Abstract
Due to its low computational burden, the affine-projection-like (APL) adaptive filtering algorithm has been extensively studied for colored signal input. Recently, a robust APL algorithm was designed by adopting the M-estimate cost function in impulsive noise environment; however, its convergence rate is very slow for sparse system identification. This paper proposed a proportionate APL M-estimate (PAPLM) algorithm, which is derived by using the proportionate matrix to heighten the convergence rate. To maintain good steady-state performance of the PAPLM algorithm, a time-varying parameter PAPLM (TV-PAPLM) algorithm is proposed, which uses a modified exponential function to adjust the time-varying parameter according to the ratio of the mean square score function to the system noise variance. Moreover, the steady-state excess mean-square error performance of PAPLM algorithm is analyzed and obtained in detail. Simulation results reveal that the proposed PAPLM and TV-PAPLM algorithms achieve fast convergence rate and good steady-state performance in sparse system identification.
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The data that support the findings of this study are available from the corresponding author on request.
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Acknowledgements
This work was in part by National Natural Science Foundation of China (Grant: 61871461, 61571374, 61433011), in part by Department of Science and Technology of Sichuan Province (Grant: 19YYJC0681, 2020JDTD0009), and in part by the National Rail Transportation Electrification and Automation Engineering Technology Research Center (Grant: NEEC-2019-A02).
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Appendices
Appendix A: Calculation of \( \text{E} \left\{ {\varphi^{2} [e(i)]} \right\} \)
Firstly, we calculate terms \( \text{E} \left\{ {\varphi^{2} [e(i)]} \right\} \) of (33) (where the time index i is ignored on the right-hand side)
where \( \text{erf} (x)\,\triangleq\, {2/{\sqrt \pi }}\int_{0}^{x} {\exp ( - t^{2} } ){\text{d}}t \), \( \sigma_{e,g}^{2} = \text{E} \left\{ {e_{a}^{2} (i)} \right\} + \sigma_{{\rm g}}^{2} \), \( \sigma_{e,s}^{2} = \text{E} \left\{ {e_{a}^{2} (i)} \right\} + \sigma_{{\rm g}}^{2} + \sigma_{\omega }^{2} = \text{E} \left\{ {e_{a}^{2} (i)} \right\} + (1 + \kappa )\sigma_{{\rm g}}^{2} \).
Appendix B: Calculation of \( \text{E} \left\{ {e_{a} (i)\varphi [e(i)]} \right\} \)
Then, since \( \upsilon (i) \) is a zero-mean contaminated Gaussian noise sequences and independent of \( e_{a} (i) \), and by using Price theorem [1, 2, 5, 23], we can obtain
where \( \varphi^{\prime}[ \cdot ] = {{\partial \varphi [ \cdot ]} \mathord{\left/ {\vphantom {{\partial \varphi [ \cdot ]} {\partial ( \cdot )}}} \right. \kern-0pt} {\partial ( \cdot )}} \).
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Song, P., Zhao, H., Zeng, X. et al. Robust Time-Varying Parameter Proportionate Affine-Projection-Like Algorithm for Sparse System Identification. Circuits Syst Signal Process 40, 3395–3416 (2021). https://doi.org/10.1007/s00034-020-01628-y
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DOI: https://doi.org/10.1007/s00034-020-01628-y