A Kalman-Filter Approach to Equalization of CDMA Downlink Channels

EURASIP Journal on Applied Signal Processing 2005:5, 611–625 c 2005 Hindawi Publishing Corporation
A Kalman-Filter Approach to Equalization of CDMA Downlink Channels Hoang Nguyen Nokia Research Center, San Diego, CA 92131, USA Email: Jianzhong Zhang Nokia Research Center, Irving, TX 75039, USA Email: Balaji Raghothaman Nokia Research Center, San Diego, CA 92131, USA Email: Received 31 July 2003; Revised 26 February 2004 An efficient method for equalization of downlink CDMA channels is presented. By describing the observed signal in terms of a state-space model, the method employs the Kalman filter (KF) to achieve an unbiased signal estimate satisfying the linear minimum mean-squared error (LMMSE) criterion. The state-space model is realized at the symbol and chip levels. With the symbollevel model, the KF is used to estimate the transmitted chips that correspond to each symbol interval; whereas at the chip level, the transmitted chips are estimated individually. The symbol-level KF has a built-in tracking capability that takes advantage of the a priori known scrambling sequence, which renders the transmitted signal nonstationary. The chip-level KF reduces the complexity of the symbol-level KF significantly by ignoring the nonstationarity introduced by scrambling. A simple method for further reducing the KF complexity is also presented. The computational complexity of the proposed technique is analyzed and compared with that of several linear approaches based on finite-impulse response (FIR) filtering. Simulations under realistic channel conditions are carried out which indicate that the KF-based approach is superior to FIR equalizers by 1–2 dBs in error-rate performance. Keywords and phrases: Kalman filter, CDMA, single-user detection, oversampling, color noise, state-space models. 1. INTRODUCTION The existence of dispersive effects of the channel, such as multipath, destroys the orthogonality of the spreading codes in CDMA systems. As a result, the conventional RAKE receiver in many cases reaches a noise floor at a fairly high frame error rate [1, 2]. There are two major categories of methods by which channel dispersion can be dealt with, namely multiuser detection and single-user detection. In multiuser detection, which generally requires knowledge of all user codes, the channel is equalized and every user is detected so that this approach is suitable for uplink CDMA. In downlink CDMA, however, all users share the same physical channel and multiuser detection is inefficient since only one user needs to be demodulated. To efficiently demodulate just the desired user, the channel is equalized in order to restore the code orthogonality inherent in the signal structure. Restoration of code orthogonality is thus motivated by the fact that to demodulate the desired user, (i) the receiver does not need to know the spreading codes of the interfering users, and (ii) the computational complexity of the detection mechanism does not increase with the number of users. Well-known methods in the literature for orthogonality restoration rely on the LMMSE criterion attained via FIR filtering, which can be realized at the chip or symbol level in adaptive or batch form. For each segment of data over which the observation sequence is considered stationary, the batch implementation requires the solution to a matrix equation which can be obtained by means of LU decomposition or matrix inversion. Frequent matrix solution imposes a heavy computational burden. In an effort to reduce this computational cost, an approximate solution to the matrix inversion problem is developed in [3] based on FFT and IFFT operations by taking advantage of the near-circulant Toeplitz structure of the observed-data autocorrelation matrix. We note parenthetically that symmetric Toeplitz equations can also be solved by means of the Levinson recursion [4, 5, 6] whose complexity is on the order of the square 612 of the system dimension. Even for stationary channels, the symbol-level LMMSE equalizer of [7] requires one matrix solution per symbol period. This equalizer in essence is a chiplevel FIR (cFIR) LMMSE equalizer that encompasses the descrambling and despreading operations, normally done outside the equalizer. Its objective, however, is to minimize the error variance of the symbol rather than chip estimate. On the other hand, matrix inversion can be avoided by means of chip-adaptive FIR equalizers. Adaptive implementations typically employ well-known algorithms such as stochastic gradient, LMS, and RLS. Adaptive FIR equalizers unfortunately tend to be unstable, sensitive to initialization, and slow to converge. Examples of stochastic gradient descent implementations of the LMMSE criterion can be found in [8, 9]. An attractive alternative to FIR filtering is to use the Kalman filter (KF) which we consider in this paper. The KF is preferred to FIR approaches for several important reasons. First, the KF has the capability to track nonstationarities which result typically from the time-variant characteristics of the channel, noise processes, and the underlying signal to be estimated. Also, the KF is well known as an optimal linear estimation method in the mean-squared error (MSE) sense. To clearly differentiate our proposed method from other applications of the KF, it is worth discussing CDMA statespace formulations proposed by several researchers. Most notable perhaps is the work of Iltis et al. [10, 11] where a state-space model is developed for estimation of the path gains and delays of multipath channels. The observation equation of this model depends nonlinearly on the path delays and is therefore linearized with respect to these variables so that the extended Kalman filter (EKF) [12] is suitable for parameter tracking. Further, in [11], multiuser detection at each time epoch is performed based on the maximum a posteriori probability (MAP) criterion evaluated from the estimated model up to the previous time epoch; in this respect, one can view the multiuser detector as one of decisionfeedback type. Similar use of the EKF with linearized statespace models for estimation and tracking of multipath gains and delays can also be found in [13, 14, 15, 16]. For the case of flat Rayleigh fading channels, relative delay estimation is no longer necessary since there is only one path to consider. If the fade process admits a Gauss-Markov model, then it is possible to employ the KF in the decision-feedback mode to track the fade [17, 18, 19]. Apart from the above applications of the KF, we discuss several state-space formulations which are more closely related to our proposed approach. In particular, the state-space models of [20, 21, 22, 23] have a resemblance to one another as seen from the fact that the measurement matrix consists of all spreading codes and channel coefficients, while the state vector consists of multiuser data symbols. Similarly, the models of [24, 25] multiplicatively lump the channel coefficients (...truncated)