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

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) 4

Not banded Singly infinite / finite Periodic A( θ ) Circulants Function theory Bottcher- Szego-Grenander Silbermann Grochenig Wiener-Hopf (Wiener L1 Lemma) Gohberg-Semencul 5

Factorization A( θ ) = U( θ ) L( θ ) A = L U or A = L P U Plemelj / G.D. Birkho fg / ... Gohberg / Kaashoek / Spitkovsky / ... Can we reach linear factors A = A 1 A 2 ... A k ? Banded but not Toeplitz “Time-varying filter” Symbol A( θ ) varies from row to row Factorization still possible? 6

[ ] A = L 1 P 1 U 1 A = L 1 σ 2 L 2 a 11 a 1n a n1 a nn A = U 1 σ 1 U 2 A = U 2 P 2 L 2 Problem for a banded doubly infinite matrix: a 11 is in the middle of A Need elimination starting from - ∞ !! A = L P U is still possible, even if A is not Toeplitz 7

when / offset

Blocks in C are offset by w from blocks in B

Which is the main diagonal of a doubly infinite matrix? Example: one-way shift S (lower diagonal of 1s) Inverse = reverse shift S T (upper diagonal of 1s) Index (S + ) = dim (nullspace(S + )) - dim(nullspace((S + ) T )) = 0 - 1 Main diagonal of shift matrix S is diagonal -1 13

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) 14

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

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

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