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

  • Authors:

  • Phillip M. S. Burt;Max Gerken

  • Affiliations:

  • Laboratory of Communications and Signals, Department of Telecommunications and Control Engineering of EPUSP, PTC/EPUSP, University of São Paulo, Av. Prof. Luciano Gualberto, trav. 3, No. 158, ...;Laboratory of Communications and Signals, Department of Telecommunications and Control Engineering of EPUSP, PTC/EPUSP, University of São Paulo, Av. Prof. Luciano Gualberto, trav. 3, No. 158, ...

  • Venue:
  • Signal Processing - Image and Video Coding beyond Standards
  • Year:
  • 2002

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Abstract

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.