Adaptive Filter Theory and Application for the Identification of Sparse Systems

Abstract

One of the most popular adaptive techniques available is the stochastic gradient algorithm, particularly a very simple implementation, the Least Mean Squares (LMS). In this paper, we focus on identifying sparse systems, as is often the case in telecommunications and acoustics applications. In this context, conventional adaptive filters, such as the LMS, are not able to exploit prior knowledge on the system sparsity, so sparse AFs have been shown to be more advantageous, as discussed in this paper. Initially, we offer a concise review on the AFs theory, including some sparse algorithms. Next, we propose a sparse AF derived from the LMS algorithm and the SparseStep approximation of the 0-pseudo-norm penalty. The proposed AF takes sparsity into account both to accelerate convergence and improve performance. Finally, the proposed filter is numerically validated by comparing it with well-known AFs.

Affine Projection Adaptive Filters for the Identification of Sparse Systems

“…This section validates the proposed SS-AP algorithm through numerical simulation under the following operating conditions: signal-to-noise ratios (SNR) of 10 and 25 dB, channel length N = 16, as in [17], and sparsity level [11] of 75%, i.e., K = 4 non-zero coefficients. Each simulation experiment consists of 100 independent Monte Carlo runs, each of which consisting of 3,000 iterations and the results are averaged.…”

“…In each simulation run, the positions of the non-zero coefficients are randomly chosen and their values follow a Gaussian distribution. We compare the performance of the proposed SS-AP algorithm with the AP, ZA-APA, RZA-APA and AP-SSI Geman-McClure in [17], here named GM-AP, all with projection order L = 3, as well as the NLMS and SS-NLMS [11]. The input signal is a correlated Gaussian sequence derived through a first-order autoregressive process given by u(n) = 0.8u(n − 1) + v(n), where v(n) is unit power Gaussian noise.…”

“…The AP algorithm belongs to the so-called data-reusing family, where, at each time instant, past data are reused. In this paper we focus on sparse AP algorithms, as a continuation of our work [11]. This paper proposes a sparse AP adaptive filter whose cost function is penalized by the SparseStep approximation of the 0 -pseudo-norm [12], which is a simple but precise continuous function.…”

“…The contributions of this paper are twofold: (a) We offer a brief discussion on the geometric interpretation of sparse AP algorithms; (b) Furthermore, we extend the sparse adaptive filter SS-LMS proposed in [11] to the AP context. Finally, the performance of the so-called SS-AP is numerically validated by comparison with other sparse AP adaptive filters.…”