As real world communication channels are stressed with higher data rates, intersymbol interference (ISI) becomes a dominant limiting factor. One way to combat this effect that has recently received considerable attention is the use of a decision feedback equalizer (DFE) in the receiver. The action of the DFE is to feed back a weighted sum of past decision to cancel the ISI they cause in the present signaling interval. This paper summarizes the work in this area beginning with the linear equalizer. Three performance criteria have been used to derive optimum systems; 1) minimize the noise variance under a "zero forcing" (ZF) constraint i.e., insist that all intersymbol interference is cancelled, 2) minimize the mean-square error (MMSE) between the true sample and the observed signal just prior to the decision threshold, and 3) minimize the probability of error (Min P e ). The transmitter can be fixed and the receiver optimized or one can obtain the joint optimum transmitter and receiver. The number of past decisions used in the feedback equalization can be finite or infinite. The infinite case is easier to handle analytically. In addition to reviewing the work done in the area, we show that the linear equalizer is in fact a portion of the DFE receiver and that the processing done by the DFE is exactly equivalent to the general problem of linear prediction. Other similarities in the various system structures are also shown. The effect of error propagation due to incorrect decisions is discussed, and the coaxial cable channel is used as an example to demonstrate the improvement available using DFE.

Decision feedback equalization

This paper surveys the literature on communication theory from 1968 to 1973. A large part of this literature, and of this paper, deals with digital communication through linear dispersive channels. Error rate bounds for pulse-amplitude modulation systems, the optimization of linear transmitters and receivers, and adaptive equalization are topics discussed initially. Important results from the literature on non-linear receivers are described next, including decision feedback, maximum-likelihood, and Viterbi algorithm approaches. New results on the conventional digital modulation formats of amplitude, phase, and frequency-shift keying are summarized. The paper concludes with sections describing the literature from this period on nonlinear and non-Gaussian problems, analog communication, and feedback communication.

A survey of the communication theory literature: 1968-1973

The following classical problem is solved and the results are applied to minimize the mean-square error e of communication and data compression systems: "a stationary random input signal x(t) is prefiltered, corrupted by noise n(t) , sampled every T seconds, and finally postfiltered to yield output \hat{x}(t) . Let the desired output z(t) = a(t) \bigotimes x(t) where a(t) is any real function of t and \bigotimes denotes convolution. Determine the linear timeinvariant pre- and postfilters which jointly minimize \epsilon = E\{\frac{1}{T}\int^{T}_{0}[z(t) - \hat{x}(t)]^{2}dt\} subject to power constraint P = E\{[k(t) \bigotimes f(t) \bigotimes x(t)]^{2}\} .'' Operator E{\dot} denotes expected value, k(t) is a real function of t , and f(t) is the prefilter impulse response. Appropriate choice of T , P,k(t) , and n(t) makes the solution applicable to amplitude, angle, pulse-amplitude (PAM), pulse-code (PCM), and differential pulsecode (DPCM) modulation systems, and data compression systems. In this analysis no restrictions are placed on the input-signal spectrum, the noise spectrum, or the passbands of the filters; furthermore, the cross correlation between signal and noise is taken into consideration. Necessary and sufficient conditions are obtained for the jointly optimum pre- and postfilters. The performance obtainable using these optimum filters is compared with that obtainable using suboptimum filters. One suboptimum filtering scheme is derived which yields virtually the same performance as optimum filters and has the practical advantage that the filter transfer characteristics are independent of noise n(t) in many cases of interest. In applying the results to PCM, DPCM, and data compression systems, the filters, sampling rate, and quantizer are jointly optimized. The performance obtainable for various filtering schemes and various communication systems is compared with the optimum attainable as calculated from information theory.

Optimum Pre- and Postfiltering of Sampled Signals with Application to Pulse Modulation and Data Compression Systems

Optimum Pre- and Postfiltering of Sampled Signals with Application to Pulse Modulation and Data. Compression Systems P-

Joint optimization of transmitting and receiving filters in pulse-amplitude-modulated stationary Gaussian channels has been considered by a large number of authors under a variety of different circumstances. In most cases it has been found that the optimum filters are strictly band limited. In the present correspondence we give a direct argument why this must be so for a broad class of joint optimization problems.

Optimum PAM filters are always band limited (Corresp.)

This paper illustrates the basic equivalence of many of the linear data transmission design techniques. It shows the unifying feature of these techniques to be a generalization of Nyquist's original ideas relating time samples and frequency domain constraints. It examines pulse amplitude (with and without constraints on the input data) and pulse shape modulation systems, and shows their relationships. It uses a number of previously-described systems to illustrate the range of possibilities of the very general design approach. This paper presents some new results on noise and channel parameter monitoring and on spectrum shifting by constraining the input data.

James W. Smith

Papers

A unified view of synchronous data transmission system design