A Duality Theorem for the Discrete Sine Transform - IV (DST - IV)

Abstract

This paper presents a new property called the Duality Theorem for the Discrete Sine Transform - IV (DST - IV). Discrete Sine Transform - IV is a finite duration discrete transform. This transform is mathematically related to the famous Discrete Fourier Transform (DFT) and is used in Image Processing applications, but it is surprising that it has escaped attention from pure mathematicians. Most of the properties of the Discrete Sine Transform - IV are quite similar to those of the DFT and Discrete Cosine Transform (DCT) although some differences persist. A formal derivation of the Duality Theorem for the Discrete Sine Transform - IV is presented which was hitherto not mentioned or derived in the literature. The Duality Theorem finds application in the computation of the discrete time - domain function from the DST - IV frequency domain and vice versa thereby reducing considerable labour involved in the evaluation of the summation and thus results in the saving of computation time and implementation cost significantly. Its usage can be successfully exploited in the arenas of Signal Processing, Image Processing, and Communication Systems, where it is common to encounter cases involving the discrete - time and the discrete frequency signals to have the same aspect or resemblance.