Page 79 - Fundamentals of Communications Systems

Signals and Systems Review 2.31

Gf ()
x
0.01

−200 200f
Figure 2.14 An example energy spectrum.

(a) Plot x(t) for one period of the signal.
(b) Find P x .
(c) What is the 98% power bandwidth, B 98 ,of x(t)?
Problem 2.16. Show that

−1 ∗ ∗
F {X (f )}= x (−t) (2.96)
Problem 2.17. For the energy spectrum shown in Figure 2.14 find

(a) E x .
(b) The 3dB bandwidth, B 3 .
(c) The 40dB bandwidth, B 40 .
(d) The 98% energy bandwidth, B 98 .

Problem 2.18. A signal has a series expansion given as
∞
8 1 2π(2n − 1)t
m(t) = cos (2.97)
π 2 (2n − 1) 2 T
n=1

(a) Characterize the Fourier series.
(b) Find the 3-dB bandwidth, B 3 .
(c) Find the 40-dB bandwidth, B 40 .
(d) Find the 98% power bandwidth, B 98 .
(e) Find the 99% power bandwidth, B 99 .

Problem 2.19. Prove that B 90 < B 98 .