Practical Signals Theory with MATLAB Applications

The Fourier Transform

In this chapter, other orthogonal sets of signals are set aside to concentrate on the complex exponential e^+j2πft. The complex Fourier series for periodic signals is logically extended into a continuous function of frequency suitable for both periodic and nonperiodic signals. A thorough study of the continuous Fourier transform reveals the properties that make this transform technique a classic tool in signals analysis. By relating the time-domain and frequency-domain behavior of signals, the Fourier transform provides a unique perspective on the behavior of signals and systems. The Fourier transform also forms the basis for the Laplace transform in Chapter 7 and the z-transform of Chapter 9.

LEARNING OBJECTIVES

By the end of this chapter, the reader will be able to:

• Explain how the Fourier series extends logically into nonperiodic signals
• Define the orthogonal basis for a Fourier transform representation
• Write the equations for the Fourier transform and its inverse
• Compute the Fourier transform of common time-domain functions
• Recognize common Fourier transform pairs
• Identify graphically the links between a signal and its Fourier transform pair
• Explain how a Fourier transform is affected by time-domain variations (shifting, scaling)
• Apply the rules of the Fourier transform to predict the Fourier transform of an unknown signal

5.1   Introduction

In Chapter ...