On orthogonal transformations

Abstract

The introduction of an orthogonal transformation pair is generally begun with the definition, followed by the proofs of the orthogonality and the associated Parseval's relation shown as one of the properties of the transform pair. This procedure has to be repeated for various transform relations. In this article, we present the generalized structure of the orthogonal transformation relation by using the vector space model. This method enables us to visualize the similarities as well as the differences among the orthogonal transformations used in signal processing. In addition to the examples, transform pairs including Fourier transform, Fourier series, discrete-time Fourier transform (DFI), Hankel transform, Hilbert transform, the sampling theorem, Legendre, Laguerre, Hermite, and Chebychev decomposition methods are tabulated and shown as special cases of the generalized model. This approach can be used for potential extension or modification of the existing orthogonal transformations. It can be also applied to the design of special-purpose orthogonal mapping techniques.