Practical Signals Theory with MATLAB Applications, 2nd Edition

Chapter 7Practical Fourier Transforms

7.1 Introduction

In this chapter, the techniques of signals analysis will be explored in the context of practical applications. Further properties of the Fourier Transform will be developed, in particular the convolution property. The ideal linear system introduced in Chapter 4 will serve as the model for these applications. A thorough analysis of the behavior of signals and systems depends on an appreciation of both the time and frequency domains. The Fourier transform will be further treated in both domains and will be found to be appropriate for both periodic and non-periodic signals.

Learning Objectives

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

• Explain how convolution and multiplication relate through the Fourier transform
• Interpret the significance of impulse response vs. transfer function
• Describe causality and its meaning for linear systems analysis
• Compute the phase and frequency response of common systems
• Identify graphically various frequency domain filtering operations
• Use the Fourier transform to accomplish frequency domain filtering
• Derive a time domain filter from a frequency description
• Explain the role of the modulation theorem in signals and systems
• Create a Bode diagram from a time domain impulse response
• Apply the Fourier transform to periodic signals graphically from first principles

7.2 Convolution: Time and Frequency

Although the role of convolution is central to the analysis of linear systems, ...