TABLE OF CONTENTS In this chapter we review the basics of signals in the frequency domain. The frequency domain description is called the spectrum. The spectral analysis of signals using Fourier series and Fourier transforms is one of the fundamental methods of communication engineering.
Let x( t) be a periodic signal with fundamental period T 0. Then we define the complex exponential Fourier series of x( t) as
where
for some arbitrary t 0. Setting t 0 = ? T 0/2, we have
The coefficients c n are called the Fourier coefficients of x( t). These are, in general, complex numbers and can be expressed as
where c n is the amplitude and ? n is the phase angle of c n.
A plot of c n versus the angular frequency ? = 2 ? f is called the amplitude spectrum of the periodic signal x( t). A plot of ? n versus ? is called the phase spectrum of x( t). These are referred to as frequency spectra of x( t). Since the index n assumes only integers, the frequency spectra of a periodic signal exist only at the discrete frequencies n ? 0. These are therefore referred to as discrete frequency...
