A STUDY OF FAST TRANSVERSAL FILTERS FOR ESTIMATION AND EQUALIZATION OF FADING DISPERSIVE CHANNELS

Abstract

In recent years, there has been an increasing interest in high speed data transmission through fading dispersive channel, such as shortwave ionospheric propagation. Fading dispersive channel is usually best described as a random linear time-varying filter and is modeled as a tapped delay line (TDL) filter with tap gains assumed to be sample functions of a zero mean complex Gaussian random process. The channel estimation problem thus comprises of an adaptive adjustment of the tap coefficients of the equavalent TDL filter according to some performance criterion. Receivers designed for decoding digital signals transmitted over fading dispersive channels generally require equalization techniques which aim to remove or reduce the intersymbol interference (ISI) in the presence of additive noise. A linear transversal filter with tapped delay line (TDL) structure is normally used to cambat the effect of ISI, but the linear equalizer is not able to cope with severe amplitude distortion in the channel and hence, in such situations a decision feedback equalizer (DFE) is used. The DFE consists of a feedforward and a feedback filter whose inputs are the decisions on the previously detected symbols. The tap coefficients of the equalizer are adjusted recursively using suitable adaptive algorithms. A wide range of adaptive algorithms have been reported over the years. Due to their high convergence rate and low computational complexity, the fast RLS algorithm are emerging as potential candidates for adaptive filtering applications. The fast transversal filter (FTF) algorithm offers the computationally most efficient realization of the fast RLS algorithm and is well suited to the channel estimation and equalization problem. However, the FTF algorithm exhibits unstable behavior due to the accumulation of round-off error in the finite precision implementation. Recently, several techniques to circumvent the problem of numerical instability in FTF algorithm have appeared in the literature, prominent among them are the reinitialization, normalization and stabialization technique using computational redundancy. This work encompasses the application of the FTF algorithm for estimation and decision feedback equalization of the fading dispersive channel and investigates the numerical performance of the FTF algorithms using various reinitialization and stabilization techniques. To slow down the growth of numerical errors in the FTF algorithm, the normalized FTF algorithm for exponentially windowed complex input signals has been derived. Using the computational redundancy technique of stabilization, two stable FTF algorithms, namely the corrected FTF and the stabilized FTF algorithm suitable for the estimation of a fading channel using QPSK data transmission system, have been presented. As the normal soft-constrained rescue reinitialization method fails to prevent the divergence of the FTF algorithm for fading channel estimation, we have proposed an alternative reinitialization criterion which restarts the algorithm when the rescue variable stabilizes at one.

A delayed version of the new reinitialization method has also been presented. The proposed reinitialization schemes are computationally more efficient as no additional computations are involved in the reinitialization process. Performance of the different FTF channel estimators have been evaluated for QPSK data trannmlRBlon system nnd the results nrn presented. It is found that the corrected and the stabilized FTF algorithms behave in a stable manner and closely track the channel variations. The performance of the proposed reinitializaton with delay is comparable to that of the corrected or stabilized FTF algorithm at low signal to noise ratio. At high signal to noise ratio, however, the latter perform better. We have next considered the application of the FTF algorithm for the decision feedback equalization. The work mainly centres round the generalized fast transversal filter (GFTF) algorithm, which is the computationally most efficient RLS implementation of an adaptive DFE. Various measures of improving the performance of the GFTF algorithm are considered. The normalized generalized fast transversal filter (NGFTF) algorithm has been derived by proper scaling of the forward, backward and gain transversal filters and residuals of the GFTF algorithm. To improve the tracking capability of the algorithm, the generalized variable forgetting factor FTF (GVFTF) algorithm has been introduced by incorporating the data sequence weighting technique. We have also applied the normalizaton technique to the GVFTF algorithm and have derived the normalized generalized variable forgetting factor FTF

(NGVFTF) algorithm. To get a more stable FTF based DFE, the idea of corrective feedback has been applied to the GFTF algorithm and the corrected GFTF (CGFTF) algorithm has been derived using the complex gradient operators. The convergence characteristics and error rate performance of the different GFTF algorithms have been studied for a variety of fixed and fading channels for the QPSK data transmission system. Simulation results have substantiated the numerical advantages of normalization and have also validated the improvement in the tracking performance of the GVFTF algorithm as a result of the data sequence weighting. The corrected GFTF algorithm is found to be more stable than the GFTF algorithm and also it gives a better error rate performance on fading dispersive channels. The GFTF algorithm uses block partitioning of data vector in order to exploit the shifting property of the data. Consequently, it involves matrix computations which are more prone to numerical problems. To overcome this problem, scalar Implementation of the GFTF algorithm has been considered and a new modular GFTF algorithm has been derived. As the modular GFTF also manifests the problem of numerical instability, an attempt has been made to stabilize the algorithm by applying the computational redundancy technique and stabilized modular GFTF algorithm has been presented. Finally the convergence characteristics and error rate performance of the modular GFTF algorithm have been studied and the simulation results are presented for fixed and fading dispersive channels. It is found that the modular GFTF algorithm gives the best performance among the various FTF based DFE's considered, both in terms

of the rate of convergence and the error rate performance. This is followed by the corrected CRT algorithm. Periodic reinitialization of the GVFTF and the NGVFTF algorithms make them viable for adaptive equalization of the fading dispersive channel but renders slight degradation in their performance.