Representing periodic waveforms with nonorthogonal basis functions

Abstract

The representation of periodic functions in a basis function expansion, f(x) \sim \sum_{k=1}^_{\infty} A_{k}g_{1}(kx)+ B_{k}g_{2}(kx) is straightforward when the basis functions g_{1}(kx) and g_{2}(lx) are mutually orthogonal for all k, l . The prototype is g_{1}(x) = \cos (x), g_{2}(x) = \sin (x) . Presented here for the first time is the method for using nonorthogonal basis functions in representing periodic waveforms f(x) . The first of a planned series of papers, this paper presents the fundamental techniques to form the representation. Conditions are given such that the coefficients A_{k} and B_{k} can be found and also that the infinite summation converges to f(x) . Minimum mean-square error finite representations are examined. Each of these aspects of function representation is of critical importance and the methods for dealing with these concerns have always, in the past, required orthogonality. By relaxing this orthogonality condition, a much wider range of basis functions can be used in signal representation. Tailor-made basis functions g_{1} and g_{2} can be used for specific purposes. Fundamental proofs of the basic properties of the representation are examined along with examples illustrating the techniques.