Page 224 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

If x(t) is odd, then a, = 0 and its Fourier series contains only sine terms:
m 2TT
x(t) = b, sin kwot w0= -
k= 1 To

D. Harmonic Form Fourier Series:
Another form of the Fourier series representation of a real periodic signal x(t) with
fundamental period To is

Equation (5.15) can be derived from Eq. (5.8) and is known as the harmonic form Fourier
series of x(t). The term Co is known as the dc component, and the term C, cos(kwot - 0,)
is referred to as the kth harmonic component of x(t). The first harmonic component
C, COS(~,~
- 8,) is commonly called the fundamental component because it has the same
fundamental period as x(t). The coefficients C, and the angles 8, are called the harmonic
amplitudes and phase angles, respectively, and they are related to the Fourier coefficients
a, and b, by

For a real periodic signal ~(r), the Fourier series in terms of complex exponentials as
given in Eq. (5.4) is mathematically equivalent to either of the two forms in Eqs. (5.8) and
(5.15). Although the latter two are common forms for Fourier series, the complex form in
Eq. (5.4) is more general and usually more convenient, and we will use that form almost
exclusively.

E. Convergence of Fourier Series:
It is known that a periodic signal x(t) has a Fourier series representation if it satisfies
the following Dirichlet conditions:
1. x(t) is absolutely integrable over any period, that is,

2. x(t) has a finite number of maxima and minima within any finite interval of t.
3. x(t) has a finite number of discontinuities within any finite interval of t, and each of
these discontinuities is finite.
Note that the Dirichlet conditions are sufficient but not necessary conditions for the Fourier
series representation (Prob. 5.8).

F. Amplitude and Phase Spectra of a Periodic Signal:
Let the complex Fourier coefficients c, in Eq. (5.4) be expressed as