s *(t- τ 1 ) 1 r(t) s *(t- )

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1 LOW COMPLEXITY ADAPTIVE RECEIVER FOR CDMA WITH MULTIPATH FADING Sridhar Gollamudi, Shirish Nagaraj, Samir Kapoor and Yih-Fang Huang Laboratory for Image and Signal Analysis Department of Electrical Engineering University of Notre Dame Notre Dame, IN USA. ABSTRACT A new receiver scheme for a DS-CDMA system similar to the IS-95 uplink is proposed in this paper that is of lower complexity and superior bit error rate performance in comparison to the conventional non-coherent Rake receiver. The proposed receiver consists of a coherent Rake receiver in the form of a transversal lter, and uses channel estimates obtained by an adaptive channel identication algorithm. Adaptation follows the Set-Membership Filtering methodology, that features fast convergence and lower computational complexity compared to least-squares techniques. I. INTRODUCTION There has been a tremendous research interest in recent years in adaptive receivers for CDMA systems that counter the eects of multipath fading and multiaccess interference (MAI). However, almost all the proposed techniques rest on two critical assumptions, namely, that the modulation is linear (e.g., BPSK) and that the spreading codes have a time period equal to the bit interval. Both these assumptions are not valid in typical multiaccess cellular CDMA systems such as the uplink in the IS-95 standard. This paper proposes an adaptive scheme that improves the performance of a CDMA system which employs M-ary orthogonal modulation and spreading sequences with no assumed (or large) periodicity, as is the case in IS-95. The Rake receiver [1] is commonly used to provide time-diversity in a multipath environment by combining multiple copies of the transmitted signal that reach the receiver at dierent delays. Since it is wasteful of This work was supported, in part, by the National Science Foundation under Grant MIP , in part, by Lucent Technologies, Whippany, NJ, USA, in part, by the Center for Applied Mathematics, University of Notre Dame, and in part, by the Tellabs Research Center, Mishawaka, IN, USA. power for the mobile transmitters to transmit pilot signals to aid phase recovery at the receiver, Rake combining and demodulation are usually carried out non-coherently. The reader is referred to [] and the references therein for details on DS-CDMA reception. Performance of a Rake receiver can be improved if the Rake ngers are combined and demodulated coherently, with the help of phase estimates on all the paths. Further improvement is possible via maximal ratio combining, that is, by weighting the Rake ngers in proportion to their signal strengths to maximize the output signal-to-noise ratio. Both these techniques can be incorporated if accurate estimates of complex channel gains are obtained. As will be shown later in this paper, the resulting Rake receiver is simply a matched lter that is matched to the multipath channel. Receiver complexity is substantially reduced due to the fact that the proposed receiver requires a single despreader and a single set of M-ary symbol correlators per each user, as opposed to one for each nger of the Rake in conventional receivers. Complex channel estimates can be obtained by an adaptive identication algorithm with the help of training signals. A novel and emerging adaptive ltering paradigm known as Set-Membership Filtering (SMF) is utilized in this paper for this purpose. SMF is an extension of the well-studied Set-Membership Identication (SMI) methodology [3, 4, 5, 6, 7, 8] for bounded-noise systems to the general parameter estimation problem for a linear-in-parameters model [9, 10, 11]. SMF algorithms oer several advantages over stochastic gradient algorithms such as the LMS algorithm and least-squares techniques such as the RLS algorithm in terms of convergence behavior, tracking characteristics and computational complexity. This paper is organized as follows. The CDMA system model is introduced in the following section. Sec- 1

2 tion III shows the equivalence of the coherent Rake receiver and a matched lter, and presents the proposed receiver scheme. SMF and its application to channel estimation are described in Section IV. Section V shows the simulation results to compare the performance of the proposed receiver with that of the conventional receiver, and concluding remarks are made in Section VI. II. CDMA SYSTEM MODEL The complex baseband received signal at a multiaccess asynchronous CDMA receiver is given by = KX k=1 r k (t) + v(t) (1) where K is the number of users, r k (t) is the kth user's received signal and v(t) is white Gaussian noise. The multipath channel for the kth user is characterized by L copies of the transmitted signal, each with dierent delays ( k;l ), gains (a k;l ) and phases ( k;l ), as shown below. r k (t) = c k;l s k (t? k;l )u k (t? k;l ) () where c k;l = a k;l e j k;l is the complex channel gain on the lth path for the kth user, s k (t) is the spreading code, and u k (t) is the information signal of the kth user. The spreading code is usually complex in order to make the interferers' phases appear random [], and is drawn from the constellation fe j 4 ; e j 3 4 ; e j 5 4 ; e j 7 4 g in an IS-95-like system. No assumptions are made on the periodicity of the spreading code. Information symbols belong to an alphabet of M mutually orthogonal waveforms f (t); : : : ; (t)g, each T seconds long. The uplink on IS-95 uses 64-ary orthogonal Walsh codes as the symbol waveforms. The information-bearing signal can expressed as u k (t) = 1X n=?1 W Ik (n)(t? nt ) (3) where I k (n) f1; ; : : : ; M g is the nth symbol index of the kth user. Conventional non-coherent Rake receiver architecture for the kth user is shown in Figure 1 1. Each nger of the Rake is time-synchronized to a multipath component. The received signal is despread on the lth nger by the complex conjugate of the spreading code, s k (t? k;l), and correlated with all the M 1 Note that the user index k is omitted throughout in the gures. symbol waveforms. The resulting correlation vector is delayed k? k;l seconds to align with the other ngers, where k = max( k;1 ; : : : ; k;l ). The correlation vectors are then summed non-coherently over all the ngers, and the maximum of the resulting correlation vector, max i y i, determines the decoded symbol. Each decoded symbol is then mapped onto the corresponding string of log M decoded bits. This receiver requires L despreaders and LM correlators per user. s *(t- τ 1 ) 1 s *(t- ) τ L Figure 1: Conventional non-coherent Rake receiver with equal-gain combining. III. PROPOSED RAKE RECEIVER If phase information k;l is available, the phase on the received signal can be compensated on each nger so that coherent Rake combining and coherent symbol detection can be performed, thereby improving bit error rate. Figure (a) shows the architecture for coherent reception using a Rake for the kth user. On the lth nger, the unspread signal is phase corrected by taking the real part after rotation by a phase angle? k;l, and then correlated with the M symbol waveforms to produce a vector of M correlations. The vector is weighted by a positive scalar k;l and time-aligned with the other ngers. The time-aligned correlation vectors are then summed over all the ngers to produce the decision statistic y = [ ; : : : ; ] T. The scalar weighting factors k;l are all unity for the so-called equal-gain combining, and are equal to the corresponding amplitudes of the signal on each nger, i.e., k;l = a k;l, for maximal-ratio combining. Maximal-ratio combining is the optimal way to combine the Rake ngers in terms of maximizing the output signal-to-noise ratio and requires knowledge of the channel gains a k;l. The detected symbol index is given by arg max i y i.

3 Since all the operations in the coherent Rake receiver prior to the decoder { despreading, phase correction, symbol correlations, delaying and Rake combining { are linear, it is easy to see that much of the processing in Figure (a) is redundant and can be eliminated. Mathematically, the operation of the receiver in Figure (a) can be expressed as, for i = 1; ; : : : ; M, y i = k;l Z (n+1)t +k nt + k Re fr(t? k + k;l ) s k(t? k )e?j k;l Wi (t? k )dt (4) j ϕ s *(t- τ )e j ϕ s *(t- τ L) e L α 1 α 1 α L α L The above expression can be rewritten as follows y i = Z (n+1)t +k nt + k Ref[! k;l r(t? k + k;l )] s k(t? k )gw i (t? k )dt (5) where! k;l = k;l e?j k;l. Without loss of generality, order the ngers so that k;1 < k; < : : : < k;l, and hence k = k;l. Equation (5) then suggests a simplied architecture for the coherent Rake receiver, shown in Figure (b). It consists of a transversal lter with complex coecients! k;l followed by a despreader and a single set of symbol correlators. Therefore, this receiver requires one despreader and M correlators for each user. Note that such a simplication is not feasible with non-coherent reception because the non-linearity of the magnitude-squared operation makes the Rake combining and correlation operations non-commutative. If maximal-ratio combining is used, then the lter coecients are given by! k;l = a k;l e?j k;l = c k;l, where c k;l is the complex conjugate of the complex channel gain c k;l. The impulse response of the transversal lter can be expressed as! k (t) = c k;l (t? k + k;l ) (6) where () is the unit impulse function. Comparing this with the impulse response of the multipath channel c k (t) = c k;l (t? k;l ); (7) it is clear that the coherent Rake receiver with maximal-ratio combining is a matched lter receiver whose impulse response is matched to that of the multipath channel. This is not an unexpected observation, since a matched lter receiver is optimal in the presence of white Gaussian noise, and the combination of multipath and multiaccess interference can be considered as L L-1 L-1 L- 1 ω L ω L-1 ω 1 s *(t- τ ) (a) (b) Figure : (a) Coherent Rake receiver, (b) Equivalent matched lter receiver. essentially white Gaussian noise in a CDMA system. This is due to the low correlation that exists between the spreading codes and delayed copies of themselves as well as with interferers' spreading codes, and the Central Limit Theorem. IV. CHANNEL ESTIMATION USING SMF The receiver proposed in Section III, though simple in structure and optimal in performance, requires knowledge of the complex channel impulse response c k (t). Knowledge of multipath delays k;l is obtained by synchronization loops and is assumed in any Rake receiver []. The complex channel gains c k;l need to be estimated, and a technique based on SMF theory is proposed in this section to obtain them eciently. Dene the kth user's signal on the lth path without channel gain as x k;l (t) = s k (t? k;l )u k (t? k;l ). Consider samples of the system at time instants nt s ; n = 0; 1; ; : : :, where 1=T s is the rate at which it is desired to update channel estimates. At time nt s, denote the augmented vector of all the multipath signals of all the 3

4 users without channel gains as x n = [x 1;1 (t); : : : ; x 1;L (t); x ;1 (t); : : : ; x ;L (t); : : : : : : ; x K;1 (t); : : : ; x K;L (t)] T jt=nt s (8) Let the corresponding vector of complex channel gains be denoted by c = [c 1;1 ; : : : ; c 1;L ; c ;1 ; : : : ; c K;L ] T (9) The channel is assumed to be constant in time, but other studies have shown that the identication algorithm in the proposed scheme is capable of tracking time-varying parameters [5, 1, 13]. The received signal from (1) can be expressed in terms of the above as r n = r(nt s ) = c T x n + v n (10) where v n = v(nt s ). The parameter vector c requires to be estimated from the observed received signal r n and training signals x n. SMF theory approaches the estimation problem (10) by specifying an upper-bound on the instantaneous estimation error for all inputoutput pairs that belong to a certain \design space" S. In other words, given a > 0, the SMF criterion is to compute ^c that satises jr? ^c T xj for all (r; x) S (11) The set of parameters that satisfy the above specication is called the feasibility set. The design space S is any set of possible input-output pairs. If the probability that (r n ; x n ) S is large, then the probability that the resulting SMF parameter estimate meets the bounded-error specication is at least that large. An example of a design space is S = f(r; x) : jr? c T xj v g (1) for some v > 0. When v n is Gaussian noise, it usually suces to let equal three times the standard deviation of v n. The class of Optimal Bounding Ellipsoids (OBE) algorithms, originally developed for SMI, have been shown to be applicable for adaptive estimation of the feasibility set [9, 10]. They provide ellipsoidal outerapproximations of the feasibility set at each iteration, and the geometric center of the ellipsoidal set estimate is taken to be the point estimate for the parameter vector. Let E n denote the ellipsoidal estimate for the feasibility set at sample n. It is characterized by its geometric center and point estimate of the parameter c n, a positive denite matrix P n that determines its shape and orientation, and a positive scalar n which, together with P n, determines the size of the ellipsoid. E n = fc : (c? c n ) H P?1 n (c? c n ) n g (13) where () H refers to complex conjugate transposition. The OBE algorithms provide update recursions for the ellipsoidal parameters c n ; P n and n to compute E n+1 when the data (r n+1 ; x n+1 ) is available at time n + 1. The recursions for c n and P n are strikingly similar to the RLS recursions [4]. The OBE algorithms also exhibit the fast convergence properties of the RLS algorithm and outperform RLS in tracking performance in non-stationary environments [5, 1]. An OBE algorithm described in [4] and extended to the complex case in [11], known as the Dasgupta-Huang OBE (D-H/OBE) algorithm, is used in this paper. An attractive feature of OBE algorithms is their selective update property. This feature requires the algorithms to compute updates of the estimates only for a small fraction of the time, leaving the updating processor idle for the remaining time, resulting in a large decrease in average computational complexity when compared with RLS. The LMS algorithm is not a serious contender due to its slow convergence rate, especially for large dimensions of the parameter vector or, in this case, large number of interferers. V. SIMULATION RESULTS An asynchronous CDMA system that uses 64-ary orthogonal signaling and complex pseudo-random spreading codes { such as the uplink of an IS-95 system { in a multipath environment was simulated to verify the performance of the proposed receiver and to compare it with the conventional non-coherent Rake receiver. The number of spreading chips per symbol was 56 and the number of samples per chip was two. A three-ray multipath model and a three nger Rake were used for the desired user. Additive Gaussian noise resulted in an E b =N o without interference of 10 db. BER estimates were obtained by averaging over N bits, where N was varied from 7000 for six interferers to 4000 for fty interferers. The proposed receiver used 360 training bits (60 symbols) before the unknown data was transmitted in order to obtain complex channel estimates. The results are plotted in Figure 3 which shows the variation of BER versus the number of interferers for the proposed and conventional receivers. The BER curve for the proposed receiver is not smooth because the identication algorithm does not always converge 4

5 to the same channel estimate. It is clear from the plots that the proposed receiver results in a superior BER performance. Moreover, the OBE channel estimation algorithm computed parameter estimates for an average of only 0.43% of the time during the 360 bit training process, thereby reducing the average computational complexity by over 99% when compared with the RLS algorithm. BER Conventional Proposed Number of interferers Figure 3: Probability of error curves. VI. CONCLUSIONS This paper has shown that signicant performance gain and large reduction in complexity are possible in realistic CDMA systems using a coherent Rake receiver that is matched to the multipath channel. The proposed scheme requires channel estimates that can be obtained using an adaptive algorithm with the help of training signals. Performance improvement has been made possible by coherent combining and demodulation, and fast estimation of channel parameters by OBE algorithms. Complexity reduction has been achieved due to a simplied Rake architecture and the selective update feature of the OBE algorithms. Further work is needed in the area of ecient blind channel identication techniques that will remove the requirement for training signals. REFERENCES [1] R. Price and P.E. Green, Jr., \ A communication technique for multipath channels," Proc. IRE, vol. 46, pp , [] A.J. Viterbi, \CDMA: Principles of spread spectrum communication," Addison-Wesley, [3] E. Fogel and Y.F. Huang, \On the value of information in system identication - Bounded noise case," Automatica, vol. 18, pp. 9-38, 198. [4] S. Dasgupta and Y.F. Huang, \Asymptotically convergent modied recursive least-squares with data dependent updating and forgetting factor for systems with bounded noise," IEEE Trans. Information Theory, vol. 33, pp , [5] J.R. Deller, Jr., M. Nayeri and S.F. Odeh, \Leastsquare identication with error bounds for realtime signal processing and control," Proceedings of the IEEE, vol. 81, no. 6, pp , June [6] J.R. Deller, Jr., \Set-membership identication in digital signal processing," IEEE ASSP Magazine, vol. 6, pp. 4-, Oct [7] J.P. Norton (editor), Intl. Journal of Automatic Control and Signal Processing (Special issues on bounded-error methods in system identication), vol. 8, Jan-Feb, 1994, vol. 9, Jan-Feb, [8] E. Walter and H. Piet-Lahanier, \Estimation of parameter bounds from bounded-error data: a survey", Mathematics and Computers in Simulation, vol. 3, No. 5 & 6, pp , December [9] S. Gollamudi, S. Nagaraj, S. Kapoor and Y.F. Huang, \Set-membership ltering and a setmembership normalized LMS algorithm with an adaptive step size," Submitted for publication in the IEEE Signal Processing Letters. [10] S. Gollamudi, S. Nagaraj, and Y.F. Huang, \SMART-A toolbox for set-membership ltering," Proc., 1997 European Conf. Circuit Theory and Design, Budapest, Hungary, Aug Sept. 3. [11] S. Gollamudi, S. Kapoor, S. Nagaraj and Y.F. Huang, \Set-membership adaptive equalization and an updator-shared implementation for multiple channel communications systems," Submitted for publication in the IEEE Trans. Signal Processing, Oct., [1] A.K. Rao and Y.F. Huang, \Tracking characteristics of an OBE parameter estimation algorithm", IEEE Trans. Signal Processing, pp , vol. 41, March [13] S. Kapoor, S. Gollamudi, S. Nagaraj and Y.F. Huang, \Tracking of time-varying parameters using Optimal Bounding Ellipsoid algorithms", Proc., 34th Annual Allerton Conf. Communication, Control and Computing, University of Illinois, Urbana-Champaign, Oct {4,