Page 81 - Fundamentals of Communications Systems

Signals and Systems Review 2.33

yt()
1.5

1.0

0.5

t
−1.0 −0.5 0.5 1.0
Figure 2.15 The rocket signal.

2.6 Example Solutions
Problem 2.24. Recall that the Fourier series is given as

∞

n
x(t) = x n exp j 2π t
T
n=−∞
Each of these waveforms in this chapter can be put in a Fourier Series rep-
resentation by the use of Euler’s formula or
1 j θ − j θ
cos(θ) = [e + e ] (2.102)
2
1
sin(θ) = [e j θ − e − j θ ] (2.103)
2 j

(a)

x(t) = 2 cos(200πt) + 5 sin(400πt)
5
= exp[ j 2π(100)t] + exp[ j 2π(−100)t] + exp[ j 2π(200)t]
j 2
5
− exp[ j 2π(−200)t]
j 2
The waveform is plotted in Figure 2.16.
1
T = 100 x 1 = 1 x −1 = 1
T = 0.01 x 2 = 5 x −2 = −5
j 2 j 2
x n = 0 otherwise