Fast transform digital filtering using non-diagonal spectral operators

The non-diagonal convolution theory is introduced as a normed commutative algebra based upon the isomorphism between time-discrete linear function spaces and arbitrary spectral spaces. The step by step recursive analysis of cyclic convolution operators towards the associated diagonal spectral operators under the fast Fourier transform outlines a clear and deep insight into the algebraic structure of the intermediate operators and operations as well as into the mechanism of reducing redundant operations. Finally, a new class of non-diagonal fast convolution algorithms for pipelined transforms and for the enhanced applicability of residue arithmetic in finite rings is presented.