Get Advances in Imaging and Electron Physics, Vol. 125 PDF

By Peter W. Hawkes

ISBN-10: 012014767X

ISBN-13: 9780120147670

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 factors prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, snapshot technological know-how and electronic picture processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in a majority of these domain names.

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R k-downsampler For every fixed integer k ≥ 2, the k-downsampling operator Dk0 , represented as is defined as follows: if σ = Dk0 (σ ) = σ0 := i si t i i is a Laurent polynomial, ski t i = ki|σ t i . i It is immediately clear that the matrix associated with Dk0 is the Hurwitz matrix R(t k , υ), namely, Dk0 (σ ) = R(t k , υ) × [σ ]. (13) B. Other Decimation Operators The downsampling operator defined above associates to a given Laurent polynomial σ the polynomial σ0 that only partially maintains the features of the signal.

M−1) , namely, the synthesis and analysis filter vectors, and represented by the two M-tuples of Hurwitz matrices (A0 , A1 , . . , A M−1 ), where, for i = 0, 1, . . , M − 1, Ai = R t M , α (i) , (C0 , C1 , . . , C M−1 ), Ci = R t M , γ (i) . Making use of formulas (39) and (41), the action of the linear operator Tα,γ can be represented as ⎤ ⎤ ⎡ ⎤ ⎡ ⎡ [σˆ (0) ] [σ (0) ] C0 ⎢ [σˆ (1) ] ⎥ ⎢ C1 ⎥ ⎢ [σ (1) ] ⎥ ⎥. ⎢ ⎥ = ⎢ . ⎥ × AT AT · · · AT ⎢ (49) × .. 0 1 M−1 ⎦ ⎣ ⎦ ⎣ .. ⎦ ⎣ .. [σˆ (M−1) ] C M−1 [σ (M−1) ] 46 BARNABEI AND MONTEFUSCO Its associated matrix is therefore ⎡ ⎤ C0 × A0T C0 × A1T · · · C0 × A TM−1 ⎥ ⎢ ⎢ C1 × A0T C1 × A1T · · · C1 × A TM−1 ⎥ ⎥.

0) α(k−1) (1) α(k−1) (k−1) · · · α(k−1) We now show that the product of a banded Hurwitz matrix, and the transpose of another banded Hurwitz matrix with the same step, turns out to be a Toeplitz matrix, whose generating function can be described explicitly. 2 Let α, β be nonzero Laurent polynomials, and k a fixed integer, k ≥ 2. Set H = R(t k , α), K = R(t k , β). Then, the product H × K T is a Toeplitz matrix, whose generating function λ is given by ∗ λ := Dk0 (αβ ∗ ) = α0 β0∗ + · · · + αk−1 βk−1 .

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


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