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