Applied Digital Signal Processing Theory and Practice

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.