A simplified variable step-size LMS algorithm for Fourier analysis and its statistical properties

Highlights

• A simplified VSS-LMS (SVSS-LMS) algorithm is proposed for Fourier analysis.

• Its performance is comparable to most of the seven typical VSS-LMS algorithms.

• In-depth performance analysis is provided for the proposed algorithm.

• Simulations are conducted to confirm the performance and the analytical results.

Abstract

In this paper, several typical variable step-size LMS (VSS-LMS) algorithms are first applied to adaptive Fourier analysis of noisy sinusoidal signals, which were developed in the context of system identification. A simplified VSS-LMS (SVSS-LMS) algorithm is then proposed to perform the same task, which, on the whole, indicates nice performance comparable to that of most of the typical VSS-LMS algorithms, and requires fewer multiplications as well as user parameters. In-depth performance analysis is provided for the proposed algorithm. Difference equations are derived that describe its dynamics, and elegant closed-form expressions are also worked out for its steady-state performance. Extensive simulations are conducted to confirm its effectiveness, and to clarify the validity of the analytical findings.

Introduction

Adaptive Fourier analysis finds applications in a lot of fields, such as digital communications, power engineering, speech signal processing, music signal processing, biomedical engineering, and control systems, see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9]. It enjoys two major advantages over the DFT and its variants [7]. First, it is able to deal with arbitrary intended frequency that does not need to be an integer multiple of the DFT fundamental frequency. Second, it is of recursive nature, inherently capable of tracking nonstationary signals.

So far, various adaptive Fourier analyzers have been proposed and applied in real-life applications, see [2], [3], [4], [5], [6], [7], [8], [9] and references therein. The LMS algorithm was applied in [2], [3], [4] to make the DFT adaptive and useful for nonstationary signals. The same LMS algorithm was discussed and analyzed in [5], [6] for real-valued signals with arbitrary frequencies like music signals generated by electronic keyboards. A set of gradient-based algorithms was proposed based on a p-power error criterion, and their statistical analysis was performed in [7]. This p-power algorithm reduces exactly to the LMS algorithm when p is set to 2. The above-mentioned gradient algorithms provide reasonably good discrete Fourier coefficient (DFC) estimates and require a small number of multiplications [2], [3], [4], [5], [6], [7]. However, their performance may become inadequate in applications that fast convergence and small estimation error are simultaneously imposed.

On the other hand, from 1990s a lot of variable step-size LMS (VSS-LMS) algorithms have been developed in the context of system identification, see, e.g., [10], [11], [12], [13], [14], [15] for some typical ones. Those algorithms enjoy fast convergence in the early stage of adaptation and present less misadjustment in the steady state. Furthermore, computational cost increase in those algorithms is quite moderate, rendering them very attractive for real applications. The use of VSS has been extended to many other signal processing applications, such as frequency estimation [16], adaptive notch filtering [17], and active noise control [18], [19]. To the best of our knowledge, there are only some preliminary results regarding the use of VSS in the realm of adaptive Fourier analysis [20], [21]. The trial presented in [20] is a simple extension of the VSS-LMS algorithm proposed in [10]. The VSS-LMS algorithm derived in [21] suffers from the risk of becoming divergent when the user parameters are not given properly. Further investigation is thus needed into the use of VSS in the context of adaptive Fourier analysis.

In this paper, first, several typical VSS-LMS algorithms derived for system identification or FIR filtering are applied to adaptive Fourier analysis of noisy sinusoidal signals. Their performance as well as computational efficiency of VSS update is compared in detail. Second, a simplified VSS-LMS (SVSS-LMS) algorithm is proposed to perform the Fourier analysis, which presents comparable or even improved performance as compared with those typical VSS-LMS algorithms, and requires less multiplications as well as user parameters. Third, in-depth performance analysis of the proposed algorithm is conducted. Difference equations that describe its dynamics and elegant closed-form expressions for its steady-state performance are derived in detail. Finally, extensive simulations are conducted to clarify the validity of the analytical findings.

The rest of the paper is organized as follows. Section 2 introduces seven (7) typical VSS-LMS algorithms, six (6) developed for system identification and one (1) derived for adaptive Fourier analysis. After comparing their computational complexity and performance, we propose a new SVSS-LMS algorithm based on the insight gained from extensive simulations. In Section 3, performance analysis of the proposed algorithm is conducted, and detailed simulations are provided to show the validity of the analytical results. Section 4 concludes the paper.