On the convergence of the LMS algorithm with a rank-deficient input autocorrelation matrix
On the convergence of the LMS algorithm with a rank-deficient input autocorrelation matrix
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DOI：
10.1016/j.sigpro.2009.05.002
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发表时间：
2009-11
期刊：
影响因子：
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通讯作者：
D. McLernon;M. Lara;A. Orozco-Lugo
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文献类型：
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作者：
D. McLernon;M. Lara;A. Orozco-Lugo
在所有关于自适应滤波的书籍和论文中，输入自相关矩阵Rxis总是被认为是正定的，因此理论Wiener-Hopf正规方程（Rxxh=rxd）具有唯一解h=hopt（“只有一个全局最优”，[B. Widrow，S. Stearns，AdaptiveSignalProcessing，Prentice-Hall，1985，p.21]）由于Rxx的可逆性（即，它是满秩的）。但是如果Rxx是半正定的而不是满秩的呢？在这种情况下，维纳-霍普夫正规方程仍然是相容的，但有无穷多个可能的解。现在，它是众所周知的，最小均方（LMS），随机梯度算法的滤波器系数，收敛（平均）的唯一Wiener-Hopf解（hopt）时，Rxx是满秩。在本文中，我们将表明，即使当Rxxis不满秩它仍然是可能的预测（收敛）行为的LMS算法的基础上的知识Rxx，rxd和滤波器系数的初始条件。
In all books and papers on adaptive filtering, the input autocorrelation matrix Rxxis always considered positive definite and hence the theoretical Wiener–Hopf normal equations (Rxxh=rxd) have a unique solution h=hopt(“there is only a single global optimum”, [B. Widrow, S. Stearns, Adaptive Signal Processing, Prentice-Hall, 1985, p. 21]) due to the invertibility of Rxx(i.e., it is full-rank). But what if Rxxis positive semi-definite and not full-rank? In this case the Wiener–Hopf normal equations are still consistent but with an infinite number of possible solutions. Now it is well known that the filter coefficients of the least mean square (LMS), stochastic gradient algorithm, converge (in the mean) to the unique Wiener–Hopf solution (hopt) when Rxxis full-rank. In this paper, we will show that even when Rxxis not full-rank it is still possible to predict the (convergence) behaviour of the LMS algorithm based upon knowledge of Rxx, rxdand the initial conditions of the filter coefficients.