Abstract
The pros and cons of a quadratic error measure in the context of various applications have often been discussed. In this tutorial, we argue that it is not only a suboptimal but definitely the wrong choice when describing the stability behavior of adaptive filters. We take a walk through the past and recent history of adaptive filters and present 14 canonical forms of adaptive algorithms and even more variants thereof contrasting their mean-square with their l 2−stability conditions. In particular, in safety critical applications, the convergence in the mean-square sense turns out to provide wrong results, often not leading to stability at all. Only the robustness concept with its l 2−stability conditions ensures the absence of divergence.
Highlights
The pros and cons of a quadratic error measure in the context of various applications have often been discussed
We provide an example of the so-called proportionate normalized LMS algorithm (PNLMS) which shows that an adaptive-filter algorithm can be mean squared error (MSE) stable but still exhibit divergence
+u+Tkμμgkgh−,kk1uxx∗k∗kTkeehaa,k,kk−1 robustness: no MSE-stability: 0 < μg,k uk μh,k xk. Based on such an algorithmic formulation, we recognize that the bipartite PNLMS algorithm is of the same kind as linearly-coupled adaptive filters with the special step-size/coupling factor choice: ν1,k = ν2,k = μg,k and ν3,k = ν4,k = μh,k
Summary
The basic concept of a quadratic error measure whose minimum can be found by differentiating and solving a resulting set of linear equations, invented by C.F.Gauss in 1795, has been the tool of choice for about 200 years. Different from MSE-stability, the robustness concept leading to l2−stability does not require any simplifying assumptions on the signals and formulates the adaptive learning process in terms of a feedback filter structure with an allpass (unitary transformation) that is lossless in the feedforward path and a feedback path that usually contains all important signal and system properties as well as free parameters such as the step-size. It has been shown [42] that depending on the correlation of the input signal, such PAP algorithm can become unstable and that, depending on the input signal statistic, situations exist in which even small step-sizes do not result in stable behavior but larger ones are required; depending on the steadystate of the predictor coefficients ak (correspondingly denoted here as linear operator A(q−1)), lower normalized step-sizes αmin may exist as well as upper bounds αmax This is not the only algorithm for which stability problems remained undiscovered for a long time.
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