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 technological know-how and electronic picture processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in most of these domain names.
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Additional info for Advances in Imaging and Electron Physics, Vol. 125
Then the product H T × K is a (k × k)-block Toeplitz matrix, whose generating function is the block Laurent polynomial = i Pi t i , where ⎡ ⎤ i(α0 )∗ β0 ··· i(α0 )∗ βk−1 ⎦. ··· ··· ··· Pi = ⎣ i(αk−1 )∗ β0 · · · i(αk−1 )∗ βk−1 Equivalently, can be represented as the polynomial matrix: = ∆k (α)∗ × ∆k (β)T . Proof. The block Toeplitz structure of the matrix H T × K is a consequence of the preceding considerations concerning formula (35). To give an 30 BARNABEI AND MONTEFUSCO explicit description of these blocks, we make use of identity (17).
M−1) . The operator Lα,γ is said to have the perfect reconstruction property whenever a nonzero constant c ∈ K and an integer h exist such that Lα,γ (σ ) = c t −h σ for every Laurent polynomial σ . This is equivalent to the fact that Lα,γ is alias-free and its associated matrix is the scalar Toeplitz matrix R(t, c t h ). 3 Let Lα,γ be the linear operator describing the action of an M-channel filter bank system relative to the two M-tuples γ = γ (0) , γ (1) , . . , γ (M−1) , α = α (0) , α (1) .
C M−1 Hence, by (27), its associated matrix is A0T × C0 + A1T × C1 + · · · + A TM−1 × C M−1 . 4, every summand in (43) is a block Toeplitz matrix, hence, so is the matrix associated with the linear operator Lα,γ . 1 Let Ai = R(t M , α (i) ), Ci = R(t M , γ (i) ), i = 0, 1, . . , M − 1, be banded Hurwitz matrices. The matrix A0T × C0 + A1T × C1 + · · · + A TM−1 × C M−1 is an (M × M)-block Toeplitz matrix, whose generating function is the polynomial matrix given by ⎡ α0(0) ∗ α0(1) ∗ ⎢ ∗ ⎢ α (0) ∗ α1(1) ⎢ 1 =⎢ ..