Page 58 - Fundamentals of Communications Systems

2.10 Chapter Two

Property 2.6 If x(t) is real and odd, i.e., x(t) =−x(−t), then X(f ) is an imaginary valued
and odd function of frequency.

Theorem 2.2 (Rayleigh’s Energy)

∞ ∞

2 2
E x = |x(t)| dt = |X(f )| df (2.33)
−∞ −∞
EXAMPLE 2.17
Example 2.1 (cont.). For the Fourier transform pair of

1 0 ≤ t ≤ T p
x(t) = X(f ) = T p exp[ j π fT p ]sinc( fT p ) (2.34)
0 elsewhere
the energy is most easily computed in the time domain

T p

2
E x = |x(t)| dt = T p (2.35)
0

EXAMPLE 2.18
Example 2.2 (cont.). For the Fourier transform pair of

1 | f |≤ W
x(t) = 2Wsinc(2Wt) X(f ) = (2.36)
0 elsewhere
the energy is most easily computed in the frequency domain

W
2
E x = |X(f )| df = 2W (2.37)
−W

Theorem 2.3 (Convolution) The convolution of two time functions, x(t) and h(t), is
defined as
∞

y(t) = x(t) ∗ h(t) = x(λ)h(t − λ)dλ (2.38)
−∞
The Fourier transform of y(t) is given as

Y (f ) = F{y(t)}= H(f )X(f ) (2.39)

Theorem 2.4 (Duality) If X(f ) = F{x(t)}, then

x(f ) = F{X(−t)} x(− f ) = F{X(t)} (2.40)