Short communicationComments on “Fractional LMS algorithm”
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 where T is the transpose operator. Assuming that the system order is exactly known and letting and be the system input vector and noisy output, respectively, the FLMS updating rule is given by [1]:where is the k-th weight at time n, and are step sizes, is the error function with , is the fractional order and Γ is the gamma function. The updating terms associated with and are the conventional and fractional derivatives of , respectively. Simple investigation on (1) yields the following remark: Remark 1 If , then the FLMS and LMS algorithms are identical. If , is complex. In particular, when v=0.5, is purely imaginary.
Raji and Qureshi [1] provide a comparative simulation study for the mean absolute deviation of when 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 . 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, , 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 , which arises from the last term of (1). Thus, we shall use Monte Carlo simulations to make our points.
Section snippets
Simulation results
Extensive simulation results are conducted to contrast (1) and the LMS algorithm in the system identification configuration. The input signal u(n) is a zero-mean white Gaussian process with unit variance. The system impulse response vector has a length of either 16 or 1. The noisy system output d(n) is obtained by adding another zero-mean white Gaussian process with variance 0.01 to . Since we already know that there will be problems with the FLMS algorithm if any of the weights are
Conclusion
There are no conditions under which the FLMS scheme is better than the LMS algorithm. If the optimum filter weights are all positive, then the two updating rules perform nearly the same. However, the FLMS algorithm is much more complicated.
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Cited by (24)
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2023, NeurocomputingCitation Excerpt :However, the fractional chain rules used in these two algorithms are wrong [17]. More importantly, compared with the classic LMS algorithm, their performance is not significantly improved or even worsened [16,15]. In order to solve the first problem, Xie et al. propose an enhanced fractional derivative [15].
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2019, Mechanical Systems and Signal ProcessingCitation Excerpt :The ANC, which can compensate for the inadequacies of the PNC, has been widely investigated and applied in engineering [10–12]. In view of the algorithms used in ANC systems, the least mean square (LMS) algorithm and its improved versions in time domain, such as variable step-size LMS (VS-LMS), filtered-x LMS (FxLMS), and variable step-size FxLMS (VS-FxLMS) are most widely used [13–15]. These modified algorithms are simple, robust and effective for stationary and nonstationary noises and can track the changes of the environment adaptively.
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2022, Electrical Engineering
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