Stochastic Signals and Systems
Analysis and Processing of Random Signals
Virginia Tech
Fall 2008
Power Spectral Density
•
For deterministic processes (signals), Fourier series and
Fourier transform represent the frequency components as
a “frequency spectrum”
•
Frequency components refer to an average rate of change
(averaged over possible realizations)
•
The power spectral density
S
(
f
)
of a signal
x
(
t
)
is a
function that gives the variation of density of power with
frequency, which can be defined as
S
(
f
) =
lim
T
→∞
1
2
T
Z
T

T
x
(
t
)
e

j
2
π
ft
dt
2
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Power Spectral Density
•
Frequencies are related to autocorrelation function
•
High autocorrelation
→
more “memory” in process, slower
changes
•
Low autocorrelation
→
less “memory” in process, faster
changes
•
The autocorrelation function
R
(
τ
)
of a power signal
x
(
t
)
is
defined as the
time average
R
(
τ
) =
lim
T
→∞
1
2
T
Z
T

T
x
(
t
)
x
(
t
+
τ
)
dt
•
The autocorrelation function and power spectral density of
a power signal are closely related
→
this relationship is
stated formally by the WienerKhinchine theorem:
S
(
f
) =
F
[
R
(
τ
)] =
Z
∞
∞
R
(
τ
)
e

j
2
π
f
τ
d
τ
Power Spectral Density
The
power spectral density
(or
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 Fall '08
 DASILVER
 Autocorrelation, power spectral density

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