Length- and cost-dependent local minima of unconstrained blind channel equalizers

Baud-rate linear blind equalizers may converge to undesirable stable equilibria due to different mechanisms. One such mechanism is the use of linear FIR filters as equalizers. In this paper, it is shown that this type of local minima exist for all unconstrained blind equalizers whose cost functions satisfy two general conditions. The local minima generated by this mechanism are thus called length-dependent local minima. Another mechanism is generated by the cost function adopted by the blind algorithm itself. This type of local minima are called cost-dependent local minima. It shall be shown that several welldesigned algorithms do not have cost-dependent local minimum, whereas other algorithms, such as the decision-directed equalizer and the stop-and-go algorithm (SGA), do. Unlike many existing convergence analysis, the convergence of the Godard algorithms (GA’s) and standard cumulant algorithms @CA’s) under Gaussian noise is also presented here. Computer simulations are used to verify the analytical results. I . INTRODUCTION LIND channel equalization is a useful tool for the reB moval of intersymbol interference (ISI) in digital communication systems when training sequences are costly or impractical. Blind (adaptive) channel equalization algorithms are usually designed to minimize cost functions based on statistics of channel output signals. If the underlying cost function of an algorithm has local minima in addition to the global one(s), undesirable local convergence becomes possible. As discovered in [4], local convergence of baud-rate blind equalization algorithms may be the result of two different causes. One cause is the standard use of finite length equalizer filter. As will be shown in this paper, this kind of local minima exist for all finite length baud-rate blind equalizers without filter parameter constraint. They are thus called lengthdependent local minima. Another kind of local minima result from poor selections of cost functions. They can exist even under the ideal “doubly infinite” [4] equalizer filters. Local minima generated by this mechanism are called cost-dependent Manuscript received April 24, 1995; revised April 30, 1996. This work was supported, in part, by NSF grants MIP9309506, MIP9457397, and MIP9210100 and the US Army Research Office. The associate cditor coordinating the review of this paper and approving it for publication was Prof. Jose Carlos M. Bermudca. Y. Li was with the Department of Electrical Engineering, University of Maryland, College Park, MD 20742 USA. He is now with the Wireless Systems Research Deparlmcnt, AT&T Laboratories Research, Holmdel, NJ 07733-0400 USA. K. J. R . Liu is wiih ihe Department of Electrical Engineering, University of Maryland, Collegc Park, MD 20742 USA. Z. Ding is with the Department of Electrical Engineering, Auburn University, Auburn, AL 36849-5201 USA. Publisher Item Identilier S 1053-587X(Y6)OX239-6. Liu, and Zhi Ding local minima. The latter kind of local minima do not exist for well-designed cost functions. Due to the recent upsurge of interest in blind equalization, many different adaptive algorithms have been proposed. The Sato algorithm (SA) [18] was the first known blind equalization algorithm. It was later generalized by Benvenite, Goursat, and Ruget [ l] into a set of what we call “BGR algorithms” (BGRA). It has been proved in [SI that both length and costdependent local minima exist for SA and the BCRA [SI. The local convergence of the decision directed equalizer (DDE) and the stop-and-go algorithm (SGA) [17] has been analyzed in [11], [14], and [IS]. Since the DDE and SGA are identical to the SA for channels with binary inputs, their cost-dependent local convergence is also given by [ 5 ] . On the other hand, the Godard algorithms (GA’s) are a different generalization of the SA by Godard [lo]. The Godard algorithm was later extended into a set of Shalvi-Weinstein algorithms (SWA’s) [25], [26]. Both the CA and SWA are well-known and effective algorithms. The analysis of GA is given in [ 3 ] , [4], and [7]. It is shown in [3] that Godard cost function also has length-dependent local minima (LDLM), but it has no cost-dependent local minima (CDLM) [7]. Based on results in [ 121 that a one-to-one correspondence exists between the minima of the GA and SWA, the SWA has the same convergence performance as the GA. Although some convex cost functions can be designed for adaptive blind equalizers under specific parameter constraints [23], [24], [27] to avoid local convergence, these algorithms tend to be rather slow due to their I , nature of the cost functions. For equalizer filters without constraints, convergence analyzes of [3]-[5], [12], [14], and [I51 have shown LDLM to exist for many known algorithms such as BGRA, SA, GA, SWA, SGA, and DDE. Therefore, a natural question is whether or not it is possible to design good cost functions so that LDLM do not exist even for the common unconstrained FIR equalizer filters. Moreover, all the above convergence analyzes of blind algorithms are based on the noiseless channel assumption for analytical simplicity. There are no analytical results under noise yet. In fact, there have been some conjecture that channel noises may help equalizer parameters to escape some shallow local minima. Since channel noise is present in all practical communication systems even it is often small, convergence analysis of blind equalization algorithms must be carried out under channel noise. In this paper, we address these two important questions. We will study whether LDLM can be eliminated by general blind equalizers under no parameter constraints. 1053-587X/96$05.00