A convolution matrix is a matrix whose multiplication by a vector computes a discrete convolution. For a vector and an input vector , where and are positive integers, the full linear convolution is represented by an matrix with entries

where , ..., and , ..., . Thus , with outside the range . The entries are constant along each negative-sloping diagonal, so has Toeplitz structure (MathWorks).

For example, the convolution of and is given by

The zero padding distinguishes full linear convolution from circular convolution, in which indices wrap around and the corresponding matrix is a circulant matrix.

See also

Circulant Matrix, Circular Convolution, Convolution, Toeplitz Matrix

Explore with Wolfram|Alpha

References

MathWorks. "convmtx: Convolution Matrix." MATLAB Documentation. https://www.mathworks.com/help/signal/ref/convmtx.html.

Cite this as:

Weisstein, Eric W. "Convolution Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ConvolutionMatrix.html

Subject classifications