Subspace-based multiuser detection for DS-CDMA systems with correlated noise

Y.Zhang, G.Bi and B.P.Ng

Abstract: Two subspace-based multiuser detectors are presented for a synchronous DS-CDMA system operating over a multipath fading channel with correlated noise. Based on the training sequence transmitted by the desired user and employing a noise whitening process, a linear MMSE detector is obtained which does not require knowledge of the code sequences and the propagation channel response. When the training sequence is not available, a matrix decomposition method is adopted to estimate the channel response. Through signature waveform estimation, a linear MMSE detector can be obtained blindly. Simulation results show that these two methods are robust against the near-far problem and the correlated noise.

1 introduction

Direct-sequence code-vision multiple access (DS-CDMA) systems have received much attention for their application to multiuser systems such as mobile cellular and personal communication systems. Since the near-far problem is the main limitation of the conventional DS-CDMA systems, a number of near-far resistant multiuser detectors have been proposed (see [ 11, and references therein). These techniques can provide significant performance gains and substantially increase the capacity of DS-CDMA systems compared with the conventional detector.

In recent years, considerable attention has been focused on the linear adaptive multiuser detection which is based on the minimum-mean-square-error (MMSE) criterion. The linear MMSE detectors can be classified into two cate- gories: the training-based adaptive detectors and the blind adaptive detectors. The training-based linear MMSE detec- tors can be implemented using the standard algorithms such as LMS or RLS [2]. The LMS algorithm requires a few hundred bits to converge to a steady state, and the convergence time grows exponentially with the number of users even though it has O(N) complexity, where N is the processing gain. With the aid of a few tens of training bits, the RLS algorithm converges to a steady state with O(N2) computational complexity [2, 31. The training-based linear MMSE detection may be the most practical approach to high capacity DS-CDMA systems. However, the decision- directed adaptation could be degraded by an extremely strong incoming interferer that the detector has not already adapted to [4]. The blind adaptive detection schemes have recently been proposed in [MI. The blind MMSE detec- tors can work when unknown data sequences are being sent, and require prior knowledge of only the spreading waveform and the timing of the user of interest. It has been

0 IEE, 2001 ZEE Proceedwgr onhe no. 20010614 DOI 10 1049hpcom 20010614 Paper fmt m i v d 4th March 2000 and m r e d form 16th January 2001 The authorj are wth the School of Electncal and Electronic Enpenng, Nanyang Technologd Umverjity, Repubhc of Singapore 639798

shown that the blind technique based on LMS or RLS suffers from a saturation effect in the steady state, which causes a significant gap between its steady state perform- ance and the performance of the true linear MMSE detec- tor [5 ] . In the subspace-based blind approach, however, the linear detectors are constructed in a closed form once the signal subspace components are computed, and it is seen in [6] that the subspace-based detector outperforms the blind MOE detector in the steady state. The blind adaptive detec- tors based on the signal subspace tracking algorithms need to estimate the channel response if they are employed for the fading channel [6, 7.

Although it is usually assumed that the ambient noise is temporally white, such an assumption may not be valid in practice due to, e.g., the interference from some narrow- band sources. Correlated noise makes it ineffective for some interference suppression methods that are based on white noise assumption, e.g. the method in [7J In [SI, a subspace method whch required two well-separated anten- nas was proposed. The basic idea of the method is that the signal is received by two well-separated antennas, so that the noise is spatially uncorrelated. By performing SVD on the cross-correlation matrix of the two received signals, the noise subspace can be obtained. Although this method provides a new approach to estimate the channel informa- tion with unknown correlated ambient noise, it requires two well-separated antennas that increase the receiver complexity.

In this paper, we consider the problem of channel estima- tion and detection in a multipath CDMA channel with cor- related noise. We first present a subspace-based training method which does not require knowledge of the code sequences of all users and the propagation channel response. The second method uses a matrix decomposition method to obtain the noise subspace so that a linear multi- user detector can be constructed blindly. In the first method, the covariance matrix of the background noise is assumed known to the receiver. It is possible that the cov- ariance matrix of background noise can be obtained through some estimation methods or training methods. Once the matrix is available, the noise whitening method can be easily applied. To remove the limitation of such an

IEE Proc-Comniun Vol 148, No 5 October 2001 316

assumption, a matrix decomposition method is proposed to obtain the noise subspace for a blind multiuser detector.

2 System model

Consider a K-user synchronous CDMA system, where the transmitted signal due to the k h user is given by

00

y k ( t ) = A k b k ( n ) s k ( t - nT) (1)

where b&z) {b&) E f l } are the information bearing sym- bols, T is the symbol duration, A, is the amplitude of the kth user and sk(t) is the signature waveform. A multipath fading channel can be described as

n=-w

L

2 = 1

where L is the total number of paths, and 2, and a, are the associated delay and complex gain of the ith path, respec- tively. We assume that the channel order L << N, where N is the code length, since the maximum delay spread of the channel is usually insignificant relative to the symbol peri- ods. The signature waveform is given by

N

( 3 ) j=1

where {ck(l), ck(2), ..., ck(N), cko] E 41) is the pre-assigned code of the kth user and T, is the chip duration. Because of the effects of intersymbol interference (ISI) caused by multipaths, the signature waveform generally has a longer support, i.e. [0, (N + L -1)g. The discrete counterpart of the signature waveform in eqn. 3 is

N

j=1

L

= C h k ( l ) C k ( i - I + 1) i = l , . . . , N + L - 1 1=1

Ignoring the first L - 1 samples and denoting 3, = [sk(L)sk(L + 1) ... sk(N)IT, the IS1 free part of the received signal rk(n) can be represented by

r k ( n ) = A k s k b l c ( n ) (4) where

hk ( 5 )

C k ( L ) ... c k ( L + 1) . . .

C k ( N ) . . . C k ( N - L + 1) and hk = [h,(l), h,(2), ..., hL(L)Ir. The partially received signal, which is IS1 free, for K users plus additive back- ground noise is given by

h‘

k= 1

where v(n) is the (either white or coloured) noise vector and is independent of user signals. Based on the concept in [9], we know that the partially received signal in eqn. 6, rather than the entire signal, can be used for the signature wave- form estimation. One benefit to be achieved is that the effect of IS1 can be minimised. The signature waveform for each user is changed from sk to S k , so that the channel can be viewed as the flat-fading channel instead of the

IEE Proc -Cornmuri , Val 148, No 5, October 2’001

multipath fading channel. Because the number of data samples in r&) is reduced to N - L, computational complexity is accordingly minimised.

3 Subspace-based linear multiuser detection

3. I Subspace decomposition Denote S = [SISz ... SK] and A = diag(A12, ..., AK2). We also assume that the background noise is white Gaussian noise, and the covariance matrix of r(n) is given by

K

R = E { r r H } = A z S k S r + C J ~ ~ N - L + ~ k = l

= SASH + a 2 I ~ - ~ + 1 (7) By performing an eigendecomposition of the covariance matrix R, we get

R = UAUH = [UsUn] [” .,I [E;] (8 )

where U = [U, U,,], A = diag(A,, A,,), A, = diag(ill, ..., AK) contains the K largest eigenvalues of R in descending order, U, = ... pd contains the corresponding eigenvectors, A, - $IN-L+I-K, and U, = &K+l ... pN] contains the ( N - L + 1 - K ) eigenvectors that correspond to the eigenvalue $. The range space of U, is called the signal subspace and the noise subspace is spanned by U,, [6].

3.2 Linear MMSE detector Suppose that we are interested in demodulating the data bit of user 1. The linear receiver for this purpose can be repre- sented by a vector w1 E dN-L+l) because the demodulation is based on the IS1 free part of the received data. By doing this, the computational complexity is accordingly mini- mised. The ith bit of user 1 is computed by

-

(9) The linear MMSE detector is based on the minimum mean-square error (MMSE) criterion, which chooses w 1 to minimise the mean-square error (MSE) between the trans- mitted data bit and the receiver output, i.e.

Assuming that the data bits of all users are independent of each other and are also independent of the noise, the linear MMSE receiver for user 1, which mininlises the MSE in eqn. 10, is then given by [6]:

w1 = R-1- s1

= UsA,lUrS1 (11)

4 Training-based multiuser detection

With a reasonable assumption that the covariance matrix of the background noise & = ~ ( n ) v ( n ) ~ = dRh is known and full rank, we multiply the received signal by w,, = to get

K

@(n) = wbr(72) = A k W b S k b k ( 1 2 . ) + wb’U(72) k = l

K

= A k s k b k (n) + v(n) (12) k = l

where j k =,Wbsk and p(n) = WbV(n). The covariance matrix of ~ ( n ) is Rh = &ZN-L+~ The effect of multiplication with

317

W, in eqn. 12 is to whiten v(n). The covariance matrix of i(n) is given by

R = E { F ( r ~ ) + ( n ) ~ } Ii

= AE~,SE + V ~ I N - L + I k=l

= SASH + v ' IN-L+~ (13) where S = bl -s2 ... JK] and A = diag(AI2, ..., A K 2 ) . The matrix R can be decomposed as shown in eqn. 8. If we sub- stitute Pk = A& into eqn. 12, i(n) becomes

K

e(n) = C P k b k ( n ) +v(n) (14) k=l

whch has the same format as r(n) in [lo]. Therefore, the signature waveform 3, can be estimated in the same way as the propagation vector p k . In addition to the fact that we use the subspace estimation method for signature wave- form estimation rather than estimation of propagation vector for adaptive array processing, as illustrated in [lo], our method has two new features. The first one is that we use only part of the signal space, as shown in eqn. 6, rather than the entire signal space. The second feature is to employ a whitening process for coloured noise, as seen in eqn. 12.

Denote b, !2 [b,(l) ... bl(m)lT and r A (i(1) ... i(m)], where m is the number of training symbols, 6, is the vector of training symbols of the desired user, and r is the matrix of m corresponding received data vectors. The signature waveform j k can be estimated by

Substitute eqn. 15 into eqn. 11, and the linear MMSE receiver for user 1, which minimises the mean-square error (MSE) between the transmitted data bit and the receiver output, is then given by

g1 = ushs(u:rrHus)-lu;rbl (15)

2 0 ~ = u , ~ ; ~ uf us A, (u; rr us) u; r b l

= us (uf rr us) - r b l (16) The algorithm is summarised by: Step I : Transform the received data vector to i(n) as in eqn. 12. Step 2: Compute the autocorrelation matrix R and perform eigenvalue decomposition on R. Step 3: Compute the linear MMSE receiver vector as in eqn. 16. Step 4: Estimate the data bit as in eqn. 9.

5

5.1 Blind channel estimation The channel response Sk can be estimated by exploiting the orthogonality between the signal subspace and the noise subspace. Since U, is orthogonal to the column space of S and S , is the column space of S, we have

unsk = U , C k h k = 0 where U, is the noise subspace, S k and c k are defined in eqn. 5 and hk, subject to llhkll = 1, is the channel response. There are (N - L + 1 - K) equations and L unknowns in eqn. 17. Therefore, if K S (N - 2L + l), eqn. 17 is generally a set of overdetermined linear equations that have unique solutions. In fact, eqn. 17 provides an efficient way to iden- tify the channel vector directly from the data matrix up to a phase ambiguity. In this case, the data bits must be differ- entially encoded and decoded, i.e. the receiver detects

Blind channel estimation and detection

(17)

/ 3 k [ i ] = b k [ i ] b k [ i - 11 (18) 318

5.2 Blind channel estimation and detection in correlated noise The idea of blind channel estimation and detection in unknown correlated noise is about the same as that used for white Gaussian noise. However, when the noise is correlated, it is impossible to get the noise subspace by performing eigendecomposition on the covariance matrix. A subspace method was proposed by using two well- separated antennas in [8]. The basic idea of this method is that because the signal is received by two well-separated antennas, the ,noise is spatially uncorrelated. The noise subspace is obtained by performing singular value decom- position (SVD) on the cross-correlation matrix of the received signals. Although it is shown that the proposed approach is effective, this approach requires the hgh cost of the multi-antenna receiver.

We now consider subspace methods for blind channel estimation and detection in a multipath CDMA channel with unknown correlated ambient noise. We adopt the matrix decomposition method used in [l 11 to obtain the noise subspace. One advantage achieved by ths method is that only one antenna is needed, which is more practical for real applications. Once the channel estimation is achieved, the signature waveform can be estimated by eqn. 5.

Since R, which is the covariance matrix of the received signa1,'is Hermitian and positive definite, there exists a non- singular matrix B such that BRBH = I . Consider the trans- formation - -

I:: 1 Br(n) = U =

where r(n) is defined in eqn. 6, u1 has dimension K, u2 has dimension (N - L + 1 - K ) , and U is a random vector with covariance matrix I. Let E1 = (C D), where C is an (N - L + l )xKmat r ixandDisan (N-L+ 1 ) x ( N - L + l - K ) m a t h Then

B-'U = r(n) = [CD] = C U I + Dua (20)

Based on the fact that the covariance matrix of U is I , we can easily prove R = CCH + DD". The matrices CCH and D p are Hermitian with rank K and ( N - L + 1 - K), respectively. Let CCH = VIAIVIH and DOH = V2A2V2H, where VI is an (N - L + 1) x K matrix with the eigenvec- tors (corresponding to nonzero eigenvalues) of CCH as columns, and A, is a diagonal matrix of the corresponding eigenvalues. S d a r l y , V2 is an ( N - L + 1) x (N- L + 1 - K ) matrix with the eigenvectors of D F as columns, and A2 is a diagonal matrix of the corresponding eigenvalues. If we define

and R I = VAVH, we have R = R,RR,, which is related to the autocorrelation matrix R. Furthermore, we have R = E + F, where E and F satisfy the following properties

E and F are Hermitian. rank[E] = K, and rank[l;l = (N - L + 1 The subspaces spanned by the column spaces of E and F

are orthogonal. Both E and F have indices equal to 1, where index[A] =

min{k : k = 1, 2 ,... , rank[Ak] = rank[Ak+']}. F has ( N - L + 1 K) nonzero eigenvalues that are equal.

By applying the above properties, the (N - L + 1 - K ) ) generalised eigenvectors xi of the matrix pencil R x = Mx

K ) .

IEE Proc.-Cummun.. Vol. 148, No. 5 , October 2001

corresponding to nonzero GE values will lie in the null space of i? = RIRRl. The vector ui = Rlxi will lie in the null space of SASH according to the definition in eqn. 7. After obtaining the noise subspace U, by applying the matrix decomposition method described above, the channel response hk defined in eqn. 5 can be estimated by solving eqn. 17, the composite signature waveform 3, can be esti- mated as in eqn. 5 and the linear MMSE detector can be constructed as in eqn. 1 1.

6 Simulation results

In this Section, we provide some computer simulation results to demonstrate the performance of the proposed training-based and blind multiuser detection methods. The simulated system is a synchronous CDMA system. Each user is assigned a Gold sequence with code length N = 3 1. The number of resolvable paths is L = 3 and one-chip delay exists among these paths. The channel is assumed to be subject to Rayleigh fading, i.e. the channel fading gains are complex Gaussian random variables and fixed over the duration of one signal frame. The fading gain of each path is the same. The simulation results are obtained by averag- ing the results for 500 runs and the channel fading is gener- ated separately for each run. The frame length of the signal is M = 300, and the signal subspace estimation is based on each data frame. The correlated noise is modelled by a second-order AR model with coeficients a = [al, a2], i.e. the noise field is generated according to

.[.I = a1v[n - 11 + azv[n - 21 + W[.] (21) where v[n] is the noise sample and w[n] is a complex white Gaussian noise.

In the first example, the AR coefficients are chosen as a = [I, 4.21, the number of users in the system is K = 10. We compare the bit error rate (BER) against signal-to-noise ratio (SNR) A I 2 / q 2 performance of the traditional matched filter and linear MMSE detector, where q2 E { v ( ~ ) ~ } . Fig. 1 shows the performance in a system with perfect power control. Fig. 2 shows the result when the strong MA1 interference is l0dB above the signal power. In contrast, the method used in [lo] with additional signal processing measures gave a reasonable performance with a noise power 6dB below the signal power. Therefore, the proposed method works well for signature waveform esti- mation with only a few training symbols in strong coloured noise.

loo I

10-4 0 2 4 6 8 10 12 14 16 18 20

SNR, dB Training bused multiuw detection: BER uguht SNR (K = IO, MAI

matched filter

Fi 1

__ noise whitening = ?&

104 0 2 4 6 8 10 12 14 16 18 20

SNR, dB Fi .2 = %dB)

~ noise whitening

Training based multiuser cletection: BER aguimt SNR (K = 10, MAI

matched filter

loo^

10-2 0 2 4 6 8 10 12 14 16 18 20

SNR, dB Fig.3 MAI = OdB)

Blind channel estimution a d detection: BER ugaht SNR (K = 5,

___ matrix decomposition matched filter

'" 0 2 4 6 8 10 12 14 16 18 20 SNR, dB

Fig. 4 MAI = IOdB) __ matrix decomposition - _. matched filter

Blind chmnel estimtltion und &teetion: BER ugaht SNR ( K = 5,

In the next example, we investigate the BER against SNR performance of the proposed blind channel estima- tion and detection method. The AR coefficients are chosen

319 IEE Proc.-Commun., Vol. 148. No. 5, October 2001

as a = [ 1, -0.21, and the number of users in the system is K = 5. As in the first example, we compare the performance in perfect power control and strong MA1 interference, whch are shown in Figs. 3 and 4, respectively. It is shown that the signature waveform can be estimated in the corre- lated noise and the linear MMSE detector is near-far resist- ant.

Finally, we compare the performance of the proposed blind channel estimation and detection method with the method in [8], which requires two well-separated antenna and performs SVD on the cross-correlation matrix of the two received signals. The number of users in the system is K = 5. The AR coefficients are chosen as U = [l, -0.21 in our method, and a’ = [l, 0.21, u2 = C1.2, -0.31 in the simula- tion for the SVD method in [8]. The BER against SNR performance with perfect power control is simulated, as shown in Fig. 5. It is shown that the proposed method per- forms almost the same as the SVD method in [8], while the receiver complexity is reduced substantially.

loo i

7 Conclusions

In this paper, we have developed two multiuser detection techniques based on the signal subspace estimation for a synchronous DS-CDMA system in a multipath fading channel with correlated ambient noise. The proposed train- ing based multiuser detection method does not require knowledge of the code sequences of all users and the chan- nel response with the aid of a short training sequence trans- mitted by the desired user and a noise whitening process. Simulation results show that this method is near-far resist- ant and offers significant performance gains compared to the conventional detector. When the training sequence is not available, a blind channel estimation and detection method is proposed, where a matrix decomposition method is adopted to obtain the noise subspace with the signal received from one antenna. Compared with the previously proposed method in [8], our method can provide a good performance with substantially reduced complexity in the receiver.

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320 IEE Proc.-Commun.. Vol. 148. No. 5, Octoher 2001