Page 57 - Fundamentals of Communications Systems

Signals and Systems Review 2.9

2.2.2 Fourier Transform
If x(t) is an energy signal, then the Fourier transform is defined as
∞

X(f ) = x(t)e − j 2π ft dt = F{x(t)} (2.28)
−∞
X(f ) is in general complex and gives the frequency domain representation of
x(t). The inverse Fourier transform is

∞

x(t) = X(f )e j 2π ft df = F −1 {X(f )}
−∞
EXAMPLE 2.15
Example 2.1 (cont.). The Fourier transform of
1 0 ≤ t ≤ T p

x(t) = (2.29)
0 elsewhere
is given as

T p
T p
− j 2π ft exp[− j 2π ft]
X(f ) = e dt = = T p exp[ j π fT p ]sinc( fT p ) (2.30)
0 − j 2π f 0

EXAMPLE 2.16
Example 2.2(cont.). The Fourier transform of

sin(2πWt)
x(t) = 2W = 2Wsinc(2Wt)
2πWt
is given as

1 | f |≤ W
X(f ) = (2.31)
0 elsewhere

Properties of the Fourier Transform
Property 2.4 If x(t) is real then the Fourier transform is Hermitian symmetric, i.e.,
∗
X(f ) = X (− f ). This implies
|X(f )|=|X(− f )| arg(X(f )) =− arg(X(− f )) (2.32)
Property 2.5 If x(t) is real and an even function of time, i.e., x(t) = x(−t), then X(f )is
a real valued and even function of frequency.