A Variable Step Size Least Mean p-Power Adaptive Filtering Algorithm
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摘要: 在
$ \alpha $ 稳定分布脉冲噪声背景下，为解决固定步长最小平均p范数(LMP)不能同时满足快收敛速度和低稳态误差的问题，该文提出一种对脉冲噪声具有鲁棒性的变步长最小平均p范数(VSS-LMP)自适应滤波算法。该算法利用改进的变形高斯函数来调节步长，采用移动平均法构造变步长函数，克服了定步长算法稳态误差高及抗噪性能差的问题。VSS-LMP算法在系统受到脉冲噪声干扰时，能维持步长稳定；当系统逐渐稳定时，能产生小步长以降低稳态误差。系统辨识仿真结果表明，在$ \alpha $ 稳定分布脉冲噪声下，VSS-LMP算法与固定步长和已有变步长算法相比，具有更快的收敛速度和更强的系统跟踪能力。Abstract: Under$ \alpha $ stable distribution impulse noise environment, in order to solve the problem that the fixed step-size Least Mean p-Power(LMP) can not satisfy the fast convergence speed and low steady-state error at the same time, a Variable Step-Size LMP (VSS-LMP) adaptive filtering algorithm with robustness to impulse noise is proposed. The algorithm uses an improved modified Gaussian function to adjust the step size, and uses a moving average method to construct a variable step size function, which overcomes the problems of high steady-state error and poor anti-noise performance of the fixed-step algorithm. When the system is disturbed by impulse noise, the VSS-LMP algorithm can maintain a stable step size; When the system is gradually stable, it can generate a small step size to reduce the steady-state error. The simulation results of system identification show that the VSS-LMP algorithm has faster convergence speed and stronger system tracking ability compared with the fixed step size and variable step size algorithm under the condition of$ \alpha $ stable distributed impulse noise. -
表 1 VSS-LMP算法框图
%初始化
μ(0)=0，初始化参数w0,a,b,β,p
For n=0,1,2,···
%计算误差
$ e(n){\rm{ = }}d(n) - {{\boldsymbol{w}}^{\rm{T}}}{\boldsymbol{x}}(n) $
%更新步长
$ \mu (e(n)) = \beta \mu (e(n - 1)) + (1 - \beta )a{\left| {e(n)} \right|^2}\exp ( - b{\left| {e(n)} \right|^2}) $
%更新权值
$ {\boldsymbol{w}}(n + 1){\rm{ = }}{\boldsymbol{w}}(n) + \mu (e(n)){\left| {e(n)} \right|^{p - 2}}e(n){\boldsymbol{x}}(n) $
End
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表 2 各算法比较
算法 步长 参数 信噪比5 dB 信噪比15 dB 信噪比25 dB 脉冲参数N4 定步长LMP算法 $\mu $ $\mu $ 0.0065 0.0050 0.0050 0.0004 文献[15] $ \mu {\rm{ = }}\beta \left\{ {\dfrac{1}{{1{\rm{ + }}\exp ( - \alpha {{\left| {e(n)} \right|}^m})}} - 0.5} \right\} $ α, β, m 0.10.015, 3 0.2, 0.015, 3 0.2, 0.02, 3 0.3, 0.0014, 3 文献[16] $ \mu = \beta \left\{ {1 - \dfrac{2}{{1 + \exp (\alpha {{\left| {e(n)} \right|}^\gamma })}}} \right\} $ α, β, γ 0.2, 0.006, 2 0.3, 0.005, 2 0.3, 0.01, 2 0.3, 0.0007, 2 文献[17] $ \mu = c{\rm{\{ }}1 - \exp ( - a{\left| {e(n)} \right|^b}){\rm{\} }} $ a, b, c 0.4, 2, 0.006 0.4, 2, 0.006 0.2, 2, 0.01 0.4, 2, 0.0006 VSS-NLMP $ \begin{gathered}\bar g(n) = \lambda \bar g(n - 1) + (1 - \lambda ){\rm{\{ }}{\left| {e(n)} \right|^{p - 2}}e(n){\boldsymbol{x}}(n){\rm{\} }} \hfill \\\mu (n) = \rho \mu (n - 1) + {\gamma _s}{\rm{\{ }}{\left\| {\bar g(n - 1)} \right\|^2}{\left| {e(n - 1)} \right|^p}{\rm{\} }} \hfill \\ \end{gathered} $ $ \lambda $, $ {\gamma _s} $, $ \rho $ 0.99, 0.97, 0.0004 0.99, 0.97, 0.004 0.99, 0.97,
0.0060.995, 0.97,
0.0008本文算法 $ \mu = \beta \mu (e(n - 1)) + (1 - \beta )a{\left| {e(n)} \right|^2}\exp ( - b{\left| {e(n)} \right|^2}) $ a, b, β 0.001, 0.02, 0.98 0.0008, 0.01, 0.98 0.0026, 0.015, 0.98 0.00008, 0.0009, 0.99
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