Convolution in the time domain Multiplication in the frequency … · 2019. 2. 28. · Convolution...
Transcript of Convolution in the time domain Multiplication in the frequency … · 2019. 2. 28. · Convolution...
Convolution in the time domain Multiplication in the frequency domain
Matrix-vector multiplicationConvolutionMultiplication
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Toeplitz Block ToeplitzDoubly infinite Doubly infiniteBanded 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. Birkhoff / ...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 rowFactorization 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[ ]
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 matrixcontains
The symbol is
for this example?when A is banded?
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