Page 55 - Fundamentals of Communications Systems

Signals and Systems Review 2.7

EXAMPLE 2.9
Consider again the signal in Example 2.4 where

1 1
x(t) = cos(2π f m t) = exp[ j 2π f m t] + exp[− j 2π f m t]
2 2
1
f T = f m x 1 = x −1 = , x n = 0 for all other n (2.20)
2

EXAMPLE 2.10
Consider again the signal in Example 2.8 where
x(t) = exp( j 2π f m t)

f T = f m x 1 = 1, x n = 0 for all other n (2.21)

EXAMPLE 2.11
Consider again the signal in Example 2.5. The Fourier series for this example is

1 τ 2πnt
x n = exp − j dt
T T
0
τ
1 exp( − j 2πnt
T
)
=
T − j 2πn
T 0
1 1 − exp( − j 2πnτ )
T
= (2.22)
T j 2πn
T
j θ
e −e − j θ
Using sin(θ) = gives
j 2
τ πnτ
sin( πnτ ) τ πnτ
nτ
T
x n = exp − j πnτ = exp − j sinc (2.23)
T T T T T
T
A number of things should be noted about this example
1. τ and the bandwidth of the waveform are inversely proportional, i.e., a smaller τ
produces a larger bandwidth signal.
2. τ and the signal power are directly proportional.
3. If T /τ = integer, some terms will vanish (i.e., sin(mπ) = 0).
4. To produce a rectangular pulse requires an infinite number of harmonics.