The Discrete Fourier Transform

Abstract

The DTFT of a discrete-time signal is a continuous function of the frequency ($\omega$), and hence, the relation between $X\left( {\text{e}^{{j}\omega } } \right)$ and $x(n)$ is not a computationally convenient representation. However, it is possible to develop an alternative frequency representation called the discrete Fourier transform (DFT) for finite duration sequences. The DFT is a discrete-time sequence with equal spacing in frequency. We first obtain the discrete-time Fourier series (DTFS) expansion of a periodic sequence. Next, we define the DFT of a finite length sequence and consider its properties in detail. We also show that the DTFS represents the DFT of a finite length sequence. Further, evaluation of linear convolution using the DFT is discussed. Finally, some fast Fourier transform (FFT) algorithms for efficient computation of DFT are described.