Convolution in the time domain Multiplication in the frequency - - PowerPoint PPT Presentation

Convolution in the time domain Multiplication in the frequency domain

Matrix-vector multiplication

Convolution Multiplication

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Toeplitz Block Toeplitz Doubly infinite Doubly infinite Banded Banded Polynomial A(θ) Matrix Polynomial A(θ) A-1 (θ) = inverse of A(θ) (not banded?) BC (θ) = B(θ)C(θ) (banded)

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Not banded Singly infinite / finite Periodic A(θ) Circulants Function theory Bottcher- Silbermann Szego-Grenander Grochenig (Wiener L1 Lemma) Wiener-Hopf Gohberg-Semencul

Factorization A(θ) = U(θ) L(θ)

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A = L U or A = L P U

Plemelj / G.D. Birkhofg / ... Gohberg / Kaashoek / Spitkovsky / ... Can we reach linear factors A = A1 A2 ... Ak?

Banded but not Toeplitz

“Time-varying filter” Symbol A(θ) varies from row to row Factorization still possible?

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A = L1 P1 U1 A = L1 σ2 L2 A = U1 σ1 U2 A = U2 P2 L2

Problem for a banded doubly infinite matrix: a11 is in the middle of A Need elimination starting from - ∞ !! A = L P U is still possible, even if A is not Toeplitz

a11 a1n an1 ann

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when / offset

Blocks in C are offset by w from blocks in B

Which is the main diagonal of a doubly infinite matrix?

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Example: one-way shift S (lower diagonal of 1s) Inverse = reverse shift ST (upper diagonal of 1s) Index (S+) = dim (nullspace(S+)) - dim(nullspace((S+)T)) = 0 - 1 Main diagonal of shift matrix S is diagonal -1

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Fredholm: Index not changed if an entry changes Index(BC) = Index(B) + Index(C) Index (S+) locates the main diagonal Index (S+) = 0 when A is centered Then inverse of upper is upper/ finite sections OK Banded permutation: Index by counting 1’s (Lindner, GS)

Matrix entries now have 4 indices (not 2) Vector entries now have 2 indices (not 1)

A is a 2D convolution matrix contains The symbol is for this example? when A is banded?

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