APPLICATIONS OF LATTICE STRUCTURE TO ADAPTIVE CHANNEL EQUALIZATION

Abstract

In data communication systems, the performance of the receiver is generally degraded due to the presence of the inter symbol interference (ISI) arising from the time-spread of the transmitted pulse. A linear transversal filter with tapped delay line (TDL) structure is normally used to combat the effect of ISI, but the linear equalizer is not able to cope with the severe distortion in the channel and hence, in such situations a decision feed back equalizer (D^E) is used. The tap coefficients of the equalizer are normally evaluated by minimizing the output mean square error (MSE), which requires the prior knowledge of the channel tap coefficients. Thus in practice the equalizer's tap coefficients are adaptively adjusted to their optimum values, using the steepest descent algorithm. The rate of convergence of MSE, in such a situation is slow and depends upon the channel characteristics. To speed up the rate of convergence, several fast algorithms have been developed during recent years and in this thesis we consider the applications of the lattice structure for the adaptive channel equalization. We first consider the comparison of the error rate performance of Kalman feed back and decision feed back equali zers for the multilevel base band data transmission system and it is found that the performance of the decision feed back equalizer (DFE) is better than the performance of the Kalman feed back equalizer in general. The DFE normally suffers from the effect of the error propagation, which Is due to the wrong

decisions being used in the feed back filter. The evaluation of the error bounds for the DFE is considered next using two different approaches. The first method uses the Gauss quadra ture rule (GQR) which requires the knowledge of the moments of the interference and in the second method, the unknown density function of the interference is approximated to a known density function for the evaluation of the probability of error. The adaptive algorithms for the equalization of the channel using DFE are considered next and the rate of «R convergence of tap gain error for the estimated gradient algorithm is analyzed and an optimum value of the step size is evaluated. The Godard and the fast Kalman algorithms are applied to the channel equalization and the rate of convergence of MSE is obtained by simulation and the results are compared. We then consider the application of the lattice structure to the channel equalization of the carrier modulated data transmission system. The complex least square lattice and the complex gradient lattice algorithms are derived using the complex gradient operator and the algorithms are simulated to study the error rate performance and the rate of convergence of mean square error for a variety of fixed channels for the quadrature phase shift keyed (QPSK) system. The performance of the lattice equalizers is compared with the performance of the complex tapped delay line equalizer and the simulation results are presented. We next consider the application of the lattice structure

to the channel equalization using decision feed back equali zation for the carrier modulated data transmission system. The equalizer structure consists of both scalar and vector lattice stages. The complex least square lattice and the complex gradient lattice algorithms suitable for the above situation are derived and simulated to obtain the probability of error at different values of signal to noise ratio . The rate of convergence of MSE for the deterministic channels for the QPSK data is also obtained. We finally consider the application of the lattice structure as a DFE for the eoualization of fading dispersive channel. A tapped delay line model of the channel is used, where the tap gain coefficients are complex valued mutually independent Gaussian random variables. The error rate perfor mance of the complex least square lattice decision feed back equalizer is obtained for the random fixed channels having different power Impulse responses and for the random fading channel with different fade rates. The performance results evaluated by simulation are presented.