ; SNR (dB)

Fig. 6. Snapshot error rate for ILSP and ILSE

scenario more difficult by adding another signal at lo”, with 3 dB SNR. Also, we reduce N = 100 to ease the computational burden. The BER’s achieved by RLSE and RPU are given in Table 1. We see that RPU yields lower BER’s than RLSE.

VIII. CONCLUSION We have presented a blind approach for the separation

of synchronous digital signals in a coherent multipath environment. We have shown that given a sufficient number of snapshots, the signal estimates obtained by this approach are unique. The block algorithms ILSP and ILSE take advantage of the FA property of digital signals to simultaneously estimate the array response and symbol sequence for each signal. We have proved that the ILSE algorithm converges to a fixed point in a finite number of iterations. A fixed point that is not a global solution can be detected by considering the magnitude of the residual. Hence, we are ensured that the global minimum will be reached by re-initializing one or more times. We may note

TALWAR et al.: SYNCHRONOUS CO-CHANNEL DIGITAL SIGNALS-PART I 1195

Alg. S1 SZ s3

RLSE 2.88 x lo-’ 5.26 x 2.59 x ’ RPU 2.66 x loF2 4.92 x loF2 2.39 x loF2

0.04

0.02 -

I I

50 100 150 200 250 0.01 i,

Symbol Period (n)

Fig. 7. Snapshot error rate profile for RPU.

that despite the difficulties associated with the existence of fixed points that are not global solutions, these algorithms are important in providing a computationally feasible alternative to complete enumeration, which is intractable. Our simulation experiments have shown that in scenarios of practical interest, the converged performance of ILSP and ILSE is virtually the same as predicted in the case where the array response matrix is completely known. We have also described recursive extensions of ILSP and ILSE, and proposed a new algorithm, RPU, that minimizes the ML criterion recursively. In contrast with block minimization, recursive minimization of the ML criterion is computationally tractable.

Our approach can be extended to asynchronous transmission and to multipath channels with large delay spread, as discussed in Section I. Since our algorithms require that the number of signals be known or accurately estimated, we have proposed in [33] a robust scheme for determining the number of incident signals in a multipath propagation environment. Determining the length of the impulse response associated with each signal is a more difficult problem, and is currently under investigation. Other directions for future work include developing computationally efficient initialization strategies in order to avoid restarts, estimating signals in the presence of non-Gaussian interference, and combining coding schemes with blind estimation.

APPENDIX A.l IDENTIFIABILITY

Lemma I : Let S d , be a matrix with f 1 elements such that the columns of S include all the n = 2dp1 possible distinct d-vectors with the first element normalized to $1. Then, the solution t for the system of equations of the form

tT[SlSZ . . . s,] = [H*l . ’ . +1]

where tT = [ t d . . . t z t l ] , is such that ti = fl for some i , and t j = 0 for j = 1 , . . . , d , j # i .

Proofi The proof is by induction. The case d = 1 is trivial. Let us consider the case d = 2. Define n1;l = t l . We have the following n = 2 equations:

Q 2 , J -- t 2 + a1,1 = f-1 n 2 , 2 =: t 2 - Ql.1 = 41.

(46) (47)

There are four possible right-hand sides, which we denote as [xl 521. Then for [xl 221 equal to:

1) [l -11 or [-1 l ] ; t * = 0 and t l = +I or -1,

2) [l 11 or [-1 -1l; t l = 0 and t2 = +I or -1,

We assume that the above theorem holds for some d = k . Then for d = k + 1, we obtain n = 2k equations of the following type:

respectively.

respectively.

Q k + l , l -- t k + l + Qk.1 = Zt1 (48) Q k + l , 2 t k + l - Q k , l = 41 (49)

(50) a k + l . n - l t k + l + “ k , ? = *I (51)

a k + l , n E tk+l - n k . 2 It1 (52)

where a k , j , for j = 1 . . . 2k-1 is of the form tk & . . . * t 2 & t l . From (48) and (49), four possibilities exist for t k + l and a k , ~ :

1) t k + l = 0 and a k , ~ = $1 or -1. 2) a k , l = 0 and t k + l = +1 or -1. Now, if t k + l = 0, from (48)-(52), we get the reduced

system { Q ~ , J = 51,. . . , a k , ? = fl} which by assumption has the solution: ti = +1 for some i , and t j = 0 for j = 1, . . . d ; j # i. If t k + l = +1 or -1, we must have { a k , ~ = 0 ; . . . , “ k , f = 0). The first two equations of this system

a k : l = t k + @ k p l , l = 0 a k , 2 t k - a k - 1 , 1 = 0

imply t k = a k - 1 , ~ = 0. Given t k = 0, we obtain the smaller system { a k - l , l = 0, . . . ak-1,: = O}. Next, we consider the first two equations of this system to show t k - 1 = 0, and continuing in this fashion, it is easily seen that t k - 2 = . . . = tl = 0. Thus, by induction, the theorem holds for all d. 0

APPENDIX A.2

Theorem 3.3: Let p denote the probability of receiving all the Ld/2 distinct (up to a sign) columns of S in N independent snapshots, N >> Ld/2. Assuming that the signals

1196

s d =

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 44, NO. 5, MAY 1996

-+1 +1 +1 . . . +1- +1 -1 $1 . . . +I +1 $1 -1 . . . +l

_+I +I +I . . . -1-

( 5 6 )

are independent, and the probability of receiving each symbol in R is equal, p is bounded by

N L d L d - 2 L p L 1 (53)

Prooj? Our sample space S is the set of all d-vectors with entries in I1 = {&I, + 3 , . . . , &(L- 1)},S = {x: xk E Cl: IC = 1 . . . d ) . Let A be the event that all the I = Ld/2 distinct vectors from S are picked in N >> 1 independent trials. We are primarily interested in showing that given a fixed value of d, the probability p of event A approaches 1 as N increases. For this reason, it suffices to compute a lower bound for p .

Consider the complement of A. A‘ is the event in which at least one of the distinct vectors has not been picked. Let d:,i = 1 . . . 1 be the event where neither a particular +x, nor -xi has been picked in N trials. Then A“ = Uf=,X. Furthermore

1

P ( k ) = P U A; 5 C P ( A ‘ ; ) (54) (1, 1 i=l

since the events 4 are not disjoint. Now, the probability that +xi and -xi is not picked in the kth trial is one minus the probability +xi or -xi is picked, i.e., (1- &). Since the trials are independent, P(AI) is (1 - 6)‘”. Then from (54), we get

The result in (53) easily follows. 0

where ST is the corresponding row of S, and can be likewise partitioned as ST = [sz S d f l s:]. The first d columns of s form a linearly independent set, and thus define t uniquely in terms of elements of Sd as follows:

The inverse of S d can be computed easily by noting that S d = eeT+2eleT-21 where e = [l 1 . . . 1IT andel = [l O...0lT, and then applying the Sherman-Morrison-Woodbury formula [341

3 - d +1 +1 . . .

1 +I 0 0 . . . -11 Remark: In situations where blind identification is not

required, the rows of s d define a useful set of d training signals sent by each of the users to learn the array response matrix A.

Hence, t can be expressed in closed form as

1 2

tT= -[(3 - d ) S l + 5 2 + . . . + Sd 31 - 52 . . . 51 - S d ] .

(60) Since S d is vector with 5 1 entries, there are a finite number of possibilities for t. The question then becomes whether there exists another vector of f l ’ s in S such that only a trivial t is possible. The answer to this question is the vector We see from (57) that t must satisfy t T s d + l = Sd+l, which yields the relation

(2 - d ) s l + 52 + “ ’ S d = s d + l . (61)

If we let k denote the number of -1’s in {Sz . . . S d } , then (61) becomes

(2 - d ) 3 l + IC(-1) + ( d - 1 - k)(+1) = Sd+I

and solving for IC, we get 1

k = ?((a - d)s1 - Sd+l + ( d - 1)). (62) 2 We consider the four different possibilities for [SI Sd+l]. In the case that [SI S d + l ] is equal to +[l 11 or -[1 I], IC equals 0 or d - 1, respectively. This implies sz = +[1 1 1 . . . 11, and then from (60), we see that tT = +[1 0 0 . . . 01. Similarly, if [SI S d + l ] = &[I: -11, then sz = . . . 1 - 1 1 . . . 11 and correspondingly tT = &[0 . . . 0 1 0 . . . 01, with a41 only in the ith position, i = 2 . . . d. In all four cases, a trivial solution is obtained. We have shown previously in Section I11 that each row t of T must be distinct. Therefore, T is an ATM.

REFERENCES

S. Talwar, M. Viberg, and A. Paulraj, “Blind estimation of multiple co- channel digital signals using an antenna array,” IEEE Signal Processing Le t t , vol. 2, no. 1, pp. 29-31, Feb. 1994. D. Gerlach and A. Paulraj, “Adaptive transmitting antenna arrays with feedback,” IEEE Signal Processing Lett., vol. 10, no. 1, pp. 150-152, Oct. 1994. G. Raleigh. S. Diggavi, V. Jones, and A. Paulraj, “A blind adaptive transmit antenna algorithm for wireless communication,” in Proc. IEEE ICC, 1995.

TALWAR et ul.: SYNCHRONOUS CO-CHANNEL DIGITAI. SIGNALS-PART I 1197

[4] G. Xu and Hui Liu, “An effective transmission beamforming scheme for frequency-division-duplex digital wireless communication systems,” in Proc. IEEE ICASSP, 1995, pp. 1729-1732.

[5] R. 0. Schmidt, “A signal subspace approach to multiple source loca- tion and spectral cstimation,” Ph.D. dissertation, Stanford University, Stanford, CA, 1981.

[61 A. Paulraj, R. Roy, and T. Kailath, “Estimation of signal parameters via rotational invariance techniqucs-ESPRIT,” in Proc. 19th Asilomar Con$ Circuits, Syst., Comput., 1985.

[7] B. Otterstcn, R. Roy, and T. Kailath, “Signal waveform estimation in sensor array processing,” in Proc. 23rd Asilomar Con$ Signals, Syst., Comput., 1989, pp. 787-791.

[SI S. Anderson, M. Millnert, M. Viberg, and B. Wahlberg, “An adaptive array for mobile communication systems,’’ IEEE Trans. Veh. Technol., vol. 40, no. I , pp. 230-236, 1991.

[9] B. Agee, “Blind separation and capture of communication signals using a multitarget constant modulus beamformer,” in Proc. IEEE MILCOM, 1989. nn. 340-346.

1131

1141

1 I I ~~

A. Swindlehurst, S. Daas, and J. Yang, “Analysis of a decision directed beamformer,” IEEE Trans. Signal Processing, vol. 43, pp. 2920-2927, Dec. 1995. R. Gooch and B. Sublett, “Joint spatial and temporal equalization in a decision-directed adaDtive antenna system,” in Proc. Asilomur Conf:, 1989, pp. 255-259. B. G. Agee, A. V. Schell, and W. A. Gardner, “Spectral self-coherence restoral: A new approach to blind adaptive signal extraction using antenna arrays,” Proc. IEEE, vol. 78, pp. 753-767, Apr. 1990. J. Cardoso, “Source separation using higher order moments,” in Proc. IEEE ICASSP, May 1989, pp. 2109-21 12. L. Tong, Y. Inouye, and R. Liu, “Waveform-preserving blind estimation - - of multiple independent sources,” IEEE Trans. Signal Processing, vol. 41, pp. 2461-2470, July 1993.

[IS] H. Liu and G. Xu, “A deterministic approach to blind symbol estimation,” IEEE Signal Processing Lett., vol. I , pp. 205-207, Dec. 1994.

[I61 H. Liu and G. Xu, “Multiuser blind channel estimation and spatial channcl pre-equalization,” in Proc. f E E E ICASSP, pp. 1756-1759, 1995,

[I71 A. J. van der Veen, S . Talwar, and A. Paulraj, “Blind identification of FIR channels carrying multiple finite alphabet signals,” in Proc. IEEE ICASSP, 1995, vol. 2, pp. 1213-1216.

[ 181 H. L. Van Trees, Delection, Estimation and Modulation Theory. New York; Wiley, vol. I, 1968.

[I91 M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Trans. Acoust., Speech., Signal Processing, vol. ASSP- 33, pp. 387-392, Apr. 1985.

[20] M. Wax and 1. Ziskind, “On unique localization of’ multiple sources by passive sensor arrays,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 996-1000, July 1989.

1211 N. V. Efimov and E. R. Rozendorn, Linear Algebra and Mulli- Dimensional Geometry.

1221 G. Golub and V. Pereyra, “The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate,” SIAM J. Num. Anal., vol. IO, pp. 413432, 1973.

[23] P. Gill, W. Murray, and M. Wright, Practical Optimization. San Diego, CA: Academic, 1981.

1241 B. Agee, “The least-squares CMA: A new technique for rapid correction of‘ constant modulus signals,” in Proc. ICASSP, pp. 953-956, 1986.

[25] B. Agce, “Maximum-likelihood approaches to blind adaptive signal extraction using narrowband antenna anays,” in Proc. Asilomar Con$, 1991, pp. 716-720.

1261 Y. Wang et al., “A matrix factorization approach to signal copy of constant modulus signals arriving at an antenna array,” in Proc. CISS, Princeton, NJ, 1994.

1271 S. Barnett, Matrices-Methods and Applications. Oxford, U.K.: Claren- don, 1990.

[28] B. Juang and L. R. Rabiner, “The segmental K-means algorithm for estimating parameters of hidden Markov models,” IEEE Trans. Acoust.,

Moscow, Russia: MIR, 1975.

ng, vol. 38, no. 9, pp. 1639-1641, Sept. 1990.

[29] W. I. Zangwill, Nonlinear Programming: A Unified Approach. Engle- wood Cliffs, NJ: Prentice-Hall, 1969.

[30] S. S. Haykin, Introduction to Aduplive Filters. New York: Macmillan, 1984.

[31] S. Talwar and A. Paulraj, “Performance analysis of blind digital signal copy algorithms,” in Proc. MZLCOM, 1994, vol. I, pp. 123-128.

[32] S. Talwar, A. Paulraj, and M. Viberg, “Blind separation of synchronous co-channel digital signals using an antenna array. Part 11. Performance analysis,” IEEE Trans. Signal Processing, submitted for publication.

[33] A. J. van der Veen, S . Talwar, and A. Paulraj, “Blind estimation of multiple digital signals transmitted over FIR channels,” IEEE Signal Processing Lett., vol. 2, pp. 99-102, May 1995.

[34] G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: John Hoplans Univ. Press, 1984.

i

3

Shilpa Talwar (S’95) received the Bachelor’s degree in mathematics and computer science from the State University of New York, Geneseo, NY, in 1989. She is currently working toward the Ph.D. degree in the scientific computing and computational mathematics program at Stanford University, Stanford, CA.

Her research interests include numerical linear algebra, array signal processing, and wireless communication systems.

Mats Viberg (S’87-M’90) was born in Linkoping, Sweden, on December 21, 1961 He received the M S degree in applied mathematics in 1985, the Lic. Eng. degree in 1987, and the Ph I) degree in electrical engineering in 1989, all from Linkoping University, Sweden

He joined the Division of Automatic Control at the Department of Electrical Engineering, Linkoping University in 1984, and from November 1989 until August 1993 he was a research associate From October 1988 to March 1989, he was on leave

at the Informations Systems Laboratory, Stanford University, Stanford, CA, as a visiting scholar From Aug 1992 until Aug 1993, Dr Viberg held a Fulbright-Hayes grant scholarship as a Visiting Researcher at the Department of Electrical and Computer Engmeering, Brigham Young University, Provo, UT, and at the Informations Systems Laboratory, Stanford University Since September 1993, he has been a professor of signal processing at the Department of Applied Electronics, Chalmers University of Technology, Sweden. His research interests are in Statistical Signal Processing and its application to sensor array signal processmg, system identification, communication, and radar systems

Dr Viberg received the IEEE Signal Processing Society’s 1993 Paper Award (Statistical Signal and Array Processing Area), for the paper ‘Sensor array processing based on subspace fitting,” coauthored with Bjom Ottersten He is currently a member of the IEEE Signal Processing Society’s technical committee on Stdtistical Signal and Array Processing

Arogyaswami Paulraj (SM’85-F’9 I ) , for a photograph and biography, see this issue, p. 1155.

00502331

Documents