A Space-Time Constant Modulus Algorithm for SDMA ... - CiteSeerX

A Space-Time Constant Modulus Algorithm for SDMA Systems C. B. Papadias and A. Paulraj

Information Systems Laboratory Stanford University, Stanford CA 94305, U.S.A. Abstract | We consider the problem of blind

recovery of p synchronous i.i.d. communication signals that are transmitted through a linear spatio-temporal channel and captured by an m? element array. Such a situation may arise for example in a Space-Division-Multiple-Access (SDMA) mobile communication system. We propose an approach of the Constant-Modulus (CM) type in order to construct optimization criteria that may allow joint blind recovery of the transmitted signals with low computational complexity. Several aspects of the proposed algorithms related to their convergence and implementation are discussed and computer simulations are provided as experimental evidence to the algorithms' performance. I. Introduction

We are interested in the problem of joint blind recovery of multiple (p) synchronous users, whose signals are impinging on an antenna array of m sensors ( m > p ). The signals are assumed to have undergone linear modulation and are transmitted through a linear channel with additive white noise. We further assume that the signals originating from the dierent users are uncorrelated between them (such a scenario is applicable for example in the case of an SDMA system). As a result of the transmission through the linear channel, the signals arriving at the antenna array are corrupted not only by inter-user (IUI) but also by intersymbol interference (ISI). Our aim will be to propose simple methods that may help to separate and equalize jointly these signals in a blind way, i.e. without the use of training signals. The simplicity requirement leads us to the consideration of linear spatio-temporal equalization structures. Figure 1 depicts a baseband representation of a linear spatio-temporal channel-equalizer structure for the case of p = 2 sources and m = 3 antennas. In this gure, aj (k); j 2 f1; : : :; pg denotes the j th transmitted signal, vi (k) the additive noise sample at antenna i, yi (k); i 2 f1; : : :; mg the noisy received signal at antenna i and zj (k); j 2 f1; : : :; pg the equalized output corresponding to the j th source, all at time instant k. Hij ; i 2 f1; : : :; mg; j 2 f1 : : :pg is a Nj  1 vector representing the linear channel from source j to antenna

v(k)

a(k)

z(k)

1

1

H11

F11

1

z(k) 2

a(k) 2

y(k)

H12

1

F12

v(k) 2

H21

F21 y(k) 2

H22

F22

v(k) 3

H31

F31 y(k)

H32

3

F32

Fig. 1. Linear spatio-temporal equalization for 2 users

i and Fij a L  1 vector that contains the coecients of the linear lter relating the ith received signal to the j th output (we have assumed the channel Hij to have Nj non-zero discrete samples and each equalizer Fij to have L coecients). The linear equations linking the above-de ned quantities are: yi (k) =

p X

HijT Aj (k) + vi (k) ; i 2 f1; : : :; mg ;

j =1 m X

zj (k) = where

i=1

FijT Yi (k) ; j 2 f1; : : :; pg;

Aj (k) = [aj (k)


aj (k?Nj +1)]T Yi (k) = [yi (k)


yi (k?L+1)]T ;

(1)

(2)

and T denotes usual matrix transpose (in this representation, all channel and equalizer samples are assumed to be taken at the symbol rate, however the extension to the fractionally-spaced case is straightforward). It has been proven in [1], that under some mild conditions on the dierent channels Hij (having to do with the \irreducible" and the \column-reduced" of the pinput { m-output matrix channel), perfect zero-forcing (ZF) equalization of the spatio-temporal channel can be achieved if the length L of each equalizer satis es   N ? p L  L = m?p ; (3) P where N = pj=1 Nj and d e denotes integer part. Therefore, provided that the above-mentioned channel

conditions are met, if L is chosen so as to satisfy (3), enough degrees of freedom are contained in the spatiotemporal equalizer so as to provide perfect ZF equalization of the spatio-temporal channel in the absence of noise. In this paper we will present a space-time (S-T) approach of the CM type that may help nd a S-T equalizer setting that leads to the recovery of the p signals in a blind way. The rest of the paper is organized as follows. Section II reviews the well-known CM approach in the case of a single user (SU). Section III contains our proposed CM approach for the case of multiple users; two algorithms are derived by stochastic-gradient minimization of the proposed cost functions. Their performance is discussed and directions for future work are given. Section IV contains some computer simulation results that show the ability of the proposed techniques to achieve blind S-T equalization. Finally, section V contains our conclusions. II. Single-user case: a review of the CMA

A popular algorithm that has been proven to be successful for the blind equalization of QAM signals is the so-called CMA 2-2 algorithm [2] [3]. In the SU case, the channel-equalizer cascade will be a single-input-singleoutput (SISO) system and the minimized cost function has the form J(F) = E(jz j2 ? 1)2 ;

 (jz(k)j ? 1) Y  (k) z(k) ; (5) 2

where  denotes complex conjugate. The CMA 2-2 al-

gorithm has recently been shown to be globally convergent [4] (in the absence of noise) if the equalizer is in nite-length (SISO channel case) or if a zeros-andlength condition holds for the case of a SIMO channel ([5]). Combined to the simplicity of the algorithm, these results make the CMA 2-2 an appealing blind technique for linear single-user equalization. III. A Space-Time CM Approach

A.1 Derivation The following cost function is a straightforward extension of the single-user CMA 2-2 cost function for the case of p users:

We propose to extend the CM criterion to the more general case of multiple users (MU). We propose two dierent MU-CM cost functions.

p ? X

J1 (F) = E

j =1




jzj j2 ? 1 2 ;

(6)

where F is the Lm  p matrix equalizer de ned as 2

f

11




f1p

1

fmp

F = 64 ... ... fm




.. .

3 7 5

;

(7)

where T



fij = Fij (0)


Fij (L ? 1)

(8)

The stochastic-gradient algorithm that minimizes (6) will have the form

F(k + 1) = F(k) ?  rF J (F; k) ; (9) where J (F; k) is the instantaneous cost function 1

1

J1(F; k) =

(4)

where E denotes statistical expectation and z(k) = F T (k)Y (k) is the equalizer output. This cost function applies as well to the cases of fractional spacing or antenna-array reception (the overall system remains SISO, whereas the channel itself can be modeled as a SIMO system). The corresponding (CMA 2-2) algorithm is given by F(k + 1) = F(k) ?

A. Algorithm I

p ? X j =1




jzj (k)j2 ? 1 2

(10)

By noting that @J1 (F; k) = 4(jz (k)j2 ? 1)z (k)Y  (k) ; j j i @ fij

(11)

we get for the instantaneous gradient

rFJ1 (F; k) = 4Y (k)Z (k) ;

(12)

where 

Y(k) = Y T (k)


YmT (k) 1

T

;

(13)

and Z (k) is de ned as 

Z (k) = (jz1 (k)j2 ? 1)z1 (k)


(jzp (k)j2 ? 1)zp (k)

(14) Therefore the stochastic gradient algorithm corresponding to the cost function (6) takes the form

F(k + 1) = F(k) ?  Y (k)Z (k) ;

(15)

where we have absorbed the factor 4 into the stepsize parameter .



A.2 Discussion Eq. (15) describes a blind stochastic gradient algorithm of the CM type that is suitable for space-time equalization in the case of multiple user signals that exhibit both IUI and ISI. (15) requires p(Lm+3) complex multiplications/iteration and corresponds to the minimization of the CM cost function (6). Before entering into the analysis of the algorithm, we outline a number of general remarks concerning its behaviour:  For p = 1, the algorithm (15) reduces to the single-

user CMA 2-2 algorithm (5).

 As the algorithm (15) operates on the outputs of a

p-input-m-output MIMO linear channel (see Fig. 1), a necessary condition in order to achieve a perfect solution (in the absence of noise) is to satisfy the length condition (3) as well as the irreducible and column-reduced conditions mentioned in Slock [1], which we repeat here for convenience: (C1): irreducible H(z): rank(H(z)) = p; 8z and rank(h(0)) = p (C2): column-reduced H(z): rank([h1(N1 ? 1)


hp (Np ? 1)]) = p where H(z) =P Nl=01 ?1 h(l)z ?l with h(l) de ned from y(k) = iN=01 ?1 h(i)a(k ? i)+ v (k) with N1 = max fN g and y(k) = [y1(k)


ym (k)]T . j =1;:::;p j P

 The following ambiguity is inherent in the cost

function (6): if the optimal solution is z(k), then any solution of the form z0 (k) with zj0 (k) = eij zj (k) where j 0 2 f1; : : :; pg also optimizes the criterion. This eect is similar to the wellknown phase-rotation ambiguity inherent in all blind equalization methods and is due to the fact that such a rotation does not aect the statistics exploited by the criterion. In the MU case, apart from the rotation eect, a user ambiguity is also present: this is due to the fact that the dierent transmitted signals share exactly the same statistical properties too. 0

 On top of the aforementioned ambiguity, the al-

gorithm (15) has also the following particularity: if any number b (b  p) of the p columns of the matrix equalizer F are identically initialized, then they will remain identical between them after any iteration of the algorithm: therefore the corresponding b algorithm outputs after convergence will also be identical. Therefore the p columns of F should be initialized with dierent settings. The problem of nding suitable (or optimal) initializations for each of the p equalizer \branches" is an

open one. A.3 Convergence analysis The following theorem can be proven about the convergence of the algorithm (15): Theorem I: If conditions (C1) and (C2) hold, the length condition (3) is satis ed, the p independent signals faj (k)g; j = 1; ?: : :; p
are i.i.d.? and
have negative kurto 4 2 2 2 2 sis K(aj ) = E jaj j ? 2E jaj j ? E(aj ) < 0 and no additive noise is present, then the algorithm (15) will converge to a setting z(k) such that each entry zj (k) is a possibly-shifted and rotated by an arbitrary scalar eij translation of any of the transmitted aj (k); j = 1; : : :; p.

Proof outline: Conditions (C1), (C2), (3) and the absence of noise guarantee the existence of ZF solutions. We rst note that each of the p columns of the recursive equation (15) operates independently of the others. Moreover, if we consider the scalar process f(k)g = f


a(k ? 1) a(k) a(k + 1)


g where a(k) = [a1(k)


ap (k)], then the channel-equalizer cascade lter from the channel input to the equalizer output zj (k) can be represented as a SISO lter with impulse response sj . Now by following the same steps as are followed in [4] it can be shown that under the negative kurtosis assumption, all the convergence points of the algorithm (15) (which are equivalent between them global minima of (6)) are of the form ?




sj =


0


0 eij 0


0


This concludes the proof of the theorem.

(16)

2

According to (16), after the algorithm has converged, each entry of z(k) will be of the form: (17) zj (k) = eij ad (k ? j ) ; where d can be any of f1; : : :; pg. Both the index d as well as the delay j will depend on the position of the single non-zero component of sj in (16). Therefore it is guaranteed that the algorithm will nd some (at least one!) of the transmitted signals, but it is not guaranteed that it will nd them all. However, if several dierent initializations are tried, all the signals should be nally found. A by-product of the proof of Theorem I is the following theorem: Theorem II: A necessary and sucient condition in order to have zj (k) = eij ad (k ? j ), where d can be any

of 1; : : :; p, and j ; j are an arbitrary phase and integer number, respectively, is to have

E(jzj j2) = E(jad j2) and jK(zj )j = jK(ad )j

(18)

This theorem holds for any j 2 f1; : : :; pg.

The proof of theorem II results again from following the Shalvi-Weinstein approach [4]. Based on Theorem II, a multiple-user kurtosis algorithm as well as a whole class of MU algorithms based on dierent constrained criteria may be proposed in the same way as in [4]. B. Algorithm II

It has been made clear from the above analysis, that the major drawback of the algorithm (15) is the fact that it may identify several versions of the same transmitted signal. An alternative approach in order to avoid this behaviour is to propose the following cost function: J2 (F) = J1 (F) + 2

p X

NX M ?1

l;n=1 ; l6=n =?NM +1

jrln()j2 ;

(19) where rln () is the cross-correlation function between users l and n de ned as rln () = E (zl (k)zn (k ? )) ;

(20)

and NM = maxj fNj g. The cost function (19) adds to the CM term J1 (F) an extra term that penalizes the correlations between any two of the equalizer outputs, as well as of a number of time-shifted versions of them (that spans the window of possible delays provided by the channel length). This should prevent the algorithm from converging to sets of signals that contain multiple times the same (or time-shifted versions of the same) signal. In order to nd a corresponding stochastic gradient algorithm we follow the same steps as in A.1: we now get instead of (11): @J2 (F) @ fij

= 4E(jzj (k)j2 ? 1)zj (k)Yi (k)+  P +4 l=1 ; l6=j N=M??N1M +1 rjl()Ezl (k ? )Yi (k) ; (21) which gives for the gradient Pp

rF J2(F) = [
1(k)



p (k)] ;

(22)

where
j (k) = 4E(jzPj (k)j2 ? 1)zj (k)Y (k)+ Pp +4 l=1 ; l6=j N=M??N1M +1 rjl()Ezl (k ? )Y (k) (23)

Therefore, the stochastic gradient algorithm that minimizes (19) is given by F(k + 1) = F(k) ?  [
b 1(k)



b p(k)] ; (24) where
b j is an estimate of
j based on instantaneous values or sample averaging. Equation (24) describes the MIMO-CMA algorithm: a new stochastic gradient algorithm derived from the space-time \constant-modulus" criterion (19) and is suitable for the spatio-temporal equalization of multiple user signals. Note that (24) too reduces to the singleuser CMA 2-2 (5) for p = 1. The parameters employed are the equalizer length L and the stepsize parameter . L should be chosen so as to satisfy (3), whereas  should be chosen so as to guarantee stable operation of the algorithm and provide a reasonable tradeo between convergence speed and steady-state error variance. Another parameter is the length and the shape (rectangular or exponential) of the window that will be used in order to estimate the quantities
b j . Finally, the number as well as the weight of the autocorrelation functions in the criterion (19) can be made variable. (24) has a low computational complexity (depending on the number of terms present in the criterion as well as the length of the averaging window), which is significantly less than the one of more complex methods, as e.g. the method proposed in [6]. We outline a number of points worth noting about the algorithm (24). The same phase-rotation ambiguity that was previously mentioned for the algorithm (15) is also present in (24): each individual user's signal may be recovered up to an arbitrary phase shift. However, due to the cross-correlation terms present in the criterion (19), it is clear that sets of output signals that contain the same or time-shifted (within the limits of the terms included) versions of the same signals are not global minima of the cost function (19). Therefore, the only global minima of (19) are phase-rotated versions of the transmitted signals of the p users. One may also note that the p columns of the algorithm (24) do not operate independently any more (as in (15)): all p equalizer outputs in uence the evolution of any of the p columns of F. This helps the algorithm avoid the convergence to sets of the same signals. IV. Computer Simulations

In this section we present some computer simulation results that show the ability of the algorithms (15) and (24) to cancel jointly the IUI and the ISI induced by the S-T channel. Figure 2 shows an example of the performance of the algorithm (15). A p = 2, m = 3 case is considered. We consider the transmission of p = 2 independent 2?PAM signals of equal magnitude (equal to 1),

a1

a2

o:[H11;H21;H31] ; *:[H12;H22;H32]

1

1

2

0

0

0

−1

−2

−1 10

20 30 dec(y1)

40

10

20 30 dec(y2)

40

2

1

1

1

0

0

0

−1

−1 10

20 30 −dec(z1)

40 1

0

0

−1

−1

8

−1 10

1

4 6 dec(y3)

20 30 −dec(z2)

40

10 2

20 e

30

40

10

0

10

−2

10

10

20

30

40

−4

10

20

30

40

10

0

1000

2000

Fig. 2. Algorithm I a1

a2

1

1

0

0

−1

o:[H11;H21;H31] ; *:[H12;H22;H32] 2 0 −2

−1 10

20 30 dec(y1)

40

10

20 30 dec(y2)

40

2

1

1

1

0

0

0

−1

−1 10

20 30 dec(z1)

40 1

0

0

−1

−1

8

−1 10

1

4 6 dec(y3)

20 30 dec(z2)

40

10

20 e

30

40

0

10

tialized to a random non-zero setting. Sample averages based on 50 samples are used to estimate the quantities
j (k) (however much smaller values have also yielded satisfactory results). The employed stepsize is  = 1  10?3. As can be seen from the plots, both signals are correctly recovered. Note moreover that the signals are recovered without a time shift: this is due to the severe constraining of the criterion (19). As several other simulation results have indicated, our theoretical claims about the algorithms' performance are veri ed: depending on its initialization, the algorithm (15) may nd some or all the transmitted user signals, whereas the algorithm (24) avoids this drawback due to the extra terms in its cost function. V. Conclusions

We have proposed a novel CM approach for the problem of blind joint S-T equalization of multiple synchronous communication signals that impinge on an antenna array. This approach leads to a number of dierent algorithms that are capable of solving the problem. We presented two algorithms of this type and analyzed their performance both theoretically and by means of computer simulations. These low-complexity and easily-implementable techniques seem to be promising for the blind MU S-T equalization problem.

−2

10 10

20

30

40

10

20

30

40

1000

2000

3000

Fig. 3. Algorithm II

through six sub-channels (m = 3) of length Nj = 3 each (a serial representation of the 9 coecients corresponding to each user is shown in the upper rightmost plot) and the SNR at the channel output is 30 dB. As can be seen from the decisions taken on the received signals y1 ; y2; y3 , severe inter-user and inter-symbol interference is present. The algorithm (15) is used, by choosing L = 4 from (3) and  = 1  10?3. The 12  2 equalizer F is initialized having only two non-zero elements equal to 1, in the positions (1; 1) and (4; 2). The algorithm's evolution in terms of average constantmodulus error for the two signals is shown in the lower rightmost plot. After taking decisions on the equalized signals z1 and z2 , it is clear that dec(z1 ); dec(z2 ) have matched a1 and a2, respectively, up to some timeshift and sign ambiguity (which has been adjusted in the plot). Therefore, in this case, joint spatio-temporal blind equalization has been achieved. The algorithm's behaviour is in accordance with our theoretical claims: at least one transmitted CM signal is found (in this case the chosen initialization has helped nd both), and the phase ambiguity is also present at each signal. Figure 3 shows a similar example on the performance of the MIMO-CMA (24). The equalizer F is now ini-

[1] [2] [3] [4] [5] [6] [7] [8] [9]

References D. T. M. Slock. \Blind Joint Equalization of Multiple Synchronous Mobile Users Using Oversampling and/or Multiple Antennas". In Proc. 28th Asilomar Conf. on Signals, Systems and Computers, Paci c Grove, CA, Oct. 31 - Nov. 2, 1994. D. N. Godard. \Self-Recovering Equalization and Carrier Tracking in Two-Dimensional Data Communication Systems". IEEE Trans. on Communications, vol. COM-28, No. 11, pp. 1867-1875, Nov., 1980. J. R. Treichler and B. G. Agee. \A New Approach to Multipath Correction of Constant Modulus Signals". IEEE Trans. on Acoustics, Speech, and Signal Processing, vol. ASSP-31, No. 2, pp. 459-472, April, 1983. O. Shalvi and E. Weinstein. \New Criteria for Blind Deconvolution of Non-Minimum Phase Systems". IEEE Trans. on Information Theory, vol. 36, pp. 312-321, March 1990. Y. Li and Z. Ding. \Global Convergence of Fractionally Spaced Godard Adaptive Equalizers". In Proc. 28th Asilomar Conf. on Signals, Systems and Computers, Paci c Grove, CA, Oct. 31 - Nov. 2, 1994. A. van der Veen, S. Talwar and A. Paulraj. \Blind Estimation of Multiple Digital Signals Transmitted over Multipath Channels". In Proc. IEEE MILCOM '95. S. Talwar, M. Viberg and A. Paulraj. \Blind Estimation of Multiple Co-Channel Digital Signals Using an Antenna Array". IEEE Signal Processing Letters, SPL-1(2), pp.29-31, Feb. 1994. B. Laheld and J.-F. Cardoso. \Adaptive Source Separation with Uniform Performance". In Proc. EUSIPCO-94, SIGNAL PROCESSING VII: Theories and Applications, pp. 183-186, Edinburgh, Scotland, Sept. 13-16 1994. C. B. Papadias and D. T. M. Slock. \New Adaptive Blind Equalization Algorithms for Constant Modulus Constellations" In Proc. ICASSP'94 Conf., Adelaide, South Australia, April 19-22, 1994.