On the uniqueness of stationary points in L₂ approximation with a rational polyphase function

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• 作者：Burt ; Phillip M.S. ; Gerken ; Max
• 关键词：Rational L₂ approximation ; Adaptive IIR filters ; Polyphase decomposition
• 刊名：Signal Processing
• 出版年：2002
• 出版时间：November, 2002
• 年：2002
• 卷：82
• 期：11
• 页码：1707-1725
• 全文大小：324 K

文摘

The question of uniqueness of L₂ approximation when using a rational polyphase function is of great importance for the practical use of polyphase IIR adaptive filters. As shown in a previous work, the polyphase structure can alleviate the problem of convergence speed in IIR adaptive filters, allowing their computational complexity gain over FIR adaptive filters to be exploited. In that work, however, a demonstration of a uniqueness property was not presented. In this paper, we initially present an example of a practical application and the gain in convergence speed that can be achieved by using a polyphase IIR adaptive filter. We then demonstrate the following uniqueness property of the approximation of a rational function with degree N by a p-phase rational polyphase function with pM+1 and M free coefficients in, respectively, its numerator and denominator: if N is less or equal to M then the two functions are identical at all stationary points of the squared L₂ norm of the error. Following the demonstration of the uniqueness property, we make some remarks on the existence of multiple polyphase decompositions of a rational function. Such situations lead to restrictions on the use of polyphase IIR adaptive filters, in order to avoid slow convergence.