Multiuser wavelet based MC-CDMA receiver with linearly constrained ...

MULTIUSER WAVELET BASED MC-CDMA RECEIVER WITH LINEARLY CONSTRAINED CONSTANT MODULUS IQRD-RLS ALGORITHM Shiunn-Jang Chern, Chung-Yao Chang and Hsiao-Chen Liu Depament of Electrical Engineering, National Sun Yat-Sen University Kaohsiung, 80424, Taiwan. Fax:(886)7-5254 199, Email:[email protected]

ABSTRACT

In this paper, a new robust linearly constrained constant modulus (LCCM) approach with inverse QRD-RLS algorithm is derived and applied to the multi-carrier code division multiple access (MC-CDMA) system. The proposed algorithm can be employed to reduce the MA1 efficiently, due to other users and combat the mismatch problem, when channel parameters could not be estimated perfectly. We show that it outperforms the conventional LCCM-gradient algorithm [5], in t e v of output SINR and improvement percentage index, for MA1 cancellation under mismatch environment. Moreover, the Wavelet-based structure can be used to further improve the performance of the proposed algorithm with pure MC-CDMA system. 1. INTRODUCTION The combination of the technology of orthogonal frequency division multiplexing (OFDM) and CDMA schemes, called MC-CDMA system, is attractive and commonly used to combat channel distortion and improve the spectral efficiency [ 1][2]. The basic idea behind this scheme is the division of the available spectrum into sub-bands of relatively narrow bandwidth, such that the sub-channels are nearly distortionless. Moreover, the multi-carrier modulation can be implemented by using the fast Fourier transform (FFT), which provides orthogonal basis between sub-channels without increasing complexities in both transmitter and receiver. We know that the discrete Wavelet multi-tone (DWMT) [3] can be utilized to form the multi-carrier modulation. The MC-CDMA system with DWMT is referred to as the WB-MC-CDMA system, where the orthonormal Wavelet filter banks [4] is used instead of the conventional Fourier-based exponential carriers in the pure MC-CDMA system. This scheme can be employed to reduce the interference and background noise, efficiently. Here the Wavelet filter banks are derived from the multistage tree-structured quadratic mirror filter (QMF). It is well known that the problems of near-far effect and multiple access interference (MAI), due to other users, in the CDMA systems may significantly degrade performance with respect to the desired user. Moreover, in wireless communication systems with fading channel, the information of channel parameters could not be estimated perfectly, it is referred to as

0-7803-7448-7/02/$17.00 02002 IEEE

the mismatch problem. To deal with the mismatch problem, a linearly constrained constant modulus (LCCM) criterion with gradient algorithm was suggested in [SI. By this approach, only the desired user's code sequence and channel coefficients are required and have to be estimated correctly. However, in fading channel by using the LCCM gradient algorithm, the weight vector could not converge to the optimal solution, due to its inherent slow convergence rate. Therefore, in this paper, a new scheme based on the linearly constrained constant modulus (LCCM) criterion with direct robust inverse QRD-RLS (IQRD-RLS) algorithm [6] is proposed. It is well known that the so-called inverse QRD-RLS algorithm has better numerical stability among the members of the IUS family and provide the faster convergence rate than the least mean squared (LMS) approaches. The fundamental idea of the linearly constrained IQRD-IUS algorithm was discussed in [SI for adaptive array processing. To further improve the performance of the LCCM -1QRD-RLS algorithm the Wavelet-based structure with the MC-CDMA system is devised. Simulation results verify the advantage of the algorithms proposed in this paper. Which can be used to compensate the drawback of the LCCM-gradient algorithm, in terms of the capability of MA1 suppression and the mismatch problem.

2. SYSTEM MODEL The transmitter model for the WB-MC-CDMA system with K simultaneous users and M carriers is depicted in Fig.1. The signature sequence of the kth (1 5 k 5 K)user is designated by a, = (...,uk(o),uk(l);. .,ak(N-1);. 3

0)

where ad13 are assumed to be independent and identically distribution (i.i.d.) random variables, such that Pr(ak(i)=-l)= Pr(adi)=l)=IR. The data signal of the kth user is given by

=(...,4(0),4(l)h(2),...)

(2)

where each data symbol is multiplied by N chips of signature sequences. From Fig. 1, the transmitted signal in discrete-time form for the kth user is denoted by

where P, is the power of the kth user in each filter branch, parameter q ( m ) is the code across the M filter branch, and g,,,(n)

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are the synthesis filters of Wavelet filter banks, for m=1,2, ...,M. Also, as illustrated in Fig.2, the h,(n) are the analysis filters, corresponding to gm(n),in the receiver. Moreover, we assume the transmitted signal undergoes independent frequency nonselective slow Rayleigh fading channel with additive white Gaussian noise (AWGN). Therefore, the received signal after Rayleigh fading channel can be expressed as

Where f m accounts for the amplitude distortion with Rayleigh distribution of the overall effects of fading channel, for the mth filter bank with respect to the kth user. And parameter, t m, is related to the phase distortion, with uniform distribution over [0,2 a 1, T,,, is the constant delay of the channel, and w(n) is the zero-mean complex Gaussian noise. As shown in Fig.2, the decision variable for the first user and ith symbol is given by

.g n=l

h, ( n ) g ,(iM + (T,

- fm) - n k J e ' , - +77

where the final term of (5) to be defined by M

+,

77 = C a1 m=I

L

(i)Zh, (n)w(iM - n)

(6)

n=l

and is the estimated delay in the mth branch of the receiver and L is the filter length. For synchronized receiver, e.g., fm = T ,, and because of the orthogonality of the filter banks, the decision variable signal can be rewritten as

+x (i)c K

M

ak,m m c

b,

k=2

(7) , (m)ah(i)a,(i)e'"'." + q

m=I

3. WEIGHT VECTOR DETERMINATION In this section, to derive the new algorithm, the LCCM criterion as suggested in [ 5 ] is used to obtain the appropriate weight vector to suppress the MA1 interference, efficiently, under mismatched environment. To have faster convergence rate and better numerical stability, the constrained IQRD-RLS algorithm proposed in [7] is utilized to devise the new adaptive LCCM algorithm, referred to as the robust LCCM-IQRD-RLS algorithm.

3.1 Robust LCCM-IQRD Algorithm In the CM criterion, the cost function, defined by the least square value of the receiver output error, is minimized subject to the linearly constraint systems, that is

In (1 0), is the forgetting factor, and Z(n) = y'(n)z(n) it can be viewed as the new input data vector from the constant module criterion. For convenience, we let e(n)=( 1-k(n)l2) and fin)=w"(n)z(n) is the real output. c is a M X P constraint matrix, and f is a PX1 response column vector. Proceed in a similar way as in [7][8], the optimal constrained solution of the LS weight vector, via Lagrange multiplier approach, based on the inverse QRD decomposition, can be derived

W=p(nMn) + ~ ( n ~ n ) r 1 s ~ ~ ( n ~ n ) ~ ~ - ' [ f - ~(1p1)( n M n ) ] where R(n) is the MxM upper triangular matrix, which is obtained via the Givens mfarion.To derive the recursive equation of (1 I), three new matrices S(n)=R-l(n)R-"(n), (n)=S(n)C and (n)=CHS(n)C are defined. After some mathematical manipulation, the recursive implementation of weight vector of (1I), referred to as the LCCM-IQRD-RLS algorithm, is obtained

In (7) the decision variable can be decomposed into three parts corresponding to the desired user, interference and background noise. Assume that the desired user is the first user and the transmitted data is bl(O), the output of the combiner of the mth branch will be

where

(8) SI (m)= 4 (O)J2P,Cl ("T,, the corresponding output signal of the mth branch due to other users, the kth user for k 2 2 ,is denoted as

e(n,n -1)=1 -W" (n-l)Z(n) (14) It is noted that the adaptation gain, k(n)=g(n)/t(n) is evaluated by the conventional inverse QRD-IUS algorithm [6] and from [7][8] (n)and -'(n) can be derived in recursive forms:

'k

(m)= f i c k ( m h , m b k

( i h ( i k ( i )

(9)

w(n) = w(n-l)+p(n)e(n,n-l) /J(n)= k(n) --

t(n)

r(n)= dr(n- I) - g(n)a(n) W (n)=41+ JX.(n a(n)Pl ) (n -1)

The overall output of the M combiners, in vector form, is given by K

r(n)q(n)

(12)

(13)

(15)

(16)

and

z=s+Cik+q k=2

To provide frequency diversity, an adaptive filter with a set of M-dimensional adjustable weight coefficients w=[wl,w2,. . .,wM]' is applied to the M different channel of filter banks to execute symbol-by-symbol detection. In next section, a linearly constrained constant modulus (LCCM) approach is developed for updating the weight coefficient to perform symbol detection.

with (n)=$(n)C. For convenience (11) can be reformulated, and after some mathematical arrangement, we have

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w(n) = w(n -1)

+ F(n)k(n)e(n, n - 1)

(18)

where F(n)=Iw (n) -'(n)C? is a projection operator. After adding the correcting term to the right hand side of (1 2), we get

Mn) = M n - l ) + F n - l ) + ~ n ~ - ' ( n ) [ f-cHw(n-l)] Next, we define a MxP matrix (n)= (n) mathematical manipulation, we obtain

qn)=kd(n-I) -k(n)v"(n)].

-I@),

(19)

after some

is simple, it is not suitable for wireless communication systems due to slow convergence rate.

-

1 fl(n)u(n)

where u(n)=C"k(n) and F(n)=p(n)(n-I). If there is no round off error accumulation, by the definition of (n), ideally, the following equation holds:

C"Q(n) = I ,

(21)

If the round off error is considered, (n) is replaced by ' (n), using the correcting term in the same manner as in (19), the correcting matrix can be formulated by

n(n) =n'(n)+C(CHC)-'[I,-c"n'(n)]

(22)

accordingly, the weight vector updated equation becomes

~ ( n=)~ ' ( -1) n +n(n)[f-C"w'(n

- I)]

(23)

with

w'(n-l)=w(n-I)+k(n)e(n,n-l)

(24)

Where w ' (n) is denoted as the case when w (n) suffers from the round off error. This completes the derivation of robust linearly constrained constant modulus IQRD-RLS algorithm.

3.2 Linearly Constrained CM Gradient Detector For comparison, the LCCM approach with gradient algorithm was briefly reviewed. In [ 5 ] , the partitioned linear interference canceller (PLIC) structure is used, where the cost function is minimized subject to a linear constraint, which defined as Iy(n)r - 11'1, s.t. C"W = 1

lml'

(29) e g (n) = -1 In (28), z.(n) is defined as z,(n)=Mz(n). Although, this approach

(25)

The overall weight vector is defined as w=w,-MHw,. where w, is the non-adaptive component of the overall weight vector corresponding to the constraint matrix and w, is adaptive to mitigate the interference. Also, M is the blocking matrix. The lower branch weight vector can be derived via the following criterion without constraint:

with the normalized stochastic gradient algorithm of the form, (26) can be solved and implemented with the following updated equation of w,:

w , ( n + l ) = w ~ ( n ) + ~ ~ ( n ) Q L i n ~ * - l ~ ( n ) M ~ n )(27) where a is a positive real relaxation factor to reduce the algorithm misadjustment noise, and it can be chosen in the interval 0 < a: < 1. The time varying step size ,,(n) and the output error e,(n) are chosen based on following equations at each iteration:

4. COMPUTER SIMULATIONRESULTS To verify the performance of the method proposed in this paper, the output SINR and the improvement percentage index (IPI) are evaluated. To see the merits of the proposed scheme, the mismatch problem in MC-CDMA system for Rayleigh fading channel is considered. For convenience, we assume that user one is the desired user, and the components related to other users are contributed to as the MAL Where the power of each user is 20dB stronger than the desired user. In this case, the desired value of SNR is about 15 dB. In the MC-CDMA system to provide the frequency diversity, the linear constraint C becomes a single constraint case, that is, the constraint matrix C is reduced to a vector, corresponding to the code sequence of desired user across each branch and channel coefficients, which is denoted as

C=[c,q, c2q.2

..'

C&,MIH

and f = 1

(30)

To model the mismatch environment, the mismatch of the components in C was introduced, where the values of errors are generated with Gaussian random variable of a zero mean and variance 0.05. Moreover, Wavelet filter banks are derived from a multistage tree-structured, that is, the Harr (dbl) QMF and Daubechies QMFs, viz., Daubechies 2 (db2), Daubechies 3 (db3), Daubechies 4 (db4) and Daubechies 5 (db5), and are used in our computer simulation. Here, the forgetting factor is chosen to be =0.995 and the IPI value is defined as

IPI =

Optimal SJNR - S N R of algorithm Optimal SINR

(3

Since in the WB-MC-CDMA system with db5 has the best performance, in Fig.3, we show the result in terms of SINR under the MA1 environment ( 6 undesired users). We learn that the proposed algorithm provides 7 dB output SINR value higher than the LCCM-gradient algorithm. Next, to fairly compare the results of WB-MC-CDMA with pure MC-CDMA systems, the IPI is used as the performance index. From Fig.4, we observed that the value of IPI with the LCCM-IQRD-RLS algorithm is less than the one with the LCCM-gradient algorithm. Moreover, to see the superiority of using the WB-MC-CDMA with different db's, the results in terms of output SINR value (dB) is given in Table I , as reference. We learned that the Wavelet-base systems have better performance than those with the pure exponential multi-carrier system. Due to the fact that the sub-band channel with Wavelet banks has better noise suppression capability, especially, when the LCCM-IQRD-RLS algorithm is employed.

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Table 1. Comparison of the output SINR values with WB-MC-CDMA and pure MC-CDMA systems ( 200 updating iterations) Exponential Wavelet QMF bank Algorithms dbl db2 db3 db4 db5 multi-camer 4.6149 LCCM-IQRD 19.5408 19.9429 19.9430 19.8483 110.004

I

I

I

I

I

CM-gradient(3.669813.7405~3.4820~3.7405~3.3999] -8.7871

I

I

5. CONCLUSIONS

In this paper, the LCCM-IQRD-RLS algorithm has been derived and applied to the MC-CDMA systems, with and without DWMT structure, for MA1 cancellation under mismatch environment. From simulation results, we leam that the proposed algorithm outperforms the conventional LCCM-gradient algorithm as suggested in [5]. Also, with the DWMT structure, the performance of the LCCM-IQRD-RLS algorithm could be improved compared with the one without using the DWMT structure. We found that the proposed algorithm is robust to the imperfect channel estimate.

Fig. 1.Transmitterfor the kth user in the WB-MC-CDMA system

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0

6. ACKNOWLEDGEMENTS The financial support of this study by the National Science Council, Republic of China, under contract number NSC-90-22 13-E-I10-024, is greatly acknowledged.

References [I] S. Hara and R. Prasad, “Overview of Multicarrier CDMA,” IEEE Communication. Mag., pp. 126133, Dec. 1997. [2] Tat M. Lok, Tan F. Wong and James S. Lehnert, “Blind Adaptive Signal Reception for MC-CDMA Systems in Rayleigh Fading Channels” IEEE Trans. Commun., vol. 47, no. 3, pp.464-471, March 1999. [3] A. Muayyadi and M.N.A. AbuRgheff, “Wavelet-Based MC-CDMA CelluIar Systems,” Proceeding of IEEE INT. Symposium on Spread Spectrum Techniques and Applications, vol. 1, pp. 145-149, Sept. 2000. [4] B.G. Negash and H. Nikookar, “Wavelet-Based Multicarrier Transmission Over Multipath Wireless Channels,” Electronics Letters, vol. 36, pp. 1787-1788, Oct. 2000. [5] J. Miguez, and L. Castedo, “A Linearly Constrained Constant Modulus Approach to Blind Adaptive Multiuser Interference Suppression,” IEEE Commun. Letfers, v01.2, no. 8, pp. 2 17-219, Aug. 1998 [6] S. T. Alexander and A. L. Ghimikar, “A Method for Recursive Least Squares Filtering Based upon an Inverse QR Decomposition,” IEEE Trans. Signal Processing, vol. 41, no.1, pp. 20-30, Jan. 1993. [7] S. J. Chern, C. Y. Chang and T. Y. Liao, “Adaptive Multiuser Interference Cancellation with Robust Constrained IQRD-RLSAlgorithm for MC-CDMA System,” Proceeding ofIEEE ISPACS’OI (accepted to present), Nov. 2001 [SI S. J. Chern and C. Y.Chang, “Adaptive Linearly Constrained Inverse QRD-FUS Beamforming Algorithm for Moving Jammers Suppression,” IEEE Tram. Antennas Propagation, (to appear), July 2002

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Fig 3. The output SINR of two algorithms in WB-MC-CDMA system with dbS-QMF bank. “rowmml

p o I C o “ I I 0 ~m4sx

1 5

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>--7-+-777

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Fig 4. The improvement percentage index of different algorithms with WB-MC-CDMA and pure MC-CDMA.

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