CHAPTER 7

FREQUENCY ANALYSIS OF SIGNALS

7.1 CHAPTER OBJECTIVES

On completion of this chapter, the reader should be able to:

1. define the Fourier series and derive equations for a given time series.

2. define the Fourier transform and the relationship between its input and output.

3. scale and interpret a Fourier transform output.

4. explain the use of frequency window functions.

5. explain the derivation of the fast Fourier transform (FFT), and its computational advantages.

6. define the discrete cosine transform (DCT) and explain its suitability for data compression applications.

7.2 INTRODUCTION

This chapter introduces techniques for determining the frequency content of signals. This is done primarily via the Fourier transform, a fundamental tool in digital signal processing. We also introduce the related but distinct DCT), which finds a great many applications in audio and image processing.

7.3 FOURIER SERIES

The Fourier series^{1} is an important technique for analyzing the frequency content of a signal. An understanding of the Fourier series is crucial to understanding a number of closely related “transform” techniques. Used in reverse, it may also be used to synthesize arbitrary periodic waveforms—those that repeat themselves over time.

A periodic waveform is one which repeats itself over time, as illustrated in Figure 7.1. Many signals contain not one, but multiple periodicities. Figure 7.2 shows some naturally occurring data: the monthly number of sunspots for the years ...