Robust Time-Varying Parameter Proportionate Affine-Projection-Like Algorithm for Sparse System Identification

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.

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)

$$ \begin{aligned} \text{E} \left\{ {\varphi^{2} [e(i)]} \right\} & = P\left( {c(i) = 1} \right)\text{E} \left\{ {\varphi^{2} \left[ {e_{a} (i) + \upsilon_{{\rm g}} (i) + \upsilon_{\omega } (i)} \right]} \right\}{\kern 1pt} {\kern 1pt} {\kern 1pt} + P\left( {c(i) = 0} \right)\text{E} \left\{ {\varphi^{2} \left[ {e_{a} (i) + \upsilon_{{\rm g}} (i)} \right]} \right\} \\ {\kern 1pt} & = p_{r} \text{E} \left\{ {\varphi^{2} [e_{s} (i)]} \right\} + (1 - p_{r} )\text{E} \left\{ {\varphi^{2} [e_{{\rm g}} (i)]} \right\} \\ & = p_{r} \int_{ - \xi }^{\xi } {e_{s}^{2} \frac{1}{{\sqrt {2\pi \sigma_{e,s}^{2} } }}} \exp \left( { - \frac{{e_{s}^{2} }}{{2\sigma_{e,s}^{2} }}} \right){\text{d}}e_{s} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + (1 - p_{r} )\int_{ - \xi }^{\xi } {e_{{\rm g}}^{2} \frac{1}{{\sqrt {2\pi \sigma_{e,g}^{2} } }}} \exp \left( { - \frac{{e_{{\rm g}}^{2} }}{{2\sigma_{e,g}^{2} }}} \right){\text{d}}e_{{\rm g}} \\ {\kern 1pt} & = p_{r} \sigma_{e,s}^{2} \left( {\text{erf} \left( {{\xi \mathord{\left/ {\vphantom {\xi {\sqrt {2\sigma_{e,s}^{2} } }}} \right. \kern-0pt} {\sqrt {2\sigma_{e,s}^{2} } }}} \right) - \xi \exp \left( { - \frac{{\xi^{2} }}{{2\sigma_{e,s}^{2} }}} \right)} \right) + (1 - p_{r} )\sigma_{e,g}^{2} \left( {\text{erf} \left( {{\xi \mathord{\left/ {\vphantom {\xi {\sqrt {2\sigma_{e,g}^{2} } }}} \right. \kern-0pt} {\sqrt {2\sigma_{e,g}^{2} } }}} \right) - \xi \exp \left( { - \frac{{\xi^{2} }}{{2\sigma_{e,g}^{2} }}} \right)} \right) \\ & \triangleq \text{H} \left[ {\text{E} \left\{ {e_{a}^{2} (i)} \right\}} \right] \\ \end{aligned} $$

(A-1)

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

$$ \begin{aligned} \text{E} \left\{ {e_{a} (i)\varphi [e(i)]} \right\} & = \text{E} \left\{ {e_{a} } \right\}\text{E} \left\{ {\varphi^{\prime } \left[ {(e_{a} + \upsilon )} \right]} \right\} \\ & = \text{E} \left\{ {e_{a} (e_{a} + \upsilon )} \right\}\text{E} \left\{ {\varphi^{\prime } [e]} \right\} \\ & = \text{E} \left\{ {e_{a}^{2} } \right\}\text{E} \left\{ {\varphi^{\prime } [e]} \right\} \\ & = \text{E} \left\{ {e_{a}^{2} } \right\}\left( {p_{r} \text{E} \left\{ {\varphi^{\prime}[e_{s} ]} \right\} + (1 - p_{r} )\text{E} \left\{ {\varphi^{\prime } [e_{{\rm g}} ]} \right\}} \right) \\ & = \text{E} \left\{ {e_{a}^{2} } \right\}\left( {p_{r} \int_{ - \xi }^{\xi } {\frac{1}{{\sqrt {2\pi \sigma_{e,s}^{2} } }}} \exp \left( { - \frac{{e_{s}^{2} }}{{2\sigma_{e,s}^{2} }}} \right){\text{d}}e_{s} } \right. + \left. {(1 - p_{r} )\int_{ - \xi }^{\xi } {\frac{1}{{\sqrt {2\pi \sigma_{e,g}^{2} } }}} \exp \left( { - \frac{{e_{{\rm g}}^{2} }}{{2\sigma_{e,g}^{2} }}} \right){\text{d}}e_{{\rm g}} } \right) \\ & = \text{E} \left\{ {e_{a}^{2} } \right\}\left( {p_{r} \text{erf} \left( {{\xi \mathord{\left/ {\vphantom {\xi {\sqrt {2\sigma_{e,s}^{2} } }}} \right. \kern-0pt} {\sqrt {2\sigma_{e,s}^{2} } }}} \right)} \right.\left. { + (1 - p_{r} )\text{erf} \left( {{\xi \mathord{\left/ {\vphantom {\xi {\sqrt {2\sigma_{e,g}^{2} } }}} \right. \kern-0pt} {\sqrt {2\sigma_{e,g}^{2} } }}} \right)} \right) \\ & \triangleq \text{E} \left\{ {e_{a}^{2} (i)} \right\}\text{T} \left[ {\text{E} \left\{ {e_{a}^{2} (i)} \right\}} \right] \\ \end{aligned} $$

(B-1)

where $\varphi^{\prime}[ \cdot ] = {{\partial \varphi [ \cdot ]} \mathord{\left/ {\vphantom {{\partial \varphi [ \cdot ]} {\partial ( \cdot )}}} \right. \kern-0pt} {\partial ( \cdot )}}$.