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 good points prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photograph technology and electronic photograph processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in a lot of these domain names.
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Extra resources for Advances in Imaging and Electron Physics, Vol. 125
M−1) σ (M−1) σˆ B. Perfect Reconstruction Transmultiplexers The above matrix description of transmultiplexers allows us to simply characterize ﬁlter vectors that yield cross-talk-free and perfect reconstruction transmultiplexers. These two kinds of systems are the analogs of alias-free and perfect reconstruction ﬁlter banks. SUBBAND SIGNAL PROCESSING 47 In particular, an M-transmultiplexer (and, hence, the corresponding operator) is said to be free from cross-talk when every output signal σˆ (i) only depends on the corresponding input σ (i) , namely, nonzero Laurent polynomials τ (0) , τ (1) .
For every integer n, let r, q be the unique integers such that 0 ≤ r < k and n = kq + r . Then the scalar generating function of the nth row of A is given by: k−1 A(n) = t kq j=0 t j (πr j ◦ t k ) . (28) Proof. Since n = kq + r , the elements of the nth scalar row of A can be found in the r th row of the qth block row, whose generating function is t q . Hence: s A(n) = t kq k−1 k−1 pr(i)j t j+ik i=−h j=0 = t kq s pr(i)j t ik tj j=0 . i=−h The assertion now follows from the fact that s i=−h pr(i)j t ik = πr j ◦ t k .
The answer to this problem is given by the next result (see Vetterli and Kovacevic, 1995). 1 Given an analysis filter vector γ = γ (0) , γ (1) , . . , γ (M−1) , 42 BARNABEI AND MONTEFUSCO alias-free reconstruction is possible if and only if the determinant of the M-decimated matrix ∆ M (γ) is different from the zero polynomial, namely, the matrix ∆ M (γ) has full rank. Obviously, the dual problem of ﬁnding an alias-canceling analysis ﬁlter vector when a synthesis ﬁlter vector is given has a similar solution.
Advances in Imaging and Electron Physics, Vol. 125 by Peter W. Hawkes