The Discrete Fourier Transform

Abstract

This chapter is primarily concerned with the definition, properties, and applications of the Discrete Fourier Transform (DFT). The DFT provides a unique representation using N coefficients for any sequence of N consecutive samples. The DFT coefficients are related to the DTFS coefficients or to equally spaced samples of the DTFT of the underlying sequences. As a result of these relationships and the existence of efficient algorithms for its computation, the DFT plays a central role in spectral analysis, the implementation of digital filters, and a variety of other signal processing applications. Study objectives After studying this chapter you should be able to: Understand the meaning and basic properties of DFT and how to use the DFT to compute the DTFS, DTFT, CTFS, and CTFT transforms. Understand how to obtain the DFT by sampling the DTFT and the implications of this operation on how accurately the DFT approximates the DTFT and other transforms. Understand the symmetry and operational properties of DFT and how to use the property of circular convolution for the computation of linear convolution. Understand how to use the DFT to compute the spectrum of continuous-time signals and how to compensate for the effects of windowing the signal to finite-length using the proper window. Computational Fourier analysis The basic premise of Fourier analysis is that any signal can be expressed as a linear superposition, that is, a sum or integral of sinusoidal signals.