A note on several orthogonal bases for digital signal processing

Various orthogonal bases for discrete signal processing are discussed with reference to definitions of signal space and signal processing functions. The discrete Fourier transform and the Walsh/Hadamard transform are formulated on a space whose elements are trigonometric polynomials. A matrix relating the two transforms is found to be sparse with more than 60% zero elements and properties similar to the discrete Fourier transform. It is thus suggested that the fast Fourier transform algorithm is applicable to the matrix. By adopting the product of the matrix and the Walsh/Hadamard transform instead of the discrete Fourier transform, a discrete Fourier transform system with a more efficient algorithm than the fast Fourier transform may be developed.