Page 244 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

we have

~(t) c, + C [(ck + cPk) cos kwOt + j(ck - C-,) sin k~,t] (5.103)
=
k- 1
Setting
a,
-5- c, + c-, =ak j(ck - c-~) = bk
Eq. (5.103) becomes

a0 m
~(t)
= - + C (ak cos kwot + bk sin kw,,t)
k=l
5.4. Determine the complex exponential Fourier series representation for each of the
following signals:
(a) x(t) = cos w,t
(6) x(t) = sin w,t
( 3
(c) x(t) = cos 2t + -

(dl x(t) = cos4t + sin 6t
(el x(t) = sin2 t
(a) Rather than using Eq. (5.5) to evaluate the complex Fourier coefficients c, using Euler's
formula, we get

Thus, the complex Fourier coefficients for cos w,t are

(b) In a similar fashion we have

Thus, the complex Fourier coefficients for sin w,t are

(c) The fundamental angular frequency w, of x(t) is 2. Thus,

Now