490 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. IT-20, NO. 4, JULY 1974 An Upper Bound on the Error Probability in Decision-Feedback I Equalization DONALD L. DUTTWEILER, MEMBER, IEEE, JAMES E. MAZO, MEMBER, IEEE, AND DAVID G. MESSERSCHMITT, MEMBER, IEEE Abstract-An upper bound on the error probability of a decision- feedback equalizer which takes into account the effect of error propaga- tion is derived. The bound, which assumesindependeltt data symbols and noise samples, is readily evaluated numerically for arbitrary tap gains and is valid for multilevel and nonequally likely data. One specific result for equally likely binary symbols is that if the worst case intersymbol interference when the first J feedback taps are set to zero is less than the original signal voltage, then the error probability is multiplied by at most a factor of 2J relative to the error probability in the absence of decision errors at high S/N ratios. Numerical results are given for the special case of exponentially decreasing tap gains. These results demonstrate that the decision-feedback equalizer has a lower error probability than the linear zero-forcing equalizer when there is both a high S/N ratio and a fast roll-off of the feedback tap gains. I. INTRODUCTION D ECISION-FEEDBACK equalization (DFE) is a technique used in the communication of discrete- time and discrete-level data over a noisy channel with a nonideal impulse response. It consists of equalizing the channel to produce a sampled causal response, and sub- tracting that part of the intersymbol interference at the sample time that is due to past data decisions. The advantage of this nonlinear receiver is that a portion of the noise enhancementinherent in linear equalization is avoided by using past decisions to subtract out a portion of the inter- symbol interference; a disadvantageis that decision errors tend to propagatebecause they result in residual intersymbol interference and a reduced margin against noise at future decisions. There exists an extensive literature spanning 50 years describing this approach (see [l] for bibliography). Rela- tively recently, explicit formulas for the signal-to-noise ratio at the DFE forward filter output have been obtained for zero-forcing [l], [4] and mean-square error criteria [2], [3], using the optimum transmitter and receiver filters. These results demonstrate that, if error propagation is neglected,the DFE always has an advantage, and often a substantial advantage, over purely linear equalization. How- ever, the presence of error propagation due to the effect of past decision errors raises the question as to whether the actual error probability of the DFE is superior to that of the linear equalizer on any or all channels. There is one instance of a successfulestimation of the effects of error propagation for the special caseof exponen- tially decreasing tap gains [5] ; the method is based on an Manuscript received December 13, 1973; revised February 4, 1974. D. L. Duttweiler and D. G. Messerschmitt are with Bell Laboratories, Holmdel, N.J. J. E. Mazo is with Bell Laboratories, Murray Hill, N.J. earlier paper [6]. There is also a collection of papers [7]- [ 121 which successfully analyze decision-directed estimators of the stochastic approximation type. The common feature of [7]-[12] is that the kth increment to the estimate is weighed by k-l, so that the effect of decisionerrors decreases with k, a property which the decision-feedbackequalizer unfortunately does not possess. The error propagation problem can be handled easily when a finite number M of past decisions enter into the current decision and M is very small. The systemcan then be modeled as a Markov chain with LM states (where L is the number of distinct magnitudes of errors), and when M is small (lessthan three or four) the stationary probabilities can be determined numerically. For larger M, it does not appear feasible to calculate the error probability exactly, except perhaps in a few special cases [5]. In this paper we derive an upper bound on the error probability of the decision-feedback equalizer which applies to an arbitrary equalized channel, even when M is very large or infinite. It will be assumedthat successive noise samples at the equalized channel output are independent (which is precisely true when the equalizer is designed by a zero-forcing criterion and the channel noise is stationary and Gaussian [l], [4]), and that the input data digits are also independent. Numerical examples wili illustrate the application of the bound. II. DERIVATION OF BOUND The typical DFE configuration is shown in Fig. 1. The forward filter is chosen to yield a causal sampled response of the form ok = B, + fJ hjB,-j + nk j=l (1) where B, is the current data digit to be detected, and nk is an additive noise term. The receiver then computes the quantity ok - 2 hjBk-j = B, + .Zk + nk j=l zk = j$l hjek-j ekMj = Bk-j - Bk-j to a threshold device in order to make a decision fik on Bk. The effect of previous decision errors on this decision- directed detection mechanismis to reduce or eliminate the margin against noise. The requirement on the forward filter that (1) be satisfied is rather stringent and will be relaxed in Section II-B. It