New Insights into Convergence Theory of Constrained Frequency-Domain Adaptive Filters
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New Insights into Convergence Theory of Constrained Frequency-Domain Adaptive Filters Feiran Yang1,2
· Gerald Enzner3 · Jun Yang2,4
Received: 1 September 2019 / Revised: 8 October 2020 / Accepted: 10 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Two kinds of update equations are commonly used for the constrained frequencydomain adaptive filter (FDAF), namely the gradient-constrained version and the weight-constrained version. The constraint is imposed only on the stochastic gradient vector in the first version, while it is imposed on the whole weight vector in the second version. It was already found that the two versions have different convergence behaviors, but a rigors analysis of the convergence behavior of the gradient-constrained FDAF is still lacking so far. This paper presents a comprehensive statistical analysis of the gradient-constrained FDAF. We set up an equivalent update equation of the gradient-constrained FDAF, which provides a close link with that of the weightconstrained version. Then, the mean and mean-square convergence behaviors of the gradient-constrained FDAF are analyzed using the new update equation, and the corresponding steady-state solutions are provided. Theoretical results confirm that the gradient-constrained FDAF will converge to a biased solution and exhibits a larger mean-square error than the weight-constrained version when, for instance, the weight vector is not initialized properly. Simulation results agree with our theoretical predictions very well.
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Feiran Yang [email protected] Gerald Enzner [email protected] Jun Yang [email protected]
1
State Key Laboratory of Acoustics, Institute of Acoustics, Chinese Academy of Sciences, 100190 Beijing, China
2
School of Electronic, Electrical and Communication Engineering, University of Chinese Academy of Sciences, 100049 Beijing, China
3
Department of Electrical Engineering and Information Technology, Ruhr-Universität Bochum, Bochum D-44780, Germany
4
Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, Beijing 100190, China
Circuits, Systems, and Signal Processing
Keywords Adaptive filtering · Frequency domain · Transient behavior · Steady-state solution · Biased solution
1 Introduction The frequency-domain adaptive filter (FDAF) is one of the most popular adaptive filtering algorithms [4–6,9,11,18,19,22,24,26], and it has been widely adopted in certain applications, e.g., echo control, adaptive beamforming, acoustic feedback cancellation and active noise control [1,2,7,9,14–16,21–23,27,29,31]. Two versions of the FDAF, i.e., the constrained FDAF and the unconstrained FDAF, are well discussed in the literature [9,19,24]. In the full-modeling situation, the constrained FDAF has a better convergence performance than the unconstrained version at the cost of requiring two extra fast Fourier transform (FFT) operations [19]. There are two kinds of update equations for the constrained FDAF. The first version, namely the grad
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