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Description

The equalization of a communication channel refers to the removal of frequency distortions introduced by the channel. For wireless systems, these distortions are usually created by multipath propagation, since depending on the transmitted frequency, signals propagating along different paths may combine coherently or destructively at the receiver. While standard equalization methods use training sequences to probe the communication channel, ``blind equalization'' has for objective to simultaneously recover an unknown transmitted signal and the channel parameters. The best known blind equalizer family is based on the constant modulus algorithm (CMA) introduced in the early 1980s. It has a linear structure and minimizes adaptively a nonlinear objective function capturing the geometry of the transmitted symbol constellation. Although the CMA works quite well for linearly modulated signals, its linear structure is ill-adapted to the equalization of nonlinearly modulated signals, including those using continuous-phase modulation, such as Gaussian minimum shift keying (GMSK). Furthermore, even in the case of non-blind equalization, linear equalizers are suboptimal and are outperformed by maximum likelihood equalizers based on the Viterbi decoder. The talk will first show how the blind equalization problem can be viewed as a hidden Markov model estimation problem, where the state of the underlying Markov chain represents the symbols stored in the channel. Then as a substitute to the Baum-Welch HMM estimation method, we will present a new technique called the expectation maximization Viterbi algorithm (EMVA). This method uses the Viterbi algorithm (VA) for maximum likelihood sequence estimation in the expectation phase. The maximization phase solves a least squares problem that can be viewed as a weighted sum of least squares problems associated to each survivor path, with each weight selected equal to the probability of the corresponding survivor path. Since the least-squares problem can be solved in closed-form, the EMV algorithm requires just a trivial increase in computational burden with respect to the ordinary Viterbi algorithm. It is also significantly faster than the Baum-Welch algorithm. The EMVA is very flexible and its implementation will be be described for linear and nonlinear modulation formats. A sliding window implementation of the EMVA for the estimation time-varying channels will be presented. It bears some similarity with the recursive least-squares (RLS) method for solving least-squares problems adaptively, with the difference that in the sliding window EMVA, the survivor paths change dynamically as the time window changes, changing the structure of the corresponding least-squares problem. It will also be shown how the EMVA can be combined with techniques based on statistical measures such as the Rissanen's minimum description length criterion or Akaike's information criterion, in order to estimate the channel order. Monte Carlo simulations show that at moderate signal to noise ratios, say above 6dBs, the channel estimate provided by the EMVA is unbiased and attains the most optimistic Cramer-Rao lower bound for the blind equalization problem, namely a bound evaluated based on assuming that the transmitted signal is known. Thus the EMVA is almost optimal.

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