Page 223 - Schaum's Outline of Theory and Problems of Signals and Systems

212 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

where c, are known as the complex Fourier coefficients and are given by

where 1," denotes the integral over any one period and 0 to To or -To/2 to T0/2 is
commonly used for the integration. Setting k = 0 in Eq. (53, we have

which indicates that co equals the average value of x(t) over a period.
When x(t) is real, then from Eq. (5.5) it follows that
*
C-, = C, (5.7)

where the asterisk indicates the complex conjugate.

C. Trigonometric Fourier Series:

The trigonometric Fourier series representation of a periodic signal x(t) with funda-
mental period T,, is given by

a0 02 23T
x(t) = - + x (a,cos ko,t + b, sin kwot) w0 = - (5.8)
k=l To
where a, and b, are the Fourier coefficients given by

x(t)cos kw,tdt (5.9a)

The coefficients a, and b, and the complex Fourier coefficients c, are related by
(Prob. 5.3)

From Eq. (5.10) we obtain

c, = +(a, - jb,) c-, = :(ak + jb,) (5.11)
When x(t) is real, then a, and b, are real and by Eq. (5.10) we have
a, = 2 Re[c,] b, = -2 Im[c,] (5.12)

Even and Odd Signals:
If a periodic signal x(t) is even, then b, = 0 and its Fourier series (5.8) contains only
cosine terms:
ffi 23T
a0
x(t) = - + x a, cos kwot o0 = -
2 k=l T"