یک مدل تحلیلی برای پیش‌بینی رفتار همگرایی الگوریتم حداقل میانگین ترکیب نُرم (lmmn)

الگوریتم کمینه میانگین ترکیب نُرم (lmmn)، الگوریتمی مبتنی بر شیب تصادفی خطا است که هدف آن کمینه‌‌سازی ترکیبی از توابع هزینه الگوریتم‌های کمینه میانگین مربعات (lms) و کمینه میانگین چهارم (lmf) است. این الگوریتم بسیاری از ویژگی‌ها و مزایای الگوریتم‌های lms و lmf را با خود به ارث برده است و از جهاتی ضعف‌های این دو الگوریتم را هم برطرف کرده است. بزرگ‌ترین مشکل الگوریتم lmmn فقدان یک مدل تحلیلی برای پیش‌بینی رفتار آن است، به‌طوری‌که کاربرد عملی آن‌را محدود کرده است. ما در این مقاله باهدف حل این مشکل، مدلی تحلیلی را ارائه می‌کنیم که قادر است رفتار میانگین مربعات خطا و میانگین خطای وزن‌ها را با دقت بالایی پیش‌بینی کند. دقت مدل استخراج‌شده از طریق آزمایش‌های متعددی تایید می‌شود.

An Analytical Model for Predicting the Convergence Behavior of the Least Mean Mixed-Norm (LMMN) Algorithm

Stochastic gradientbased adaptation algorithms have received a great attention in various applications. The most wellknown algorithm in this category is the Least Mean Squares (LMS) algorithm that tries to minimize the secondorder criterion of mean squares of the error signal. On the other hand, it has been shown that higherorder adaptive filtering algorithms based on higherorder statistics can perform better in many applications, particularly in the presence of intense noises. However, these algorithms are more prone to instability and also their convergence rates decline in the vicinity of their optimum solutions. In attempt to make use of the useful aspects of these algorithms, it has been proposed to combine the secondorder criterion with higherorder ones, e.g. that of the Least Mean Fourth (LMF) algorithm. The Least Mean MixedNorm (LMMN) algorithm is a stochastic gradientbased algorithm which aim is to minimize an affine combination of the cost functions of the LMS and LMF algorithms. This algorithm has inherited many properties and advantages of the LMS and the LMF algorithms and mitigated their weaknesses in some ways. These advantages are achieved at the cost of the additional computation burden of just one addition and four multiplications per iteration. The main issue of the LMMN algorithm is the lack of an analytical model for predicting its behaviour, the fact that has restricted its practical application. To address this issue, an analytical model is presented in the current paper that is able to predict the meansquareerror and the meanweightserror behaviour with a high accuracy. This model is derived using the Isserlis rsquo; theorem, based on two mild and practically valid assumptions; namely the input signal is stationary, zeromean Gaussian and the measurement noise are additive zeromean with an even probability distribution function (pdf). The accuracy of the derived model is verified using several simulation tests. These results show that the model is of a high accuracy in various settings for the noise rsquo;s power level and distribution as well as the unknown filter characteristics. Furthermore, since the LMF and the LMS algorithms are special cases of the more general LMMN algorithm, the proposed model can also be used for predicting the behaviour of these algorithms.