Signal representation using Fourier series - Continuous and Discrete Time Signals and Systems - page 141

CHAPTER

4

Signal representation using
Fourier series

In Chapter 3, we developed analysis techniques for LTIC systems using the
convolution integral by representing the input signal x ( t ) as a linear combi-
nation of time-shifted impulse functions δ ( t ). In Chapters 4 and 5, we will
introduce alternative representations for CT signals and LTIC systems based on
the weighted superpositions of complex exponential functions. The resulting
representations are referred to as the continuous-time Fourier series (CTFS)
and continuous-time Fourier transform (CTFT). Representing CT signals as
superpositions of complex exponentials leads to frequency-domain characteri-
zations, which provide a meaningful insight into the working of many natural
systems. For example, a human ear is sensitive to audio signals within the fre-
quency range 20 Hz to 20 kHz. Typically, a musical note occupies a much
wider frequency range. Therefore, the human ear processes frequency com-
ponents within the audible range and rejects other frequency components. In
such applications, frequency-domain analysis of signals and systems provides
a convenient means of solving for the response of LTIC systems to arbitrary
input signals.
In this chapter, we focus on periodic CT signals and introduce the CTFS used
to decompose such signals into their frequency components. Chapter 5 considers
aperiodic CT signals and develops an equivalent Fourier representation, CTFT,
for aperiodic signals. The organization of Chapter 4 is as follows. In Section 4.1,
we define two- and three-dimensional orthogonal vector spaces and use them
to motivate our introduction to orthogonal signal spaces in Section 4.2. We
show that sinusoidal and complex exponential signals form complete sets of
orthogonal functions. By selecting the sinusoidal signals as an orthogonal set of
basis functions, Sections 4.3 and 4.4 present the trigonometric CTFS for a CT
periodic signal. Section 4.5 defines the exponential representation for the CTFS
based on using the complex exponentials as the basis functions. The properties
of the exponential CTFS are presented in Section 4.6. The condition for the
existence of CTFS is described in Section 4.7. Several interesting applications
of the CTFS are presented in Section 4.8, which is followed by a summary of
the chapter in Section 4.9.

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