# convmtx

Convolution matrix

## Description

example

A = convmtx(h,n) returns the convolution matrix, A, such that the product of A and an n-element vector, x, is the convolution of h and x.

## Examples

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Computing a convolution using conv when the signals are vectors is generally more efficient than using convmtx. For multichannel signals, convmtx might be more efficient.

Compute the convolution of two random vectors, a and b, using both conv and convmtx. The signals have 1000 samples each. Compare the times spent by the two functions. Eliminate random fluctuations by repeating the calculation 30 times and averaging.

Nt = 30;
Na = 1000;
Nb = 1000;

tcnv = 0;
tmtx = 0;

for kj = 1:Nt
a = randn(Na,1);
b = randn(Nb,1);

tic
n = conv(a,b);
tcnv = tcnv+toc;

tic
c = convmtx(b,Na);
d = c*a;
tmtx = tmtx+toc;
end

t1col = [tcnv tmtx]/Nt
t1col = 1×2

0.0007    0.0263

t1rat = tcnv\tmtx
t1rat = 36.4320

conv is about two orders of magnitude more efficient.

Repeat the exercise for the case where a is a multichannel signal with 1000 channels. Optimize conv's performance by preallocating.

Nchan = 1000;

tcnv = 0;
tmtx = 0;

n = zeros(Na+Nb-1,Nchan);

for kj = 1:Nt
a = randn(Na,Nchan);
b = randn(Nb,1);

tic
for k = 1:Nchan
n(:,k) = conv(a(:,k),b);
end
tcnv = tcnv+toc;

tic
c = convmtx(b,Na);
d = c*a;
tmtx = tmtx+toc;
end

tmcol = [tcnv tmtx]/Nt
tmcol = 1×2

0.2141    0.1012

tmrat = tcnv/tmtx
tmrat = 2.1160

convmtx is about three times as efficient as conv.

## Input Arguments

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Input vector, specified as a row or column.

Data Types: single | double

Length of vector to convolve, specified as a positive integer.

• If h is a column vector of length m, A is (m+n-1)-by-n, and the product of A and a column vector, x, of length n is the convolution of h and x.

• If h is a row vector of length m, A is n-by-(m+n-1), and the product of a row vector, x, of length n with A is the convolution of h and x.

## Output Arguments

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Convolution matrix of input h and the vector x, returned as a matrix.

## Algorithms

• convmtx uses the function toeplitz to generate the convolution matrix.

• convmtx handles edge conditions by zero padding.