INTERACTING MULTIPLE MODELS FOR SINGLE-USER CHANNEL ESTIMATION AND EQUALIZATION M.H. Jaward and V. Kadirkamanathan Department of Automatic Control and Systems Engineering The University of Sheffield Mappin Street, Sheffield S1 3JD, UK ABSTRACT In this paper, a blind sequence estimation algorithm based on in- teracting multiple model is introduced to estimate the channel and the transmitted sequence corrupted by ISI (intersymbol interfer- ence) and noise. The proposed algorithm avoids the exponential growth complexity caused by increasing channel memory length. The performance of the IMM (interacting multiple model) based equalizer is studied and compared with the well known algorithm like DDFSE (Delayed Decision-Feedback Sequence Estimation). 1. INTRODUCTION The interest in this paper is the problem of detection of digital data in the presence of intersymbol interference (ISI) and additive noise. Throughout this work the assumption made is that after some processing (matched filtering, for instance), the continuous time received signals are sampled at the baud (symbol) rate. Thus results in a discrete time model of the channel. Our objective is to produce a reliable decision of the input sequence based on the received data in the absence of channel characteristics. As discussed in [1], various approaches to data detection can be broadly divided into symbol by symbol and sequence estima- tion. The first class contains linear and decision-feedback detec- tors. These schemes have low complexity and undesirably high er- ror rates. Another approach to data detection is given by maximum- likelihood sequence estimation (MLSE) [2]. The trellis-based Viterbi algorithm [3] solves the MLSE problem recursively when the memory of the channel is finite. The symbol error rate of the Viterbi algorithm is often much lower than error rates of the symbol by symbol detectors. However, the total storage (com- plexity) of the algorithm is proportional to the number of states of the trellis which grows exponentially with the channel memory length. When the channel memory becomes large, the algorithm becomes impractical. In this case reduced state algorithms like RSSE (Reduced-State Sequence Estimation) [4, 5] and DDFSE (Delayed Decision-Feedback Sequence Estimation) [6] are used. These algorithms assume some past decisions as correct while es- timating several most recent symbols. Another set of equalizers use a hidden Markov model (HMM) formulation for blind (or semi blind) equalization for input se- quences governed by Markov chains. Either they use off-line [7] or on-line EM algorithm [8, 9] to maximize the Kullback-Leibler (KL) measure to calculate the HMM model. [10, 11] use a HMM estimator together with a sequence estimation for stochastic max- imum likelihood (ML) equalization. While on-line methods over- come the memory and computational cost involved in the off-line EM algorithm based methods, they still need to use some kind of state reduction algorithm to reduce the state complexity of the state trellis [9]. Tugnait et al. [12] presents a comprehensive review of single-user channel estimation and equalization techniques. Here we propose an alternative approach which utilizes the interacting multiple model (IMM) algorithm. The paper is organized as follows: In Section 2 we define our signal model and formulate the equalization problem as a state esti- mation under model uncertainty problem. In Section 3, we review the IMM algorithm. We derive our IMM based equalizer in Sec- tion 4. Simulation studies are presented in Section 5 and finally some conclusions are drawn in Section 6. 2. PROBLEM FORMULATION Let denote the symbol emitted by the digital source at time , where is the symbol duration. This discrete time signal is modulated, filtered, sent through the communication channel, filtered and demodulated. The resulting signal is continuous and is given by (1) where is the symbol period, is the additive white noise inde- pendent from the emitted symbols, is the composite channel response encompassing the effects of the transmitting filter, recep- tion filter, channel response and modulation/demodulation (which is assumed to be linear). The composite channel is assumed to be FIR, with a duration of approximately . In general, can take on possible values. For simplicity we use binary trans- mission ( ) and symbols transmitted are either -1 and 1. The extension to the general case is straightforward. Symbol rate sampling is used which results in an equivalent discrete-time rep- resentation, (2) This can easily be extended to multi rate sampling. To overcome the phase ambiguity in the channel coefficients, differential decod- ing is used. Under the assumption that is perfectly known, the opti- mum receiver is composed of a filter matched to the pulse fol- lowed by a symbol rate sampler and a Viterbi decoder that searches for the path with minimum metric in the trellis diagram of a finite state machine of the equivalent channel.