Short communicationA novel combination scheme of proportionate filter☆
Introduction
Generally, information is distributed unevenly in many real-world scenarios, such as network echo cancellation (NEC) and acoustic echo cancellation (AEC). A network echo path with hundreds or thousands of coefficients is a typical impulse response with obvious sparsity, therein only a small number of coefficients have active values and the others are zeros or close to zeros. Although an acoustic echo path is not as sparse as the network echo path, it has a small fraction of big coefficients distributed in a small region and the others are relatively small. The common standard adaptive algorithms, such as the least-mean-squares (LMS) algorithm [1], normalized LMS (NLMS) algorithm [1], [2], affine projection algorithm (APA) [3], [4] and affine projection sign algorithm (APSA) [5], [6], [7], suffer a deteriorative convergence rate in these scenarios because they set the same step size for each coefficient. Therefore, how to exploit the sparse characteristic of such echo paths to achieve faster convergence rate is an attractive and interesting question to many researchers. In the past decade, various proportionate algorithms, such as proportionate NLMS (PNLMS) [8], [9], proportionate APA (PAPA) [10], [11] and real-coefficient proportionate APSA (RP-APSA) [12], have been developed to improve the convergence rate for sparse echo paths. These proportionate algorithms assign the adaptive step size gain in proportion to the every coefficient individually and thus the overall convergence rate of the filter can be improved. When the echo impulse response is dispersive, however, these proportionate filters converge much slower than the standard filters. An effective scheme to solve this problem is a family of being called improved proportionate filters (such as PNLMS++ [13], improved PNLMS (IPNLMS) [14], improved PAPA (IPAPA) [15], [16] and real-coefficient improved proportionate APSA (RIP-APSA) [12]) which possesses the advantages of the standard filters in dispersive impulse response and the proportionate filters in sparse impulse response. And the IPNLMS algorithm is more popular and reliable than the PNLMS whatever the nature of the impulse response is.
Like the standard adaptive filters, the proportionate filters including improved versions also encounter the same problem that the conflict between the step size and steady-state error. That is, they require a tradeoff between fast convergence rate and small steady-state error. One popular method to address the problem is adaptive combination of two proportionate filters, in which one filter with a large step size offers a fast convergence rate and the other one with a small step size ensures a small steady-state error [17], [18], [19], [20], [21]. Nevertheless, the conventional convex combination schemes with two proportionate filters have high computational burdens due to the two proportionate filters running at the same time, and they also show poor convergence behavior at the intersection of the fast filter and the slow one. In order to overcome these problems, this paper proposes a family of combined-step-size (CSS) proportionate filters, which uses a modified sigmoidal activation function to adaptively combine two different step sizes of one proportionate filter.
Notation: Boldface letters denote vectors or matrices, n is the time index, superscript T stands for vector or matrix transpose, L is the filter length, M represents the projection order, sgn[•] denotes sign function, wopt is an L × 1 unknown weight vector that needs to be estimated, and w(n) is the estimate of wopt at iteration n.
Section snippets
Proposed algorithms
The update equations of the weight vectors of the standard proportionate filters are, respectively, where μ is the step size, δ is a small positive constant, x(n) is an L × 1 input vector, X(n) is an L × M input matrix, G(n) = diag{g0(n),…, gL-1(n)} is a diagonal proportionate matrix, I is an M × M identity matrix, e(
Simulations
The conventional combination of two proportionate filters (including the CPNLMS, CIPNLMS, CPAPA, CIPAPA, CRP-APSA and CRIP-APSA) with coefficients feedback as in [20] are also compared with the proposed algorithms, respectively. The method of the coefficients feedback is [20]:
The parameters M, ρ, η, τ, ε, and δ are set to 2, 0.01, 0.01, 0, 0.01, and 10–6, respectively. As in [22], C = 2 is a good choice for the proposed CSS algorithms. Three impulse responses (512
Conclusion
A family of combined-step-size (CSS) proportionate filters has been proposed in this paper, which well solves this conflicting requirement of fast convergence rate and small steady-state error of the proportionate filters. The proposed CSS proportionate filters have lower computational complexities and better transient performances than the corresponding convex combination proportionate filters since they only require one filter update and the CSS is essentially a variable step size. The good
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This work was supported in part by the National Natural Science Foundation of China under Grants 61671392, 61271341, and U1562218.