计算机应用   2017, Vol. 37 Issue (1): 197-199,205  DOI: 10.11772/j.issn.1001-9081.2017.01.0197 0

### 引用本文

WEI Dandan, ZHOU Yi, SHI Liming, LIU Hongqing. Block-sparse adaptive filtering algorithm based on inverse hyperbolic sine function against impulsive interference[J]. JOURNAL OF COMPUTER APPLICATIONS, 2017, 37(1): 197-199,205. DOI: 10.11772/j.issn.1001-9081.2017.01.0197.

### 文章历史

Block-sparse adaptive filtering algorithm based on inverse hyperbolic sine function against impulsive interference
WEI Dandan, ZHOU Yi, SHI Liming, LIU Hongqing
Chongqing Key Laboratory of Signal and Information Processing(Chongqing University of Posts and Telecommunications), Chongqing 400065, China
Abstract: Since the existing block-sparse system identification algorithm based on Mean Square Error (MSE) shows poor performance under impulsive interference, an Improved Block Sparse-Normalization Least Mean Square (IBS-NLMS) algorithm was proposed by introducing the inverse hyperbolic sine cost function instead of MSE. A new cost function was constructed and the additive value was obtained by steepest-descent method. Furthermore, a new vector updating equation for filter coefficients was deduced. The adaptive update of the weight vector was close to zero in the presence of impulsive interference, which eliminated the estimation error of adaptive updating based on the wrong information. Meanwhile, mean convergence behavior was analyzed theoretically and then the simulation results demonstrate that in comparison with the Block Sparse-Normalization Least Mean Square (BS-NLMS) algorithm, the proposed algorithm has higher convergence rate and less steady-state error under non-Gaussion noise impulsive interference and abrupt change.
Key words: adaptive filter    non-Gaussion noise    inverse hyperbolic sine function    block sparse system    system identification
0 引言

1 IBS-NLMS自适应滤波算法

 图 1 系统辨识下自适应滤波器系统框图 Figure 1 Block diagram of adaptive filter for system identification
1.1 BS-NLMS自适应滤波算法

 ${\left\| w \right\|_{2,0}}\mathop = \limits^\Delta {\left\| {\;{{\left[ {{{\left\| {{w_{\left[ 1 \right]}}} \right\|}_2}\;{{\left\| {{w_{\left[ 2 \right]}}} \right\|}_2} \cdots {\kern 1pt} \,\,{{\left\| {{w_{\left[ {\text{N}} \right]}}} \right\|}_2}} \right]}^{\text{T}}}} \right\|_0}$ (1)

 $w(\mathrm{n}+1)=w(\mathrm{n})+\frac{\mu \mathrm{e}(\mathrm{n})u(\mathrm{n})}{u(\mathrm{n})u{{(\mathrm{n})}^{\operatorname{T}}}+\varepsilon }+\kappa \mathbf{g}\left( w(\mathrm{n}) \right)$ (2)

 ${{g}_{\mathrm{j}}}(w)\overset{\Delta }{\mathop{=}}\,\left\{ \begin{array}{*{35}{l}} 2{{\alpha }^{2}}{{\mathrm{w}}_{\mathrm{j}}}-2\alpha {{\mathrm{w}}_{\mathrm{j}}}/\left\| w\left[ \left\lceil \mathrm{j/p} \right\rceil \right] \right\|, & {} \\ {} & 0＜\left\| w\left[ \left\lceil \mathrm{j/p} \right\rceil \right] \right\|\le 1/\alpha \\ 0, & 其他 \\ \end{array} \right.$ (3)

1.2 基于非线性函数的IBS-NLMS算法

 $J\text{(}\mathsf{w})=E[arsinh({{(\alpha \left( e(n) \right)/\left\| \mathsf{u}(n) \right\|)}^{2}}/2\alpha )]$ (4)

w(n+1) =w(n)+

 \begin{align} & \mathsf{w}(n+1)=\mathsf{w}(n)+ \\ & \mu \mathsf{u}(n)e(n)/\sqrt{{{\left\| \mathsf{u}(n) \right\|}^{4}}+{{\alpha }^{2}}e{{(n)}^{4}}}+\kappa \mathbf{g}\left( \mathsf{w}(n) \right) \\ \end{align} (5)

 图 2 小误差下的代价函数 Figure 2 Cost function for small error
2 性能分析

 $v\left( n \right)\text{ }={{v}_{g}}\left( n \right)\text{ }+{{v}_{im}}\left( n \right)$ (6)

 $f(v)=\frac{1-{{P}_{r}}}{\sqrt{2\text{ }\!\!\pi\!\!\text{ }\sigma _{g}^{2}}}\exp (-\frac{{{v}^{2}}}{2\sigma _{g}^{2}})+\frac{{{P}_{r}}}{\sqrt{2\text{ }\!\!\pi\!\!\text{ }\sigma _{\Sigma }^{2}}}\exp (-\frac{{{v}^{2}}}{2\sigma _{\Sigma }^{2}})$ (7)

 $\mathsf{c}\left( n\text{ }+\text{ }1 \right)\text{ }=\mathsf{ c}\left( n \right)-\frac{\mu \mathsf{u}(n)e(n)}{\sqrt{{{\left\| \mathsf{u}(n) \right\|}^{4}}+{{\alpha }^{2}}e{{(n)}^{4}}}}\text{+}\kappa \mathbf{g}$ (8)

 $E\text{ }\left[ \mathsf{c}\left( n\text{ }+\text{ }1 \right) \right]\text{ }=\text{ }E\text{ }\left[ \mathsf{c}\left( n \right) \right]\text{ }-\mu \mathsf{H}$ (9)

E[·]表示对{v(n),c(n),u(n)}三者的期望值，其中E[·]记为E{v(n),c(n),u(n)}

 $\mathsf{H=}{{E}_{\left\{ \mathsf{v}(n),\mathsf{c}(n),\mathsf{u}(n) \right\}}}\left[ \mu \mathsf{u}(n)e(n)/\sqrt{{{\left\| \mathsf{u}(n) \right\|}^{4}}+{{\alpha }^{2}}e{{(n)}^{4}}}-\kappa \mathbf{g} \right]= \\ {{E}_{\left\{ \mathsf{c}(n) \right\}}}[{{\mathsf{H}}_{c}}]$ (10)

 ${{\mathsf{H}}_{c}}={{E}_{\left\{ \mathsf{v}(n),\mathsf{u}(n) \right\}}}\left[ \mu \mathsf{u}(n)e(n)/\sqrt{{{\left\| \mathsf{u}(n) \right\|}^{4}}+ \\ {{\alpha }^{2}}e{{(n)}^{4}}}-\kappa \mathbf{g}|\mathsf{c}(n) \right]$ (11)

 ${{\mathsf{H}}_{c}}\approx \varphi '(\sigma _{e}^{2}(n))\mathsf{U\Lambda }{{\mathsf{D}}_{\Lambda }}{{\mathsf{U}}^{\text{T}}}\mathsf{c}(n)$ (12)

 $E\text{ }\left[ \mathsf{c}\left( n\text{ }+\text{ }1 \right) \right]\text{ }=(\mathsf{I}-\mu {{A}_{\varphi }}(\sigma _{e}^{2}(n))\mathsf{U\Lambda }{{\mathsf{D}}_{\Lambda }}{{\mathsf{U}}^{T}})\text{ }E\text{ }\left[ \mathsf{c}\left( n \right) \right]\text{ }$ (13)

 $E\text{ }{{\left[ \mathsf{C}\left( n\text{ }+\text{ }1 \right) \right]}_{ i}}=(\mathsf{I}-\mu {{A}_{\varphi }}(\sigma _{e}^{2}(n)){{\lambda }_{i}}{{\mathrm{I}}_{i}}(\mathsf{\Lambda }))\text{ }E\text{ }{{\left[ \mathsf{C}\left( n \right) \right]}_{i}}$ (14)

3 实验测试和结果分析

 图 3 不同参数α值下的IBS-NLMS算法收敛及跟踪曲线 Figure 3 Convergence and tracking curve for IBS-NLMS algorithm with different α

 图 4 不同概率冲激噪声下的算法NMSD学习曲线 Figure 4 NMSD learning curve with impulsive noise of probability praba

 图 5 系统冲激响应在第20000次迭代时从w变为-w Figure 5 Change of system impulse response from w to -w in iteration 20000
4 结语

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