Comments on “Fractional LMS algorithm”

Abstract

The purpose of this note is to point out that the recently proposed fractional least mean squares (FLMS) algorithm, whose derivation is based on fractional derivative, is not suitable for adaptive signal processing. Our claims are verified via extensive simulation results with comparison with the least mean squares (LMS) algorithm, indicating that the new method does not have any advantages over the classical one.

Introduction

The least mean squares (LMS) algorithm is a standard solution for adaptive signal processing applications and numerous LMS variants have been developed in the literature. A recent series of papers [1], [2], [3], [4], [5], [6], [7], [8] suggest a modification of the LMS algorithm, referred to as fractional LMS (FLMS), in an adaptive system identification framework. Denote the unknown stationary impulse response vector by ${\mathit{w}}^{\star }={[w_0^{\star }{w}_1^{\star }\cdots w_{K-1}^{\star }]}^T\in {\mathbb{R}}^K$ where ^T is the transpose operator. Assuming that the system order is exactly known and letting $\mathit{u}(n)={[u(n)u(n-1)\cdots u(n-K+1)]}^T\in {\mathbb{R}}^K$ and $d(n)\in \mathbb{R}$ be the system input vector and noisy output, respectively, the FLMS updating rule is given by [1]:

$$\begin{array}{c}{w}_k(n)=w_k(n-1)+{\mu }_{1}e(n)u(n-k)\hfill \\ +{\mu }_{2}e(n)u(n-k)\frac{w_k^{1-v}(n-1)}{\Gamma (2-v)},k=0,1,\dots ,K-1\hfill \end{array}$$

where $w_k(n)$ is the k-th weight at time n, ${\mu }_1>0$ and ${\mu }_2>0$ are step sizes, $e(n)=d(n)-{\mathit{w}}^T(n)\mathit{u}(n)$ is the error function with $\mathit{w}(n)={[w_0(n)w_1(n)\cdots w_{K-1}(n)]}^T$, $0<v<1$ is the fractional order and Γ is the gamma function. The updating terms associated with ${\mu }_1$ and ${\mu }_2$ are the conventional and fractional derivatives of $e^2(n)$, respectively. Simple investigation on (1) yields the following remark:

Remark 1

• If ${\mu }_2=0$, then the FLMS and LMS algorithms are identical.

• If $w_k(n-1)<0$, $w_k^{1-v}(n-1)$ is complex. In particular, when v=0.5, $w_k^{1-v}$ is purely imaginary.

Raji and Qureshi [1] provide a comparative simulation study for the mean absolute deviation of $\mathit{w}(n)$ when ${\mathit{w}}^{\star }$ is made up of fixed positive weights, and conclude that the FLMS scheme converges faster than the LMS algorithm. Nevertheless, they do not analyze the stochastic behavior of the FLMS algorithm with respect to mean weights and mean square deviation (MSD) of $\mathit{w}(n)$. Founded on [1], several relevant works [2], [3], [4], [5], [6], [7], [8] then appear in the literature, still without a proper stochastic analysis of the algorithm behavior. In fact, all algorithms presented in [2], [4], [6], [7], [8] are based on (1). While in [3], [5], a magnitude sign on the last term of (1), namely, $w_k^{1-v}(n-1)$, is included to produce a positive real number. It is mentioned in [3] that “fractional power of a negative entry can cause problems”, which correspond to Remark 1(ii).

The purpose of this note is to show that the FLMS idea is not useful by examining (1) in a more thorough manner. However, it does not appear to be easy to analyze the stochastic behavior of (1). For example, it is difficult, if not impossible, to obtain the expected value of $w_k(n)w_k^{1-v}(n-1)$, which arises from the last term of (1). Thus, we shall use Monte Carlo simulations to make our points.