A Novel Discrete Transform

Abstract

The notion of a discrete transform is of great importance in solving many problems in Science and Engineering. For instance, the ordinary discrete Fourier transform (DFT) is one of the most important and distinguished elements in the class of discrete transform which is extensively used in Digital Signal and Image processing. In this paper, we ourselves define a novel discrete transform. To define this, let 1,80,fli,..., fiN-11 be a given sequence of N complex numbers. For a given positive integer p and a complex parameter a-, we define our transform by the sequence a = la 0 (P), al (P), ?, aN_i (P)}, where for all unity. We show that this transform holds the properties of the linearity and periodicity. The naive computation of this new transform requires a complexity of O(pN2) which is computationally prohibitive for large values of N and p. For relatively small values of p, We further develop a fast algorithm with the complexity of 0(NlogN). The naive and fast algorithms are both implemented in MATLAB and we explore the performance of the fast algorithm by means of numerical examples. N-1 ak(P)= Efl.J (0- + Wik)P j=0 k = 0,1,...,N- 1 and w=e-i2" is an Nth root of