Robust Total Least Mean M-Estimate Normalized Subband Filter Adaptive Algorithm Under EIV Model in Impulsive Noise

Abstract

In order to solve the problem of deteriorating performance of the conventional subband adaptive filtering algorithm when processing the EIV model with impulsive noise, this paper proposes the robust Total Least Mean M-Estimate normalized subband filter (TLMM-NSAF) adaptive algorithm based on the M-estimation function. In addition, we conduct a detailed theoretical performance analysis of the TLMM-NSAF algorithm, which allows us to determine the stable step size range and theoretical steady-state mean squared deviation of the algorithm. To further improve the algorithm's performance, we propose a new variable step size method. Finally, we compared the algorithm with other competition algorithms in applications of system identification and acoustic echo cancellation. Simulation results have demonstrated the superiority of our proposed algorithm, as well as the consistency between the theoretical values and the simulated values.

Appendices

Appendix A

Section 4 of the paper states that the error $e_{i,D} \left( t \right)$ satisfies the equation $e_{i,D} \left( t \right) = {\varvec{h}}^{T} \tilde{x}_{i} \left( t \right)\varvec{ - w}^{T} \left( t \right)\tilde{x}_{i} \left( t \right)$, and that ${\varvec{w}}\left( t \right) \approx {\varvec{w}}_{0}$ when the number of iterations of the algorithm is sufficiently large. From this, (31) can be simplified to:

$$ \begin{aligned} {\frac{{2E\left[ {\tilde{\varvec{w}}(t)^{T} \hat{\varvec{g}}_{TLMM - NSAF} ({\varvec{w}}(t))} \right]}}{{E\left[ {\hat{\varvec{g}}_{TLMM - NSAF} ({\varvec{w}}(t))^{T} \hat{\varvec{g}}_{TLMM - NSAF} ({\varvec{w}}(t))} \right]}}} \hfill \\ { \approx \frac{{2E\left[ {\frac{{e_{i,D}^{2} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{2} \left\| {\tilde{x}} \right\|^{2} }}} \right]}}{{E\left[ {\frac{{e_{i,D}^{2} \left\| {\tilde{x}} \right\|^{2} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{4} \left\| {\tilde{x}} \right\|^{4} }} + \frac{{2e_{i,D}^{3} \tilde{x}^{T} {\varvec{w}}_{0} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{6} \left\| {\tilde{x}} \right\|^{4} }} + \frac{{e_{i,D}^{4} {\varvec{w}}_{0}^{T} {\varvec{w}}_{0} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{8} \left\| {\tilde{x}} \right\|^{4} }}} \right]}}} \end{aligned} . $$

(A1)

From the paper [23] we have $\left\| {\tilde{x}} \right\|^{2} \approx (L - 2)\sigma_{{i,\tilde{\varvec{x}}}}^{2}$ and the length $L$ of the subband adaptive filter is large, so (A1) can be further simplified as:

$$ \begin{array}{*{20}l} {\frac{{2E\left[ {\tilde{\varvec{w}}(t)^{T} \hat{\varvec{g}}_{TLMM - NSAF} ({\varvec{w}}(t))} \right]}}{{E\left[ {\hat{\varvec{g}}_{TLMM - NSAF} ({\varvec{w}}(t))^{T} \hat{\varvec{g}}_{TLMM - NSAF} ({\varvec{w}}(t))} \right]}}} \hfill \\ { \approx 2\frac{{E\left[ {\frac{{e_{i,D}^{2} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{{2}} \left\| {\tilde{x}} \right\|^{2} }}} \right]}}{{E\left[ {\frac{{e_{i,D}^{2} \left\| {\tilde{x}} \right\|^{2} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{{4}} \left\| {\tilde{x}} \right\|^{4} }}} \right]}} = 2\left( {\left\| {{\varvec{w}}_{0} } \right\|^{2} + \theta_{i} } \right)} \hfill \\ \end{array} . $$

(A2)

Appendix B

Using (36) we have:

$$ \begin{aligned} & E\left[ {{\varvec{r}}({\varvec{w}}_{0} ){\varvec{r}}({\varvec{w}}_{0} )^{T} } \right] \\ { = } & \sum\limits_{i = 0}^{N - 1} {E\left[ {\frac{{e_{i,D}^{2} \tilde{x}_{i} \left( t \right)\tilde{x}_{{\text{i}}} \left( t \right)^{T} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{4} \left\| {\tilde{x}_{i} \left( t \right)} \right\|^{4} }} + \frac{{e_{i,D}^{3} \left( t \right){\varvec{w}}_{0} \tilde{x}_{i} \left( t \right)^{T} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{6} \left\| {\tilde{x}_{i} \left( t \right)} \right\|^{4} }}} \right.} \left. { + \frac{{e_{i,D}^{3} \left( z \right)\tilde{x}_{i} \left( z \right){\varvec{w}}_{0}^{T} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{6} \left\| {\tilde{x}_{i} \left( z \right)} \right\|^{4} }}\frac{{e_{i,D}^{4} \left( z \right){\varvec{w}}_{0} {\varvec{w}}_{0}^{T} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{8} \left\| {\tilde{x}_{i} \left( z \right)} \right\|^{4} }}} \right] \\ = & \frac{1}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{4} }}\sum\limits_{i = 0}^{N - 1} {\left\{ {\frac{{E\left[ {e_{i,D}^{2} \left( t \right)\tilde{x}_{i} \left( t \right)\tilde{x}_{i} \left( t \right)^{T} } \right]}}{{\left\| {\tilde{x}_{i} \left( t \right)} \right\|^{4} }} + \frac{{E\left[ {e_{i,D}^{3} \left( t \right){\varvec{w}}_{0} \tilde{x}_{i} \left( t \right)^{T} } \right]}}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{2} \left\| {\tilde{x}_{i} \left( t \right)} \right\|^{4} }}} \right.} \\ \quad & \left. { + \frac{{E\left[ {e_{i,D}^{3} \left( t \right)\tilde{x}_{i} \left( t \right){\varvec{w}}_{0}^{T} } \right]}}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{2} \left\| {\tilde{x}_{i} \left( t \right)} \right\|^{4} }} + \frac{{E\left[ {e_{i,D}^{4} \left( t \right)} \right]{\varvec{w}}_{0} {\varvec{w}}_{0}^{T} }}{{\left\| {\overline{\varvec{w}}_{0} } \right\|^{4} \left\| {\tilde{x}_{i} \left( t \right)} \right\|^{4} }}} \right\} \\ \end{aligned} $$

(B1)

Using the properties of the Gaussian distribution and Assumption 1, it is easy to verify that

$$ \begin{aligned} E\left[ {e_{i,D}^{2} \left( t \right)} \right] = & E\left[ {\left( {v_{i} \left( t \right) - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} } \right] \\ = & E\left[ {v_{i} \left( t \right)^{2} } \right] + E\left[ {\left( {{\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} } \right] \\ = & \sigma_{i}^{2} \parallel \overline{\varvec{w}}_{0} \parallel^{2} \\ \end{aligned} $$

(B2)

and

$$ \begin{gathered} E\left[ {e_{i,D}^{4} \left( t \right)} \right] = \, E\left[ {\left( {v_{i} \left( t \right) - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{4} } \right] \hfill \\ = 3\left\{ {E\left[ {\left( {v_{i} \left( t \right) - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} } \right]} \right\}^{2} \hfill \\ = 3\sigma_{i}^{4} \parallel \overline{\varvec{w}}_{0} \parallel^{4} \hfill \\ \end{gathered} $$

(B3)

Due to the mathematical definition of expectation, when $a$ and $b$ do not satisfy the independence condition, $E[ab] \ne E[a]E[b]$. Then the expected term in B1 is calculated as:

$$ \begin{aligned} & E\left[ {e_{i,D}^{2} \left( t \right)\tilde{x}_{i} \left( t \right)\tilde{x}_{i} \left( t \right)^{T} } \right] \\ = & E\left[ {\left( {v_{n} - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} {\mathbf{R}} + \left( {v_{n} - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} \tilde{x}_{i} \left( t \right){\text{u}}_{i}^{T} \left( t \right)} \right. \\ \quad & + \left( {v_{n} - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} {\text{u}}_{i} \left( t \right)\tilde{x}_{i} \left( t \right)^{T} \\ \quad & \left. { + \left( {v_{n} - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} {\text{u}}_{i} \left( t \right){\text{u}}_{i}^{T} \left( t \right)} \right] \\ = & \sigma_{i,in}^{2} \parallel \overline{\varvec{w}}_{0} \parallel^{2} \left( {{\mathbf{R}} + \sigma_{i,in}^{2} {\text{I}}} \right) \\ \end{aligned} $$

(B4)

and

$$ \begin{aligned} & E\left[ {\left( {v_{i} \left( t \right) - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{3} \tilde{x}_{i} \left( t \right){\varvec{w}}_{0}^{T} } \right] \\ = & E\left[ {v_{i} \left( t \right)^{3} {\text{u}}_{i} \left( t \right){\varvec{w}}_{0}^{T} - 3v_{i} \left( t \right)^{2} \left( {{\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right){\text{u}}_{i} \left( t \right){\varvec{w}}_{0}^{T} } \right. \\ \quad & \left. { + 3v_{i} \left( t \right)\left( {{\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{2} {\text{u}}_{i} \left( t \right){\varvec{w}}_{0}^{T} - \left( {{\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{3} {\text{u}}_{i} \left( t \right){\varvec{w}}_{0}^{T} } \right] \\ = & - 3\sigma_{i,in}^{4} \parallel \overline{\varvec{w}}_{0} \parallel^{2} {\varvec{w}}_{0} {\varvec{w}}_{0}^{{\text{T}}} \\ \end{aligned} $$

(B5)

in the same way, we have

$$ E\left[ {\left( {v_{i} \left( t \right) - {\text{u}}_{i}^{T} \left( t \right){\varvec{w}}_{0} } \right)^{3} {\varvec{w}}_{0} \tilde{x}_{i} \left( t \right)^{T} } \right] = - 3\sigma_{i,in}^{4} \parallel \overline{\varvec{w}}_{0} \parallel^{2} {\varvec{w}}_{0} {\varvec{w}}_{0}^{{\text{T}}} $$

(B6)

where ${\mathbf{R}} = E\left[ {x_{i} \left( t \right)x_{i}^{T} \left( t \right)} \right]$. Substituting (B3) − (B6) into (B1) results in (36).