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By John Ch. E.

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Extra resources for [Article] On the Suggested Mutual Repulsion of Fraunhofer Lines

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As we have already seen, the matrix associated with the linear operator describing the action of an M-channel filter bank system is a block Toeplitz matrix. Hence, the alias cancelation condition is equivalent to requiring that this matrix is also a scalar Toeplitz matrix, whose generating function is the distortion function of the system. 1, we are able to characterize alias-free systems and give an explicit expression of their distortion function. 2 Let Lα,γ be the linear operator describing the action of an M-channel filter bank relative to the two M-tuples γ = γ (0) , γ (1) , .

We note that, when Q is the linear operator describing the combined action of the analysis and synthesis operators of an M-channel filter bank system, the preceding definition gives the usual notion of alias-free system, as given, for example, in Vaidyanathan (1993). As we have already seen, the matrix associated with the linear operator describing the action of an M-channel filter bank system is a block Toeplitz matrix. Hence, the alias cancelation condition is equivalent to requiring that this matrix is also a scalar Toeplitz matrix, whose generating function is the distortion function of the system.

A. Matrix Description A maximally decimated M-channel filter bank consists of a size M analyisis operator Cγ , followed by a size M synthesis operator Aα , according to the 38 BARNABEI AND MONTEFUSCO following scheme: It is therefore represented by the linear operator Lα,γ = Aα Cγ . Hence, an M-channel filter bank is uniquely identified by the two M-tuples of Laurent polynomials γ = γ (0) , γ (1) , . . , γ (M−1) , α = α (0) , α (1) . . , α (M−1) , namely, the analysis and synthesis filter vectors, and represented by the two M-tuples of Hurwitz matrices (C0 , C1 , .

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