Robust Multitask Diffusion Affine Projection M-Estimate Algorithm: Design and Performance Analysis

Abstract

The distributed estimation performance of multitask diffusion affine projection (AP) algorithm (MD-APA) will be greatly reduced under the impulsive noise interference. To overcome this defect, a robust MD-APA is derived by using M-estimate function (MD-APM) to resist the impulsive noise interference. The mean performance, mean square performance and steady-state performance of MD-APM algorithm are studied, and the convergence range of the step-size and the theoretical steady-state MSD are obtained. In addition, the computational complexity of MD-APM algorithm is analyzed in detail. Simulation experiments show that the proposed MD-APM algorithm has better estimation performance compared with MD-APA and MD-APSA under the impulsive noise interference, and its theoretical steady-state mean square deviation (MSD) can provide accurate prediction.

Appendices

Appendix A

The block column vector ${\boldsymbol{z}} = {\text{col}} \left\{ {{\boldsymbol{z}}_{1} ,{\boldsymbol{z}}_{2} ,...,{\boldsymbol{z}}_{N} } \right\}$, where the size of each block ${\boldsymbol{z}}_{k}$ is $M \times 1$, $k = 1,2,...,N$. Then the block maximum norm of the block column vector ${\boldsymbol{z}}$ is defined as [31, 39]

$$ \left\| {\boldsymbol{z}} \right\|_{b,\infty } \triangleq \mathop {\max }\limits_{1 \le k \le N} \left\| {{\boldsymbol{z}}_{k} } \right\|_{2} $$

(A-1)

Accordingly, the maximum block norm of an arbitrary block matrix is defined as

$$ \left\| {{\boldsymbol{\Theta}}} \right\|_{b,\infty } \triangleq \mathop {\max }\limits_{{{\boldsymbol{z}} \ne {\boldsymbol{0}}}} \frac{{\left\| {{\boldsymbol{\Theta z}}} \right\|_{b,\infty } }}{{\left\| {\boldsymbol{z}} \right\|_{b,\infty } }} $$

(A-2)

where ${{\boldsymbol{\Theta}}}$ is a block matrix of $MN \times MN$ and the size of each block is $M \times M$.

Consider the $N \times N$ block diagonal Hermitian matrix ${{\boldsymbol{\varOmega}}} = {\text{diag}} \left\{ {{{\boldsymbol{\varOmega}}}_{1} ,{{\boldsymbol{\varOmega}}}_{2} ,...,{{\boldsymbol{\varOmega}}}_{N} } \right\}$, where each block ${{\boldsymbol{\varOmega}}}_{k}$ is a Hermitian matrix of $M \times M$, thus we obtain [39]

$$ \rho ({{\boldsymbol{\varOmega}}}) = \mathop {\max }\limits_{1 \le k \le N} \rho ({{\boldsymbol{\varOmega}}}_{k} ) = \left\| {{\boldsymbol{\varOmega}}} \right\|_{b,\infty } $$

(A-3)

where $\rho ( \cdot )$ represents the spectral radius of a matrix.

$$ \begin{aligned} &- 2{\mathbb{E}}\left\{ {\mu {{\boldsymbol{\nu}}}^{T} (i){\boldsymbol{C}}^{T} (i){\boldsymbol{G}}^{T} \sum \left[ {{\boldsymbol{G}} - \mu {\boldsymbol{GZ}}(i)} \right]{\tilde{\boldsymbol{w}}}(i)} \right\} \\&\quad = 2\sum_{n = 1}^{P - 1} {{\mathbb{E}}\left\{ {\mu {{\boldsymbol{\nu}}}^{T} (i){\boldsymbol{C}}^{T} (i){\boldsymbol{G}}^{T} \sum \left\{ {\prod\limits_{l = 0}^{n - 1} {\left( {{\boldsymbol{G}} - \mu {\boldsymbol{GZ}}(i - l)} \right)} } \right\}\mu {\boldsymbol{GC}}(i - n){{\boldsymbol{\nu}}}(i - n)} \right\}} \\&\quad = 2\sum_{n = 1}^{P - 1} {{\mathbb{E}}\left\{ {{\text{Tr}} \left( {\sum \left\{ {\prod\limits_{l = 0}^{n - 1} {\left( {{\boldsymbol{G}} - \mu {\boldsymbol{GZ}}(i - l)} \right)} } \right\}\mu^{2} {\boldsymbol{GC}}(i - n){{\boldsymbol{\nu}}}(i - n){{\boldsymbol{\nu}}}^{T} (i){\boldsymbol{C}}^{T} (i){\boldsymbol{G}}^{T} } \right)} \right\}} \\&\quad { = }2\sum_{n = 1}^{P - 1} {{\text{bvec}}\left( {{\mathbb{E}}\left\{ {\left\{ {\prod\limits_{l = 0}^{n - 1} {\left( {{\boldsymbol{G}} - \mu {\boldsymbol{GZ}}(i - l)} \right)} } \right\}\mu^{2} {\boldsymbol{GC}}(i - n){{\boldsymbol{\nu}}}(i - n){{\boldsymbol{\nu}}}^{T} (i){\boldsymbol{C}}^{T} (i){\boldsymbol{G}}^{T} } \right\}^{T} } \right)}^{T} {{\boldsymbol{\sigma}}} \\&\quad = 2\sum_{n = 1}^{P - 1} {\delta^{T} (n)} {{\boldsymbol{\sigma}}} \\ \end{aligned} $$

(A-4)

Appendix B

By using Assumption 3 and defining a probabilistic event $\left| {e_{k} (i)} \right| < \xi_{k}$, we can get

$$ \begin{aligned} P_{e,k} (i) \triangleq P\left\{ {\left| {e_{k} (i)} \right| < \xi_{k} } \right\} \\ = p_{k} P\left\{ {\left| {e_{s,k} (i)} \right| < \xi_{k} } \right\} + (1 - p_{k} )P\left\{ {\left| {e_{v,k} (i)} \right| < \xi_{k} } \right\} \\ \end{aligned} $$

(B-1)

where $e_{s,k} (i) \triangleq {\boldsymbol{x}}_{k}^{T} (i){\tilde{\boldsymbol{w}}}_{k} (i) + v_{k} (i) + \theta_{k} (i)$, $e_{v,k} (i) \triangleq {\boldsymbol{x}}_{k}^{T} (i){\tilde{\boldsymbol{w}}}_{k} (i) + v_{k} (i)$, and ${\tilde{\boldsymbol{w}}}_{k} (i) \triangleq {\boldsymbol{w}}_{k}^{ * } - {\boldsymbol{w}}_{k} (i)$.

Since $e_{s,k} (i)$ and $e_{v,k} (i)$ are zero-mean Gaussian variables [55] so that

$$ \begin{aligned} P_{e,k} (i) &= p_{k} P\left\{ {\left| {e_{s,k} (i)} \right| < \xi_{k} } \right\} + (1 - p_{k} )P\left\{ {\left| {e_{v,k} (i)} \right| < \xi_{k} } \right\} \\& = p_{k} \frac{1}{{\sqrt {2\pi \sigma_{{e_{s,k} }}^{2} (i)} }}\int_{{ - \xi_{k} }}^{{\xi_{k} }} {\exp \left( { - \frac{{e_{s,k}^{2} (i)}}{{2\sigma_{{e_{s,k} }}^{2} (i)}}} \right)} de_{s,k} (i) \\&\quad+ (1 - p_{k} )\frac{1}{{\sqrt {2\pi \sigma_{{e_{v,k} }}^{2} (i)} }} \int_{{ - \xi_{k} }}^{{\xi_{k} }} {\exp \left( { - \frac{{e_{v,k}^{2} (i)}}{{2\sigma_{{e_{v,k} }}^{2} (i)}}} \right)} de_{v,k} (i) \\&\quad\triangleq p_{k} {\text{erf}} \left( {\frac{{\xi_{k} }}{{\sqrt {2\sigma_{{e_{s,k} }}^{2} (i)} }}} \right) + (1 - p_{k} ){\text{erf}} \left( {\frac{{\xi_{k} }}{{\sqrt {2\sigma_{{e_{v,k} }}^{2} (i)} }}} \right) \\ \end{aligned} $$

(B-2)

where $\sigma_{{e_{s,k} }}^{2} (i) = {\text{Tr}} \left( {{\boldsymbol{W}}_{k} (i){\boldsymbol{R}}_{k} (i)} \right) + \sigma_{{s_{k} }}^{2}$, $\sigma_{{e_{v,k} }}^{2} (i) = {\text{Tr}} \left( {{\boldsymbol{W}}_{k} (i){\boldsymbol{R}}_{k} (i)} \right) + \sigma_{{v_{k} }}^{2}$, ${\boldsymbol{W}}_{k} (i) \triangleq {\mathbb{E}}\left\{ {{\tilde{\boldsymbol{w}}}_{k} (i){\tilde{\boldsymbol{w}}}_{k}^{T} (i)} \right\}$, and ${\boldsymbol{R}}_{k} (i) \triangleq {\mathbb{E}}\left\{ {{\boldsymbol{x}}_{k} (i){\boldsymbol{x}}_{k}^{T} (i)} \right\}$.

Using (9), ${\mathbb{E}}\left\{ {q\left[ {e_{k} (i)} \right]} \right\}$ can be computed as

$$ \begin{aligned} {\mathbb{E}}\left\{ {q\left[ {e_{k} (i)} \right]} \right\} &= 1 \times P_{e,k} (i) + 0 \times \left( {1 - P_{e,k} (i)} \right) \\& = p_{k} {\text{erf}} \left( {\frac{{\xi_{k} }}{{\sqrt {2\sigma_{{e_{s,k} }}^{2} (i)} }}} \right) + (1 - p_{k} ){\text{erf}} \left( {\frac{{\xi_{k} }}{{\sqrt {2\sigma_{{e_{v,k} }}^{2} (i)} }}} \right) \\ \end{aligned} $$

(B-3)

Since $\hat{\sigma }_{{e_{k} }}^{2} (i)$ is the variance of error signal without impulsive noise, $\xi_{k}$ is calculated by $\kappa \sqrt {\hat{\sigma }_{{e_{k} }}^{2} (i)}$, thus we get

$$ \xi_{k} = \kappa \sqrt {\hat{\sigma }_{{e_{k} }}^{2} (i)} = \kappa \sqrt {\sigma_{{e_{v,k} }}^{2} (i)} $$

(B-4)

At steady-state, $\sigma_{{e_{s,k} }}^{2} (\infty ) \approx \sigma_{{s_{k} }}^{2}$ and $\sigma_{{e_{v,k} }}^{2} (\infty ) \approx \sigma_{{v_{k} }}^{2}$, hence, we get

$$ \begin{aligned} {\mathbb{E}}\left\{ {q\left[ {e_{k} (\infty )} \right]} \right\} & = P_{e,k} (\infty ) \\ & = p_{k} {\text{erf}} \left( {\frac{{\kappa \sqrt {\sigma_{{v_{k} }}^{2} } }}{{\sqrt {2\sigma_{{s_{k} }}^{2} } }}} \right) + (1 - p_{k} ){\text{erf}} \left( {\frac{\kappa }{\sqrt 2 }} \right) \\ \end{aligned} $$

(B-5)