This concise paper investigates various aspects of the application of unbiased linear minimum mean-square error (ULMMSE) estimators to the equalization of channels used for digital data transmission. One application is to the equalization of the channel to a response free from intersymbol interference. Previous results regarding this application are clarified and extended. The other application is to the equalization of the channel to a finite short memory response. This stems from an interest in reducing the complexity of the Viterbi algorithm. Various new results on the optimum ULMMSE equalizer and its performance are presented. In both applications, the optimum ULMMSE equalizer is a stable, realizable, finite dimensional recursive filter.

The matrices which are symmetric about both diagonals (persymmetric matrices) occur often in communication and information theory problems. There have been a number of problems whose solution involves the eigenvectors of persymmetric matrices. This paper summarizes a number of properties of the eigenvectors of persymmetric matrices and demonstrates how they can be applied to characterize and simplify the solution to a number of communication problems.

The method of inversion of linear discrete time systems utilizing quantized feedback is examined, and a stability concept is developed. The application in mind is decision feedback equalization (DFE). Sets of necessary conditions and sufficient conditions for stability are derived. A lower bound for the probability of recovery from error, and an upper bound for the mean recovery time are obtained. It is suggested that a stability index for each system could be defined. The bounds derived do not depend on the nature of the noise or the channel impulse response except through its length.

An equalizer based on the minimum mean-square error (MSE) criterion, but with direct solution of the resulting equations, is described. The direct solution is based on the algorithms devised by Levinson and Trench. With the availability of large-scale integration (LSI) bipolar computing elements, these algorithms are competitive with iterative procedures. A method is considered for the estimation of the required parameters, and for automatic adaption to a changing channel. It is shown that the delay can be adjusted to achieve minimum MSE with respect to that parameter. Simulations undertaken show the robust performance of the algorithm, and that the equalizer performance is not adversely affected by operation in decision-directed mode.