IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 49, NO. 2, FEBRUARY 2001 239 Error Propagation and Recovery in Decision-Feedback Equalizers for Nonlinear Channels John Tsimbinos, Member, IEEE, and Langford B. White, Member, IEEE Abstract—Nonlinear intersymbol interference is often present in communication and digital storage channels. Decision-feedback equalizers (DFEs) can decrease this nonlinear effect by including appropriate nonlinear feedback filters. Although various applica- tions of these types of equalizers have been published in the liter- ature, the analysis of their stability and error recovery has not ap- peared. In this letter, we consider a DFE with a nonlinear feedback filter based on a discrete Volterra series. We extend error propa- gation, error probability, stability, and error recovery time results for th-order nonlinear channels. Index Terms—Decision-feedback equalizers, error analysis, non- linear distortion, nonlinear filters, Volterra series. I. INTRODUCTION N ONLINEAR intersymbol interference (ISI) caused by channel nonlinearities can significantly increase the error rate in communication and digital storage channels. A decision-feedback equalizer (DFE) is aimed at decreasing the effect of this nonlinear process by incorporating nonlinear feedforward and feedback filters. The nonlinear feedback filter may be of the form of a discrete time Volterra series, a look-up table, or a neural network [1]–[5]. Basic details of DFEs for nonlinear channels, and their applications have been sparsely published in the literature. However, to the best of the authors’ knowledge, no work has appeared on the error propagation and recovery of such equalizers. In this letter we consider a DFE with a nonlinear feedback filter based on a discrete th-order Volterra series, a natural nonlinear extension of the linear finite impulse response (FIR) filter. We determine the effect of the nonlinear extensions on the error propagation analysis, error probability analysis, stability analysis, and error recovery time. II. NONLINEAR CHANNEL MODEL The discrete time Volterra model [6] is a natural extension of the widely used linear FIR channel model. Such a model has been used to extend the DFE to the nonlinear channel case [4], [5]. The th-order discrete time Volterra model adopted in this letter is given by (1). For the purpose of our analysis we as- sume only postcursor terms, and that any precursor terms can be dealt with by prefiltering. is the channel output for an Paper approved by R. A. Kennedy, the Editor for Data Communications, Modulation, and Signal Design of the IEEE Communications Society. Manu- script received November 6, 1999; revised April 15, 2000. This paper was pre- sented in part at the Fifth International Symposium on Signal Processing and its Applications (ISSPA), Brisbane, Australia, August, 1999. J. Tsimbinos is with the Defence Science and Technology Organization, Sal- isbury, SA 5108, Australia (e-mail: john.tsimbinos@dsto.defence.gov.au). L. B. White is with the University of Adelaide, Adelaide, SA 5005, Australia (e-mail: lwhite@eleceng.adelaide.edu.au). Publisher Item Identifier S 0090-6778(01)01306-X. Fig. 1. Nonlinear channel equalizer with Volterra feedback filters. input and white Gaussian noise . are the Volterra kernels, and , are the associ- ated memories. (1) III. DFE WITH VOLTERRA FEEDBACK FILTER Nonlinear intersymbol interference is a function of previous as well as current samples, necessitating the need for the DFE to make tentative decisions. Such an equalizer is shown in Fig. 1. We separate the nonlinear ISI components not containing the current sample , and subtract them from the system output to obtain a reasonable tentative decision . For a second-order nonlinear system, a tentative decision would be , where denotes quantization. The final decision of such an equalizer is given by (2) We define the first-order error variable as , the second-order error variable as for , , and . The 0090–6778/01$10.00 © 2001 IEEE