Fast Computation of Dft of Real-Valued Symmetric Data

Abstract

Formulation of fast algorithms for the computation of the discrete Fourier transform (DFT) has been a subject of research for quite some time. It is well known that when a symmetry is present in a signal, it reduces the information content and hence we may expect that the use of symmetry will reduce the complexity in the evaluation of the DFT of the signal. In this paper, the DFT of a symmetric signal is evaluated using the well-known radix-2 fast Fourier transform (FFT). Two different kinds of symmetry, the even-symmetry and the odd-symmetry, are considered. Symmetry RelationsLet z ( n ) be a causal signal of length N . z ( n ) possesses even-symmetry (referred to as positive symmetry [l]) when z ( n ) = z ( N -1 -n), n = O,1,2, .., ( N -1).(1) N-1This symmetry is also termed as the reflection symmetry, reflected about the line n = 7.In the frequency domain, the above relation c m be expressed as X ( k ) = w-"(N