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

Abstract

Abstract-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.

Adaptive Criteria Optimization as a Least Squares Problem

“…The results presented by Li et al [5] also were encouraging. In their work, blind criteria minima are also divided in two classes: length-and cost-dependent.…”

“…as can be promptly confirmed by an inspection of equations (3,4,5,7). In (10), k(n) contains information about or an estimate of the transmitted signal and G(.)…”

“…The last two decades have seen a great deal of work that launched and scrutinized the foundations of blind equalization theory. The works of Lucky, Sato, Godard and many others brought out new criteria, which have been the subject of important theoretical analysis [1] - [5]. A remarkable need arose: to understand not only the criteria per se, but also their interconnections.…”

Equalizadores Cegos Particionados Baseados na Técnica de Bussgang

“…Em [6] o problema da convergência local para equalizadores implementados com filtros de comprimento finito é novamente abordado. Porém, desta feita, um dos exemplos de má convergência é apresentado através do algoritmo Stop-and-Go.…”

“…onde t i é definido como em (11). Note que não foi seguido o mesmo parâmetro (10) utilizado na seção anterior com a finalidade de realizar exatamente a mesma simulação da referência em questão [6]. A partir da Figura 6 podemos constatar que com a aplicação da filtragem particionada o resultado obtido foi o da convergência global.…”