Vol.20 No.4 JOURNAL OF ELECTRONICS July 2003

A N O V E L D E T E C T O R F O R U P L I N K C D M A S Y S T E M W I T H U N K N O W N I N T E R F E R E N C E 1

Wu Lili Huang Hui Liao Guisheng

(Key Lab for Radar Signal Processing, Xidian University, Xi'an 710071)

Abstract In this letter, the detection of asynchronous DS-CDMA signal with multipath fading and interference from neighboring cells is studied. A novel multiuser detector based on Gibbs sampler is proposed, in which Gibbs sampler is employed to perform the Bayesian multiuser detec- tion according to the linear group-blind decorrelator output. Since Gibbs sampler is dependent of parameter estimation that can be improved by the output of the detector, an enhanced Gibbs sam* pler based detector using the improved parameter estimation is put forward. The novel multiuser detection technique has the advantages of high performance and wide applications. Computer simulations show its effectiveness.

Key words Gibbs sampler; Asynchronous DS-CDMA signal; Bayesian multiuser detection; Parameter estimation

I. I n t r o d u c t i o n

In digital communications, the received signal y is a function of the transmitted sym- bol sequence b = [bl ,b2, . . . ,bK] T. The design of optimal digital communication receiver is based on the Bayeisan formulation of the problem. In this approach, we are interested in computing the marginal posterior probability distribution of each symbol, say bk, condi- tioned on the observation denoted as y, i.e., p(bk[y), 1 < k < K. However, direct evaluation is computationally infeasible, especially when the dimension K is large.

Recently, a completely different paradigm for solving the above problem is used by Markov Chain Monte Carlo (MCMC) sampling methods [11. Gibbs sampler [1], a famous MCMC algorithm, has been successfully applied to the detection of synchronous DS-CDMA signal[2,al.

In this letter, the detection of asynchronous DS-CDMA signal with interference from neighboring cells (unknown interference) and multipath fading is studied. We are motivated to perform the optimal multiuser detection according to the linear group-blind decorrelator output [4]. And a novel group-blind multiuser detector based on Gibbs sampler is proposed, based on which we also design an improved detector where more accurate parameter esti- mation is achieved to enhance the detection performance.

II. G i b b s S a m p l e r

Suppose we are interested in the Bayesian estimation of an unknown bk, which maxi- mizes the marginal posterior probability p(bk [y). The basic idea behind MCMC is to gener- ate ergodic random samples from the joint posterior distribution p(b]y) and then to average the appropriate samples to estimate marginal posterior probability p(bk[y). Usually, it is difficult to directly draw samples from p(bly ). Gibbs sampler generates samples from the objective distribution using the full conditional distribution.

1Manuscript received date: December 20, 2002; revised date: January 10, 2003.

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Given any initial value b(°) = tvlrh(°), ~2 h(°), • .- , b(~ ), Gibbs sampler iterates according to the following procedure.

At the t-th iteration

sample

sample

sample

b~ t) ..~ p(bl lb~t-1), h( a , '" . , b~ ' U, y) b~ t) ,.~ p(b21b~ t), b~t-1), . . . , b~ -1) ' y)

~ ' ( O K I O 1 ~o 2 , ' ' "

The Gibbs sampler requires an initial transient period to converge to equilibrium. The initial period of length no is known as the "burn-in" period. The first no samples should always be discarded.

I I I . S y s t e m M o d e l

We consider multiuser detection of asynchronous DS-CDMA signal with multipath fad- ing and additive white Gaussian noise. The signature sequences of the first/7/users within the cell are known to the receiver. Those of the rest (K - / ~ ) users from neighboring cells are unknown. The received signal can be expressed in the following form [5].

r[i]= Hb[i] + nil] (1)

where, H denotes the channel response matrix, b[i] is composed of the transmitted symbols from the K users and n[i] is the noise vector. The output of linear group-blind decorrelator

is derived in Ref.[5] and given in the following expression:

o[i]= DUr[i] = _ab[i] + ffIn[i] (2)

where, b[i] = [bl [i], b2 [i],. . . , bR[i]] T is a vector containing the i-th symbols of the /~ known users, and ~l =diag{cq, . . . ,aR} is the ambiguity induced by the blind channel estimator, which can be coarsely estimated by a simple method [~]. The eigendecomposition of the autocorrelation matrix of r[i] gives the estimation of noise covariance a 2. The output of linear group-blind decorrelator can be expressed in a real form written as follows:

where

[ Re(O[il) ] y[i] = [Im(O[il)] '

The covariance of vii t is given by

where

y[i] = ~ob[i] + v[i] (3)

[ Re(DZn[ i l ) ] vii] = [ Im(~:}Hn[i] ) ] ,

[ Re(A) ] ~o = [ i m ( ~ ) J

O [o coy{v i i i ) = ~ = T

Q = Re(/))TRe(D) + Im(/))TIm(/9)

(4)

IV. A N o v e l Mul t iuser D e t e c t o r

First, we propose a novel nonlinear group-blind multiuser detector based on Gibbs sampler. For convenience, channel coder is assumed not to be included in transmitter and

No.4 A NOVEL DETECTOR FOR UPLINK CDMA SYSTEM 291

symbols are BPSK modulated. The symbol sequences of different users are independent. We have [1 ]

p(b[i]ly[i]) cx p(y[illb[i]) = (27r)~lEi1/2 exp - (y[i] - ,~ob[il)rN-l(y[i] - ~,ob[i]) (5)

Gibbs sampler is employed to draw random samples from p(b[i]ly[i]). To implement Gibbs sampler, we need the full conditional distribution derived from the posterior distri- bution. It can be derived from Eq.(5) as

p(bk[i] = ÷llb-k[i] ,y[i]) =(1 + exp [-- l(y[i] _ ¢pb[i]_k) T~_I (y[i] - ~ab[i] -k)

"~- X(y[i] __ ~ob[i]+k)T~__l (y[/] __ ~ob[i]-l-k)] } -1 (6)

where

b_k[/] b[i] \ bk[i]

The k-th

b[i] -k \ bk[i] : b [ i ] +k \ bk[i] : b-k[i]

The expression "A \ a" means to strike out the element a from a vector A. components in b[i] +k and b[i] -k are +1 and -1, respectively.

Gibbs sampler draws T samples from posterior distribution in Eq.(5). samples are discarded. We have

T

where

The first To

1 y~ Jg(~) (7) ~(bk[i] = +l[y[i]) ~ T - To Vki

t~-To+l

(t) ~ 1, b~t)[i] = +1

ki = ( 0 , b~t)[i] = - I

The marginal posterior probability in Eq.(7) provides the optimal estimation of the i-th symbol of the k-th user according to the linear group-blind decorrelator output y[i].

The complexity of above detector is O(/~T), which is a substantial complexity reduction compared with the direct !mplementation of the Bayesian symbol estimation when/~ is large, whose complexity is o(2K).

We notice that the full conditional distribution in Eq.(6) is dependent on ~I whose estimation error degrades the performance of the proposed detector. Therefore, we give an improvement described as follows.

2 Once the estimation b[i](i = 1, 2 , . . . , L) given by the above detector is got, the Least

Square (LS) estimation of ~I, indicated by ~ILS, is obtained by substituting ~[i] (i = 1, 2, . - . , L) into Eq.(2). Then Gibbs sampler is again employed to draw random samples according to

the full conditional distribution that uses more accurate estimation ALS. By doing so, better detection performance is achieved. The computational complexity of the improved detector is twice that of the unimproved one.

V. Simulation Results

In the simulated system, Gold sequences of length 31 are used as the spreading se-

quences. K = 20,/~ = 15. Each user's channel has 3 paths and a delay spread up to one

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symbol. The path gains in each user's channel and the initial transmission delays are gen- erated randomly and fixed for all simulations. All users are transmitted with equal power. The users' symbols are BPSK modulated and no channel coder is included in transmitter. SNR is defined by the ratio of desired user's transmission power to additive noise power. The decision of linear group-blind decorrelator output is taken as the initial value of Gibbs sampler that generates 200 random samples and the subsequent 100 ones are used to make inference. We use 10 a to 106 independent symbols, depending on the SNR value, to esti- mate the BER at a tested SNR. In Fig.l, the experimental BER curves of linear group-blind decorrelator, the Gibbs Sampler Based Detector (GSBD) and its improvement are plotted. It is obvious that the latter two outperform the linear counterpart to a great extent and the improved BER is about half that of the unimproved detector.

10-~

r , A

10 -2

10 -3

iff

[ 0 0 , , , , , , , , f

I I I I I I I I I

5 6 7 8 9 10 11 12 13 14 SNR (da)

Fig.1 BER performance of three detectors

VI. Conc lus ion

A novel multiuser detector based on Gibbs sampler is proposed in this letter. Gibbs sampler, a famous MCMC method, is employed to perform the Bayesian multiuser detection according to the linear group-blind decorrelator output. And then we put forward a more accurate parameter estimation method to improve the performance of the proposed detector. This multiuser detection technique has the advantages of low complexity and high detection performance. It can be used in many applications.

References

[1] A. E. Gelfand, A. F. W. Smith, Sampling-based approaches to calculating marginal densities, J. Amer.

Star. Assoc., 85(1990)3, 398-409.

[2] X. Wang, R. Chen, Adaptive Bayesian multiuser detection for synchronous CDMA with Gaussian and

impulsive noise, IEEE Trans. on SP, 47(2000)7, 2013-2027.

[3] Y. Huang, P. M. Djuric, Multiuser detection of synchronous code-division multiple-access signals by

perfect sampling, IEEE Trans. on SP, 50(2002)7, 1724-1734.

[4] X. Wang, A. Host-Madsen, Group-blind multiuser detection for uplink CDMA, IEEE d. on Select.

Areas Commun., 17(1999)11, 1971-1984.

[5] P. Spasojevic, X. Wang, et al., Nonlinear group-blind multiuser detection, IEEE Trans. on Commun.,

49(2001)9, 1631-1641.