Signals and Systems - Electrical Engineering

CHAPTER 4 Frequency Analysis: The Fourier Series..................................................

A Mathematician is a device for

turning coffee into theorems.

Paul Erdos (1913–1996)

mathematician

4.1 Introduction

In this chapter and the next we consider the frequency analysis of continuous-time signals and
systems—the Fourier series for periodic signals in this chapter, and the Fourier transform for both
periodic and aperiodic signals as well as for systems in Chapter 5. In these chapters we consider:

n Spectral representation—The frequency representation of periodic and aperiodic signals indicates
how their power or energy is allocated to different frequencies. Such a distribution over frequency
is called thespectrum of the signal. For a periodic signal the spectrum is discrete, as its power
is concentrated at frequencies multiples of a so-calledfundamental frequency, directly related to
the period of the signal. On the other hand, the spectrum of an aperiodic signal is a contin-
uous function of frequency. The concept of spectrum is similar to the one used in optics for
light, or in material science for metals, each indicating the distribution of power or energy over
frequency. The Fourier representation is also useful in finding the frequency response of linear
time-invariant systems, which is related to the transfer function obtained with the Laplace trans-
form. The frequency response of a system indicates how an LTI system responds to sinusoids of
different frequencies. Such a response characterizes the system and permits easy computation of
its steady-state response, and will be equally important in the synthesis of systems.
n Eigenfunctions and Fourier analysis—It is important to understand the driving force behind the
representation of signals in terms of basic signals when applied to LTI systems. For instance, the
convolution integral that gives the output of an LTI system resulted from the representation of its
input signal in terms of shifted impulses. Along with this result came the concept of the impulse
response of an LTI system. Likewise, the Laplace transform can be seen as the representation of
signals in terms of general eigenfunctions. In this chapter and the next we will see that complex

Signals and Systems Using MATLAB®. DOI: 10.1016/B978-0-12-374716-7.00007-7
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