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 gains prolonged articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photo technology and electronic picture processing, electromagnetic wave propagation, electron microscopy, and the computing tools utilized in these kind of domain names.
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Extra info for Advances in Imaging and Electron Physics, Vol. 125
Basic Linear Operators This subsection is devoted to the analysis of some linear operators on the subspace P of Laurent polynomials, frequently used in practical applications, whose associated matrices reveal a recursive structure. r Duality As we have already remarked, the duality map T : P → P can be represented as follows: T(σ ) = σ ◦ t −1 , (9) where σ is any Laurent polynomial. Using identity (8), we get: [T(σ )] = [σ ∗ ] = R(t −1 , υ) × [σ ], (10) 14 BARNABEI AND MONTEFUSCO Figure 6. The action of the counter-identity matrix R(t −1 , υ).
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 =⎢ .. ⎢ . ⎣ ∗ ∗ (0) (1) α M−1 α M−1 · · · α0(M−1) · · · α1(M−1) .. . · · · α (M−1) M−1 ∗ ⎤ ⎡ γ0(0) γ1(0) ⎥ ⎢ (1) ⎢ γ1(1) ⎥ ⎢ γ0 ⎥×⎢ . ⎥ ⎢ .. ⎦ ⎣ ∗ γ0(M−1) γ1(M−1) ∗⎥ ⎤ (0) · · · γ M−1 (1) · · · γ M−1 .. . (M−1) · · · γ M−1 ⎥ ⎥ ⎥ ⎥, ⎥ ⎦ or, equivalently, = ∆ M (α)∗ × ∆ M (γ)T . The preceding result represents one of the most interesting achievements of the present algebraic approach, as it gives a description of a ﬁlter bank system both in terms of the associated matrix (time-domain formulation) and of generating function (polyphase formulation).
0 ⎣ .. ⎦ ⎣ .. ⎦ αk−1 σˆ k−1 ⎡ ⎡ ⎤ σ0 ⎢ σ1 ⎥ ∗ ⎥ γ1∗ · · · γk−1 ]×⎢ ⎣ ... ⎦ . σk−1 (38) 33 SUBBAND SIGNAL PROCESSING On the other hand, the analysis–synthesis operator is associated with the matrix R(t k , α)T × R(t k , γ ). 4, this matrix is a (k × k)block Toeplitz matrix, whose generating function is the polynomial matrix: = ∆k (α)∗ × ∆k (γ )T . 1 Let Q : P → P be a linear operator associated with the block Toeplitz matrix R(t, Ω), where Ω is the square polynomial matrix ⎡ ω0,0 ⎢ ω1,0 Ω=⎢ ⎣ ...
Advances in Imaging and Electron Physics, Vol. 125 by Peter W. Hawkes