Variable regularisation efficient μ-law improved proportionate affine projection algorithm for sparse system identification

Abstract

For sparse system identification, a m-law memorised improved proportionate affine projection algorithm (MMIPAPA) can achieve faster convergence rate than the standard affine projection algorithm. However, the MMIPAPA with constant regularisation parameter requires a tradeoff between fast convergence speed and low steady-state error. To address the problem, proposed are two kinds of variable non-identity regularisation matrices for the MMIPAPA with a negligible additional computational cost and a stability condition for the step-size choice. Simulation results show the good misalignment performance of the proposed algorithms for both coloured and speech input.Introduction: System identification is one of the most important applications of adaptive filtering algorithms. For sparse systems, proportionate normal least-mean-square (PNLMS) has been designed to improve the convergence speed by updating each filter coefficient in proportion to the magnitude of its estimate [1]. Following this approach, many proportionate-type algorithms have been derived, such as improved PNLMS (IPNLMS) [2] and m-law PNLMS (MuPNLMS) [3]. However, the affine projection algorithm (APA) is preferred over least-mean-square (LMS) owing to its faster convergence speed, particularly for coloured input. By taking into account the history of the proportionate factors, a memorised improved proportionate APA (MeIPAPA) has been developed to reduce the computational complexity [4]. Recently, m-law MeIPAPA (MMIPAPA) has been proposed to further improve the overall convergence rate [5]. To meet the conflicting requirements of the fast convergence rate and the low steady-state misalignment, a variable explicit regularisation algorithm for the APA (VR-APA) has been designed [6]. The regularisation matrix for the VR-APA is a variable scalar weighted identity matrix, derived by letting the L 2 -norm of the a posterior error vector equal that of the system noise vector. In this Letter, however, to equate the component of the a posterior error energy vector with the system noise variance instead, we propose two kinds of variable non-identity regularisation matrices for the MMIPAPA, termed the VR-MMIPAPA and the improved VR-MMIPAPA. The proposed algorithms are expected to further improve the overall performance for sparse system identification with a negligible additional computational cost.

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“…Recently, the adaptive filtering algorithms have been utilised to solve many engineering problems such as signal processing, electronics engineering, audio, speech, and language applications, and wind energy conversion systems [28][29][30][31]. In these algorithms, a compromise should be taken into account between the complexity and convergence speed of the algorithm.…”

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“…In addition, it has less computational analysis than the Kalman filter and the recursive least square algorithms. The APA has been applied to solve different engineering problems [22][23][24].…”