# Download Advances in Imaging and Electron Physics, Vol. 125 by Peter W. Hawkes PDF

By Peter W. Hawkes

Advances in Imaging and Electron Physics merges long-running serials-Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. The sequence positive aspects prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photograph technology and electronic snapshot processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in these types of domain names.

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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 , .