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Iterated Rational Filter Banks—Underlying Limit Functions

T. Blu

Proceedings of the IEEE Signal Processing Society Digital Signal Processing Workshop, Utica IL, USA, September 13-16, 1992, pp. 1.8.1-1.8.2.



The term “Rational Filter Bank” (RFB) stands for “Filter Bank with Rational Rate Changes”. An analysis two-band RFB critically sampled is shown with its synthesis counterpart in figure 1. G stands typically for a low-pass FIR filter, whereas H is high-pass FIR. We are interested, in this paper in the iteration of the sole low-pass branch, which leads, in the integer case (q = 1), to a wavelet decomposition.

Kovacevic and Vetterli have wondered whether iterated RFB could involve too, a discrete wavelet transform. Actually, Daubechies proved that whenever p/q is not an integert and G is FIR, this could not be the case. We here show that despite this discouraging feature, there still exists, not only one function (then shifted), as in the integer case, but an infinite set of compactly supported functions φs(t). More importantly, under certain conditions, these functions appear to be "almost" the shifted version of one sole function. These φs are constructed the same way as in the dyadic case (p = 2, q = 1), that is to say by the iteration of the low-pass branch of a synthesis RFB, but in this case the initialization is meaningful.


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