MATLAB: Generating Toeplitz Matrix which Matches the Convolution Shape Same

convolutionconvolution-matrixMATLABmatrixtoeplitz

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

Best Answer

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;
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