Page 256 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

5.14. Verify Parseval's identity (5.21) for the Fourier series, that is,

r m w

then Ckejkw~r = x C-e-ikW = C C*_keikwo' (5.131)
k- -m
k= -w
where * denotes the complex conjugate. Equation (5.131). indicates that if the Fourier
coefficients of x(t) are c,, then the Fourier coefficients of x*(t) are c?,. Setting x,(O =x(O
and x2(t) =x*(t) in Eq. (5.1301, we have d, = c, and ek = c?, or (e-, = c:), and we obtain

5.15. (a) The periodic convolution f(r) = x ,(t ) @ x2(t) was defined in Prob. 2.8. If d, and
en are the complex Fourier coefficients of x,(r) and x2(t), respectively, then show that
the complex Fourier coefficients ck of f(t) are given by

where To is the fundamental period common to x,(t), x2( t), and f(t).
(b) Find the complex exponential Fourier series of f(t) defined in Prob. 2.8(c).

(a) From Eq. (2.70) (Prob. 2.8)

Let

Then

Since

we get

which shows that the complex Fourier coefficients c, of f(t) equal Todkek.