Online Blind Equalization Algorithm for Satellite Channel Based on Echo State Network
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摘要: 针对非线性卫星信道，该文提出了两种基于回声状态网络(ESN)的在线盲均衡算法。利用ESN良好的非线性逼近能力，将发送信号的高阶统计量(HOS)代入ESN，结合常模算法(CMA)和多模算法(MMA)构造盲均衡的代价函数，并采用递归最小二乘(RLS)算法对ESN输出权值进行迭代寻优，实现了Volterra卫星信道下常模和多模信号的在线盲均衡。实验表明，该文算法可以有效降低非线性信道对发送信号产生的畸变，相较于传统的Volterra滤波方法，有更快的收敛速度和更低的均方误差值。
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关键词:
- Volterra卫星信道 /
- 回声状态网络 /
- 常模算法 /
- 多模算法 /
- 递归最小二乘算法
Abstract: Two online blind equalization algorithms based on Echo State Network (ESN) in this paper are proposed for the nonlinear satellite channel. These two algorithms take advantage of the good nonlinear approximation of ESN to bring the High-Order Statistics (HOS) of the transmitted signal into the ESN, and constructing cost function of blind equalization by combining Constant Modulus Algorithm (CMA) and Multi-Modulus Algorithm (MMA). Then, the Recursive Least Squares (RLS) algorithm is used to iteratively optimize the network output weights, and the online blind equalization of the constant modulus signals and the multi-modulus signals over the channel of Volterra satellite are realized. Experiments show that the proposed algorithms can effectively reduce the distortion of the transmitted signal by the nonlinear channel. Compared with the traditional Volterra filtering method, they have faster convergence speed and lower mean square error. -
表 1 ESN-RLS-CMA算法
步骤 1 均衡器初始化：随机生成(${{\text{W}}_{{\rm{res}}}},{{\text{W}}_{{\rm{in}}}}$)，初始化
${\text{u}}(0)$,${{\text{W}}_{{\rm{out}}}}$和$\lambda $; ${\text{P}}(0) = {\delta ^{ - 1}}{\text{I}}$($\delta $是一个很小的正数)；步骤 2 For：n=1, 2,···, N； (1) 更新储备池状态：${\text{u}}(n) = f({{\text{W}}_{{\rm{res}}}}{\text{u}}(n - 1) + {{\text{W}}_{{\rm{in}}}}x(n))$； (2) 计算$y\left( n \right) = {{\text{W}}_{{\rm{out}}}}\left( {n - 1} \right){\text{u}}\left( n \right)$； (3) 由式(7)得到${\tilde{\text U}}(n,n)$，通过式(11)计算自相关矩阵${\text{P}}(n)$； (4) 按照式(12)更新ESN的输出权值${{\text{W}}_{{\rm{out}}}}(n)$； (5) 根据文献[14]的方法调整$\lambda $值。 End； 步骤 3 迭代直到网络收敛为止。 
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表 2 ESN-RLS-MMA算法
步骤 1 均衡器初始化：随机生成(${{\text{W}}_{{\rm{res}}}},{{\text{W}}_{{\rm{in}}}}$)；初始化
${\text{u}}(0)$,${{\text{W}}_{{\rm{out}}}}$,$\lambda $($0 \ll \lambda < 1$),${{\hat{\text R}}^{ - 1}}(0){\rm{ = }}\delta {\text{I}}$($\delta $是一个很小的正
数)；设置$\gamma {\rm{ \!=\! }}3{\rm{E}} \{ s_{\rm{R}}^2(n)\} \!-\! {R_{{\rm{MMA}}}}$，门限值T=$3{\rm{E}}\{ {\left|\! {s(n)}\! \right|^2}\} $；步骤 2 For：n=1,2,···,N； (1) 更新储备池状态：${\text{u}}(n) = f({{\text{W}}_{{\rm{res}}}}{\text{u}}(n - 1) + {{\text{W}}_{{\rm{in}}}}x(n))$； (2) 计算$y(n) = {{\text{W}}_{{\rm{out}}}}(n - 1){\text{u}}(n)$； (3) 通过式(30)计算${{\hat{\text R}}^{ - 1}}(n)$； (4) 计算：${d_{\rm{R}}}(n) = \left[ {\gamma + {R_{{\rm{MMA}}}} - y_{\rm{R}}^2(n)} \right]{y_{\rm{R}}}(n)$,
${d_{\rm{I}}}(n) = \left[ {\gamma + {R_{{\rm{MMA}}}} - y_{\rm{I}}^2(n)} \right]{y_{\rm{I}}}(n)$$d(n) = {\gamma ^{{\rm{ - }}1}}\left[ {{d_{\rm{R}}}(n) + j{d_{\rm{I}}}(n)} \right]$； (5) If ${\left| {y(n)} \right|^2}$>T； $d(n) = 0$ End； (6) 根据式(32)更新${{\text{W}}_{{\rm{out}}}}(n)$。 End； 步骤 3 迭代直到网络收敛为止。
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表 3 取不同储备池规模N时两种算法的MSE值(dB)
算法 N=20 N=50 N=100 N=200 N=300 ESN-RLS-CMA –22.56 –28.12 –29.06 –28.41 –28.72 ESN-RLS-MMA –18.12 –29.58 –30.62 –29.10 –29.29
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表 4 本文算法与5阶Volterra滤波算法的运算复杂度对比
算法 运算复杂度 Volterra O(24M5+16M3+8M) ESN-RLS-CMA O(4N3+18N2+10N) ESN-RLS-MMA O(4N3+19N2+10N)
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