Page 297 - Schaum's Outline of Theory and Problems of Signals and Systems

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

5.64. Derive the harmonic form Fourier series representation (5.15) from the trigonometric Fourier
series representation (5.8).
Hint: Rewrite a, cos kw,t + b, sin kw,t as

cos kw,t + bk ,/, sin kw,t
,/,
(4 + b:)
and use the trigonometric formula cod A - B) = cos A cos B + sin A sin B.

5.65. Show that the mean-square value of a real periodic signal x(r) is the sum of the mean-square
values of its harmonics.
1
Hint: Use Parseval's identity (5.21) for the Fourier series and Eq. (5.168).

5.66. Show that if

then

Hint: Repeat the time-differentiation property (5.55).

5.67. Using the differentiation technique, find the Fourier transform of the triangular pulse signal
shown in Fig. 5-38.
wd/2 ]
sin( wd/2) 2
Ar..Ad[

-d 0 d I
Fig. 5-38

5.68. Find the inverse Fourier transform of

Hint: Differentiate Eq. (5.155) N times with respect to (a).