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IEEE SIGNAL PROCESSING LETTERS, VOL. 2 NO. 5, MAY 1995

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Blind Estimation of Multiple Digital Signals Transmitted over FIR Channels Alle-Jan van der Veen, Shilpa Talwar and Arogyaswami Paulraj

Abstract | Using oversampling and the nite-alphabet where property of digital communication signals, it is possible to 2 3 2 3 blindly identify an FIR channel carrying a superposition of x1 (t) h11 (t)


h1d (t) such signals, provided they have the same (known) period, .. 75 ; and certain rank conditions on the data and channel ma- x(t) = 64 ... 75 ; H(t) = 64 ... . trices are satis ed. In particular, this technique allows to x (t) h (t)


h (t) M M 1 Md separate nite alphabet signals, remove intersymbol inter2 3 s (t) ference, and synchronize the signals. An algorithm is pro1 posed and tested on simulated data. 6 .. 7 s(t) = 4 . 5: sd (t) I. INTRODUCTION

In the context of blind identi cation of channels carrying digital communication signals, a number of algorithms have been proposed to (A) estimate a single FIR channel carrying one signal. In one class of these algorithms, initiated by Tong, Xu and Kailath [1], the signal is recovered by oversampling the channel output (viz. a.o. [1{4]). In the context of array signal processing, another scenario which admits blind identi cation is the case where (B) M antennas receive a superposition of d  M synchronized nitealphabet (FA) input signals via memoryless channels. An algorithm to recover the signals was recently proposed by Talwar, Viberg and Paulraj [5]. In this communication, we show that the above two scenarios can be combined to (C) blindly identify multiple FIR channels carrying d unsynchronized digital FA input signals that have the same symbol rate and alphabet. See gure 1. The new identi cation algorithm combines algorithms for (A) and (B). In the context of array signal processing, the algorithm can be used to separate, synchronize and recover a number of incoming digital signals, arriving from dierent or possibly the same directions, and distorted by multipath with nite delay spread.

II. DATA MODEL

We describePa digital signal s(t) as a sequence of dirac pulses, s(t) = 1 ?1 sk (t ? k) : For convenience, the symbol rate is normalized to T = 1. An array of M sensors, with outputs x1(t);


; xM (t), receives d digital signals s1 (t);


; sd (t) through independent channels hij (t). Each impulse response hij (t) is a convolution of the shaping lter of the i-th signal and the actual channel from the i-th input to xj (t), including any propagation delays. This data model is written compactly as x(t) = H(t)  s(t), A.J. van der Veen was with Information Systems Laboratory, Stanford University, Stanford, CA 94305. He is presently with the Department of Electrical Engineering, Delft University of Technology, Delft, The Netherlands. S. Talwar is with the Scienti c Computing Program, Stanford University, Stanford, CA 94305. A. Paulraj is with Information Systems Laboratory, Stanford University, Stanford, CA 94305. This research was partly supported by ARPA, contract no. F49620-91-C-0086, monitored by the AFOSR.

If we sample each xi (t) over N symbol periods at a rate P 2 N , where P is the oversampling factor, then we can construct a data matrix X as X = [2x0


xN ?1 ] 3 x(0) x(1)


x(N ? 1) 7 6 x(1 + P1 )
x( P1 ) 7 6 := 64 .. 7 .. 5: . . x( PP?1 )



x(N ? 1 + PP?1 ) j

The k-th column xk of X contains the MP samples taken in the k-th symbol period. If we assume that all hij (t) are FIR lters of length at most L 2 N : hij (t) = 0 ; t 2= [0; L), then at most L consecutive symbols of each signal play a role in x(t) at any given moment, and X can be factored as j

X = HST ; where

2 6 := 664

H(0) H(1)
H( P1 ) H .. . H( PP?1 )
2H : MP  dL; ... s0 6 6 ... ... 6 6 ST := 6




H(L ? 1)


.. .


H(L ? 1+ PP?1 ) sN ?2

3 7 7 7 5

3 77 sN ?2 7 7 . . . 77 5 sN ?1

... ... 6 4 s?L+2 s?L+3 ... s s?L+1 s?L+2 N ?L ST : dL  N; block-Toeplitz.

The blind identi cation problem is to estimate H and ST from X. Note that for such a factorization to be unique, it is necessary that H and ST have full column rank and row rank, respectively, which implies a.o. MP > dL. If this condition does not hold, we can extend X to a block-

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IEEE SIGNAL PROCESSING LETTERS, VOL. 2 NO. 5, MAY 1995 h11 x1 (t) s1 (t)

sd (t)

P1

h1d

hM 1

xM (t)

hMd

Fig. 1. Channel model. Input signals s1 (t); nized dirac-pulse sequences.






P1

x1k

A. Computation of S

xMk

; sd (t) are synchro-

Hankel matrix, by left-shifting and stacking m times, 2 3 . x0 x1 . . xN ?m .. .. 7 6 X = 64 x. .1. x. .2. .. . . x. N ?2 75 . x xm?1 . . N ?2 xN ?1 X : mMP  (N ? m + 1) : The augmented data matrix X has a factorization X = HS 3 2 ... s 2 3 6 sm?1 sN ?1 7 N ?2 0 . H. 6 . 7 . . . . . . . 66 . . 77 66 . . . sN ?2 7 7 = 4 H 56 . . . 77 64 s?L+2 s?L+3 . . . 5 H 0 . . . sN ?L?m+1 s?L+1 s?L+2 H : mMP  d(L + m ? 1) : block-Hankel; S : d(m + L ? 1)  (N ? m + 1) : block-Toeplitz: Now, necessary conditions for X to have a minimal-rank factorization X = HS are that H is a `tall' matrix and S is a `wide' matrix, which for L > 1 leads to MP > d ? 1) (1) m  d(L MP ? d N > (d + 1)m + d(L ? 1) ? 1 : Moreover, if H and S have full rank, then row(X ) = row(S ) and col(X ) = col(H).

III. BLIND IDENTIFICATION

taking advantage of the nite-alphabet property of the signals. In general, estimating H is computationally easier (for large N), but estimating S directly might be more accurate. In this letter, we focus on estimating S . Let G be a matrix whose columns constitute a basis for ker(X ). If H has full column rank, then G has dimensions (N ? m + 1)  (N ? (d + 1)m ? d(L ? 1) + 1) =: mG  NG . Moreover, X G = 0 ) S G = 0. Using the fact that S is block-Toeplitz, we obtain

S G = 0 , SGT = 0 where S

:= [2s?L+1

s?L+2




3sN ?1 ] ;

0

6 G 6 6 6 G ... 6 GT := 66 ... 6 6 ... G 4 0

7 7 7 7 7 7 7 7 7 5

(2)

GT : (N + L ? 1)  NG (L + m ? 1) :

If GT is a wide matrix, then ker(GT ) is d-dimensional and determines S, but only up to a left invertible d  d matrix A. This is because Y = AS also satis es Y GT = 0. To identify S, we impose the nite alphabet structure on S using the ILSP algorithm presented in [5]. ILSP is an iterative algorithm that solves the factorization (Y = AS : A; S full rank; [S]ij 2 FA), where FA is a pre-speci ed nite alphabet. In its simplest formulation, the algorithm consists of alternating projections: starting, e.g., with S (0) = Y , where in our case Y is a matrix containing any basis for ker(GHT ),  Project S (k) onto row(Y ): S (k)0 := S (k) Y y Y ,  Project each [S (k)0 ]ij onto the closest member of the alphabet, resulting in S (k+1) . The iteration generally converges very rapidly. B. Detection of d and L If H and S have full column rank and row rank, respectively, then the rank of X is dX := d(m + L ? 1). In principle, d can be estimated by increasing m by one, and looking at the increase in rank of X . This property provides a useful detection mechanism even if the noise level is quite high since it is independent of the actual (observable) channel length L^ . Furthermore, it still holds if all channels do not have equal lengths. If they do, then L can be estimated from the estimated rank of X , d^X , and the estimated number of signals, d,^ by L^ = d^X =d^ ? m + 1.

Our identi cation scheme is based on determining a factorization X = HS , such that either S is a blockToeplitz matrix with a speci ed row span, or H is a blockHankel matrix with a speci ed column span. However, such subspace information alone results in an ambiguity, since C. Remarks X = (HD?1 )(DS ) is a factorization with the same sub- If the channels do not have equal lengths, then H is spaces for D = diag[A;


; A] and A any invertible d  d not full rank and the above algorithm has to be modi ed. matrix. This ambiguity is resolved in a second step, by Omitting the details, we mention that it is sucient to set

VAN DER VEEN ET AL.: BLIND ESTIMATION OF MULTIPLE DIGITAL SIGNALS

L equal to the minimal channel length. This reduces the number of shifts of G in (2), so that ker(GT ) is larger than before. Besides the symbol sequences, this space now also contains a number of shifted copies of them. After processing this larger space by ILSP, it is straightforward to detect the shifted copies and remove them.

2

10

: m=2 +: m=3 : m=4 N =50; P =5

1

10

0

10

IV. SIMULATION RESULTS

In a simulation of a multiray wireless channel, we consider d = 2 unsynchronized BPSK signals, each modulated by a raised-cosine shaping lter (modulation parameter 0:35, and truncated to a length of 6 symbol periods). The signals are received by M = 2 antennas spaced by half a wavelength. The simulated channel consists of four paths per signal, where each path is speci ed by an angle-ofarrival , delay , and complex damping factor p: s1 : (i ; i; pi) 2 f(?10 ; 0; 1e?2:72j); (?2 ; :3; :8e:61j); (?120 ; 1:2; :4e:93j); (160; 2:1; :4e1:64j)g ; s2 : (i; i ; pi) 2 f(10; :5; 1e?3:07j); (15 ; :9; :9e1:17j); (?40 ; 1:5; :5e?1:06j); (150; 2:8; :3e1:83j)g : The resulting channel length is L = 9. We took N = 50 sampling intervals, with P = 5 times oversampling, and set the SNR to 10 dB per signal. The singular values of the resulting data matrix X are plotted in gure 2(a), for blocking factors m = 2; 3; 4. The numerical rank of X is about d^X = 10; 12; 14, respectively, so that the number of signals is detected as d^ = 2 and the observed channel length is L^ = 4, rather than 9. Figure 2(b) shows the singular values of GT , for m = 3, d^X = 12, and L^ = 3; 4. For L^ = 4, the number of small singular values is indeed equal to d^ = 2; for L^ = 3, we also obtain two shifted copies of the two signals, as expected. For the choice (L^ = 3, d^ = 4), all bits were estimated correctly, and the estimated symbol standard deviations (before classi cation as +1 or ?1) was about 0.20. For L^ = 4, however, not all bits were estimated correctly, and the standard deviations were about 0.60. In general, underestimating L and overestimating d yields signi cantly more accurate results. This is because ILSP now gets a larger responsibility in separating both the signals and their echos, which is favorable since the nite alphabet property is quite powerful. At this noise level, the alternative algorithm which rst identi es H and then estimates S = HyX gave bit errors of about 15%. More details are provided in [6].

3

(a)

−1

10

0

5

10

15

20

25

30

35

40

45

50

1

10

+ : ^L^ = 3 :L=4 m=3; d^X =12

0

10

(b)

−1

10

0

10

20

30

40

50

60

Fig. 2. (a) Singular values of , for dierent blocking factors m. (b) Singular values of GT . X

References

[1] L. Tong, G. Xu, and T. Kailath, \A new approach to blind identi cation and equalization of multipath channels," in 25-th Asilomar Conf. on Signals, Systems and Computers, pp. 856{ 860 vol.2, 1991. [2] H. Liu, G. Xu, and L. Tong, \A deterministic approach to blind equalization," in 27-th Asilomar Conf. on Signals, Systems and Computers, 1993. [3] D.T.M. Slock, \Blind fractionally-spaced equalization, perfectreconstruction lter banks and multichannel linear prediction," in Proc. IEEE ICASSP, pp. IV:585{588, 1994. [4] E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, \Subspace methods for the blind identi cation of multichannel FIR lters," in Proc. IEEE ICASSP, pp. IV:573{576, 1994. [5] S. Talwar, M. Viberg, and A. Paulraj, \Blind estimation of multiple co-channel digital signals using an antenna array," IEEE Signal Processing Letters, vol. 1, pp. 29{31, Feb. 1994. [6] A.J. van der Veen, S. Talwar, and A. Paulraj, \Blind identi cation of FIR channels carrying multiple nite alphabet signals," to appear in IEEE ICASSP-95, May 1995.