Ill-convergence of Godard blind equalizers in data communication systems

Godard algorithms form an important class of adaptive blind channel equalization algorithms for QAM transmission. In this paper, the existence of stable undesirable equilibria for the Godard algorithms is demonstrated through a simple AR channel model. These undesirable equilibria correspond to local but nonglobal minima of the underlying mean cost function, and thus permit the ill-convergence of the Godard algorithms which are stochastic gradient descent in nature. Simulation results confirm predicted misbehavior. For channel input of constant modulus, it is shown that attaining the global minimum of the mean cost necessarily implies correct equalization. A criterion is also presented for allowing a decision at the equalizer as to whether a global or nonglobal minimum has been reached. I N band-limited data communications systems that are widely used today, each transmitted symbol is extended by the distortion of the analog channel over a much longer interval than its original duration, hence causing the undesirable intersymbol interference (ISI) effect. Adaptive equalizers are currently the primary devices used by the receiver to combat IS1 introduced in telephone and radio channels. Successful blind equalizers do not require a known training sequence for adequate initialization as conventional adaptive equalizers do. Blind equalization has very important applications in data transmission systems, particularly where sending a training sequence is unrealistic or costly. Among a number of blind equalizer schemes that have been introduced [I]-[3], a special class of blind equalizer proposed by Godard [3] is now well accepted and has been proposed for many applications, including the equalization of QAM (quadrature amplitude modulated) digital signals. The Godard family of blind equalizers, indexed by a parameter p, generalizes the pioneering structure presented by Sato [2] (which is recovered as the special case when p = 1). The first indication that the blind Sato scheme can lead to false parameter convergence under adaptation and consequently Paper approved by the Editor for Channel Equalization of the IEEE Communications Society. Manuscript received August 17, 1989; revised February 27, 1990. This work was supported by the National Science Foundation under Grant MIP-8608787 and MIP-8921003, by the Australian Telecommunications and Electronics Research Board, and by the ANU Centre for Information Science Research. This paper was presented at the 23rd Conference on Information Sciences and Systems, Baltimore, MD, March 1989. Z. Ding is with the Department of Electrical Engineering, Auburn University, Auburn, AL 36849. R. A. Kennedy and B. D. 0. Anderson are with the Department of Systems Engineering, Australian National University, Canberra, ACT 2601, Australia. C.R. Johnson, Jr. is with the School of Electrical Engineering, Cornell University, Ithaca, NY 14853. IEEE Log Number 9100828. to poor performance of the equalizer was given by Mazo [4]. There he showed in the somewhat special case of a noiseless channel having no IS1 that an overparametrized Sato equalizer can lead to convergence to undesirable equilibria in the adaptation process. Macchi and Eweda [5 ] achieved similar results using a different analytical approach. A more significant contribution of [5] was to show that in the Sato scheme one has almost sure convergence to the ideal parameter setting once the eye diagram has opened. (See Kumar [6] for a similar result.) However, global convergence to one of the desirable equilibria (from an initially closed eye) of the Sato scheme has only been established for a nonpractical situation of an infinite number of equalizer parameters and for some specific continuous "symbol" distributions-the heuristic being that an alphabet of M-ary symbols may be approximated by a uniform distribution [I]. Nonetheless, this result (which is one of many found in [I]) is remarkable given the difficulty of the general problem of establishing global convergence only to one of the desired equilibria. In contrast, our work is directed towards showing that generally a practical Godard MA equalizer with finite parameters never has this ideal global convergence property when the symbol distribution is discrete, as in all QAM digital systems. Godard [3] also considered the problem of false equilibria, and showed that for an infinitely parameterized equalizer with infinite delay, false equilibria can exist but these were later shown by Foschini [7] to be locally unstable and are thus insignificant. More recently Treichler et al. [8], [9] provided an alternative development and interpretation of a special case of the Godard family where p = 2, and labeled it the constant modulus algorithm or CMA. (While CMA has now been extended into a class of algorithms sometimes labeled CMA Version p-q for various integers p and q, conventionally CMA refers to the original CMA Version 2-2, identical to the Godard p = 2 algorithm. In this paper we follow the same convention unless otherwise stated.) Stimulated by [8], [9], Johnson, Dasgupta, and Sethares [lo] established local convergence properties of real CMA in a neighborhood of the desired equilibrium using averaging methods [I 11. This work relates closely to open-eye convergence results found in [5] but uses a different analysis technique. In this paper, after some background results (Section 11) we show that in principle it is possible to test for the illconvergence of any of the Godard schemes without explicit knowledge of the actual input sequence, which would appear essential to be able to do in practice. Then we establish 009M778191$01.00 O 1991 IEEE 1314 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 39, NO. 9, SEPTEMBER 1991 (Section 111) the possibility of ill-convergence by deriving a set of undesired stable equilibria for the entire family of real Godard equalizers (when IS1 is present), including the important CMA scheme. Our results are subsequently generalized (Section IV) to the family of complex Godard equalizers. Our techniques stress constructive procedures providing a clear picture of why these blind schemes fail, thus complementing the earlier results for the p = 1 Sato special case [4], (51. Our results also stand in contrast to those of Godard [3] concerning false equilibria for the y = 2 case. His results presuppose an infinite length equalizer which can overparameterize the channel inverse; our (FIR) equalizers are constrained in length to exactly match that of the AR channel inverse such that we cannot encounter this particular type of false equilibrium. His theory is also incomplete in the sense that it only shows the existence of a particular class of false equilibria, which were later shown to be locally unstable for p = 2 by Foschini [7] (and thus of no practical concern), while the theory also fails to recognize the existence of the false equilibria exhibited in this paper. Finally, we mention an important work by Verdu [12], which motivated much of our work here. A. Godard Equalizers Fig. 1 shows the diagram of a data communication system where a Godard equalizer is used. A sequence of i.i.d., digital signals {nk E C ) is sent by the transmitter through a channel exhibiting linear distortion thus generating the output sequence {.rk E C ) . The objective of the equalizer is to recover by inversion (modulo a delay) the original sequence from the received sequence { x k ) . For a channel of AR(iz ) structure, with parameter vector given by H A [ H o H I . . . H,,]', i.e., ~ n p u l sequence