Algebraic theory of finite fourier transforms

Abstract

The finite Fourier transform is discussed from the viewpoint of finite dimensional algebras over commutative rings. Three main consequences of this approach are developed. First we assume, G to be an arbitrary finite Abelian group of order n and R to be a commutative integral domain. It is shown that the group algebra R[G] is isomorphic to the “pointwise multiplication” algebra R n if and only if R contains 1/ n together with a primitive m -th root of unity, where m is the exponent of G . Matrix representations of the isomorphisms are derived and each is called a generalized finite Fourier transform. The finite Walsh and Hadamard transforms are of course special cases. More important is the result that Fourier transforms may be defined over finite fields of appropriate characteristic. The second development involves the systematic use of tensor products of algebras in order to induce factorization of the fourier trans-form matrix. This results in a general decomposition formula from which all possible versions of the Fast Fourier Transform algorithm may be derived. Specifically both the Cooley-Tukey and Good algorithms are induced by a single functional congruence, the solutions to which define all algorithms of the Fast Fourier Transform type.