Page 59 - Fundamentals of Communications Systems

Signals and Systems Review 2.11

Theorem 2.5 Translation and Dilation If y(t) = x(at + b), then

1 f b
Y (f ) = X exp j 2π f (2.41)
|a| a a
Theorem 2.6 Frequency Translation Multiplying any signal by a sinusoidal signal
results in a frequency translation of the Fourier transforms, i.e.,
1 1
x c (t) = x(t) cos(2π f c t) ⇒ X c (f ) = X( f − f c ) + X( f + f c ) (2.42)
2 2
1 1
x c (t) = x(t) sin(2π f c t) ⇒ X c (f ) = X( f − f c ) − X( f + f c ) (2.43)
j 2 j 2
Definition 2.6 The correlation function of a signal x(t)is
∞

∗
V x (τ) = x(t)x (t − τ)dt (2.44)
−∞
Three important characteristics of the correlation funtion are
∞ 2

1. V x (0) = |x(t)| dt = E x
−∞
2. V x (τ) = V (−τ)
∗
x
3. |V x (τ)| < V x (0)

EXAMPLE 2.19
Example 2.1 (cont.). For the pulse
1 0 ≤ t ≤ T p

x(t) = (2.45)
0 elsewhere
the correlation function is
⎧
|τ|
⎨
T p 1 − |τ|≤ T p
V x (τ) = T p (2.46)
⎩
0 elsewhere
Definition 2.7 The energy spectrum of a signal x(t)is
∗ 2
G x (f ) = X(f )X (f ) =|X(f )| (2.47)
The energy spectral density is the Fourier transform of the correlation
function, i.e.,

G x (f ) = F{V x (τ)} (2.48)
The energy spectrum is a functional description of how the energy in the signal
x(t) is distributed as a function of frequency. The units on an energy spectrum