Spectral filtering using the fast Walsh transform

This paper describes a method of doing spectral filtering using the fast Walsh transform (FWT) rather than the fast Fourier transform (FFT). Rather than using the Walsh transform to find Fourier coefficients which can then be filtered by ordinary means, as was done in [2], we find a new filter function, expressed as a matrix, that does the same filtering operation in the Walsh domain as the filter function matrix in the Fourier domain. This new filter matrix, called the Walsh gain matrix (G
w
), is block-diagonal and real while the Fourier gain matrix (G
f
) is complex diagonal. The block-diagonal structure of G
w
and a condition that causes G
w
to be real are proven. An off-line method for finding G
w
given G
f
is presented. Using the block-diagonal structure of G
w
it is proven that spectral filtering via FWT requires fewer multiplications than spectral filtering via FFT for
N \leq 64
where N is the length of the sequence of samples of the input signal (N is a power of 2). A special condition on G
f
gives a G
w
such that spectral filtering via FWT becomes better, in terms of multiplications, than spectral filtering via FFT for
N \leq 128
.