Generating Toeplitz Matrix which Matches the Convolution Shape Same

Given a filter vH I'm looking for vectors vR and vC such that:

toeplitz(vC, vR) * vX = conv(vX, vH, 'same');

For instance, for vH = [1, 2, 3, 4] and length(vX) = 7; the matrix is given by:

mH =

     3     2     1     0     0     0     0
     4     3     2     1     0     0     0
     0     4     3     2     1     0     0
     0     0     4     3     2     1     0
     0     0     0     4     3     2     1
     0     0     0     0     4     3     2
     0     0     0     0     0     4     3

[Royi Avital]

Anyone?

[Steven Lord]

The convmtx function from Signal Processing Toolbox comes close to doing what you want. I don't know offhand if there's a function in any MathWorks product that comes closer.

Is there a reason you don't want to simply call conv? [If you're expecting multiplying by the convolution matrix to be faster than calling conv, make sure you time the two operations using timeit to test if you're correct in your expectation.]

[Royi Avital]

I just want to generate the matrix specifically with the toeplitz() function.

I acutally can create larger matrix (Matching the full option) and then take a subset of that. But I wonder if I miss something about generating it directly with toeplitz().

Answer 1

[Matt J]

I am specifically asking about using the function toeplitz().

If it must be with toeplitz, then:

nH=numel(vH);
nX=numel(vX);

ic=ceil( (nH+1)/2);
kC=vH(ic:end);
kR=vH(ic:-1:1);


[vC,vR]=deal(sparse(1,nX));

vC(1:length(kC))=kC;
vR(1:length(kR))=kR;

[Matt J]

But note that interpMatrix will be much, much faster than toeplitz for building a sparse mH:

vH=1:12;
vX=rand(1,6000);

 nH=numel(vH);
 nX=numel(vX);
 ic=ceil( (nH+1)/2);
 kC=vH(ic:end);
 kR=vH(ic:-1:1);
 vC=sparse(1,1:numel(kC),kC,1,nX);
 vR=sparse(1,1:numel(kR),kR,1,nX);
tic;
 mH1=toeplitz(vC,vR);
toc; %Elapsed time is 0.652667 seconds.

tic;
 nH=numel(vH);
 nX=numel(vX);
 ic=ceil( (nH+1)/2);
 mH2 = interpMatrix(vH,ic , nX,1);
toc; %Elapsed time is 0.004266 seconds.

>> isequal(mH1,mH2)

ans =

  logical

   1

Answer 2

[Chris Turnes]

I'm very late to this question, but this is what I'd do:

% Even-sized filter smaller than input:
[vC, vR] = get_toeplitz_vectors(7, 1:4)

vC = 7×1

3 4 0 0 0 0 0

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vR = 1×7

3 2 1 0 0 0 0

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% Odd-sized filter smaller than input:
[vC, vR] = get_toeplitz_vectors(7, 1:5)

vC = 7×1

3 4 5 0 0 0 0

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vR = 1×7

3 2 1 0 0 0 0

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% Even-sized filter bigger than input:
[vC, vR] = get_toeplitz_vectors(3, 1:4)

vC = 3×1

3 4 0

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vR = 1×3

3 2 1

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% Odd-sized filter bigger than input:
[vC, vR] = get_toeplitz_vectors(3, 1:5)

vC = 3×1

3 4 5

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vR = 1×3

3 2 1

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function [vC, vR] = get_toeplitz_vectors(n, vH)
vC = zeros(n,1);
vR = zeros(1,n);
nh = numel(vH);
f = 1 + floor(nh/2);
vC(1:(nh-f+1)) = vH(f:nh);
vR(1:f) = vH(f:-1:1);
end

Answer 3

[Matt J]

Using interpMatrix (Download),

nH=numel(vH);
nX=numel(vX);

ic=ceil( (nH+1)/2);

mH = interpMatrix(vH,ic , nX,1);

[Royi Avital]

Hi Matt,

I can generate the matrix in other ways.

I am specifically asking about using the function toeplitz().

[Royi Avital]

@Matt, Have you seen - https://www.mathworks.com/matlabcentral/answers/483310 ?

[Royi Avital]

@Matt, I meant - https://www.mathworks.com/matlabcentral/answers/519408 ?

Answer 4

[Matt J]

Using func2mat (Download),

mH=func2mat(@(vX) conv(vX, vH, 'same'),  ones(length(vX),1));

[Royi Avital]

Hi Matt,

I can generate the matrix in other ways.

I am specifically asking about using the function toeplitz().