preferred to another.
The obvious definition of the PSD of a stationary random process is the ensemble average of the power spectral
densities for all the members in the ensemble (compare with (1.13)):
S
x
(
ω
)
=
lim
T
→∞

X
T
(
ω
)

2
T
where the bar denotes an ensemble average and
X
T
(
ω
)
is the Fourier transform of the truncated random process
x
(
t
)Π(
t/T
)
:
X
T
(
ω
)
=
integraldisplay
∞
−∞
x
(
t
)
e
−
jωt
dt
For a complex process, we have

X
T
(
ω
)

2
=
X
∗
T
(
ω
)
X
T
(
ω
)
=
integraldisplay
T/
2
−
T/
2
x
∗
(
t
1
)
e
jωt
1
dt
1
integraldisplay
T/
2
−
T/
2
x
(
t
2
)
e
−
jωt
2
dt
2
=
integraldisplay
T/
2
−
T/
2
integraldisplay
T/
2
−
T/
2
x
∗
(
t
1
)
x
(
t
2
)
e
−
jω
(
t
2
−
t
1
)
dt
1
dt
2
so that
S
x
(
ω
)
=
lim
T
→∞
1
T
integraldisplay
T/
2
−
T/
2
integraldisplay
T/
2
−
T/
2
x
∗
(
t
1
)
x
(
t
2
)
e
−
jω
(
t
2
−
t
1
)
dt
1
dt
2
The ensemble averaging and integrating operations may be interchanged, yielding
S
x
(
ω
)
=
lim
T
→∞
1
T
integraldisplay
T/
2
−
T/
2
integraldisplay
T/
2
−
T/
2
ρ
x
(
t
2
−
t
1
)
e
−
jω
(
t
2
−
t
1
)
dt
1
dt
2
where the general complex form of the autocorrelation function
ρ
x
(
τ
)
has been used with the implicit assumption of
widesense stationarity, so that the particular time interval span in question is irrelevant.
The integral can be evaluated using a change of variables, with
τ
≡
t
2
−
t
1
and
τ
′
= (
t
2
+
t
1
)
/
2
. The corresponding
region of integration will be diamond shaped, with only the
τ
coordinate appearing within the integrand.
S
x
(
ω
)
=
lim
T
→∞
1
T
integraldisplay
T
−
T
integraldisplay

τ

/
2
−
τ

/
2
ρ
x
(
τ
)
e
−
jωτ
dτ
′
dτ
=
lim
T
→∞
integraldisplay
T
−
T
ρ
x
(
τ
)(1
− 
τ

/T
)
e
−
jωτ
dτ
=
integraldisplay
∞
−∞
ρ
x
(
τ
)
e
−
jωτ
dτ
Consequently, we find that the PSD for a widesense stationary random process is the Fourier transform of its autocor
relation. This is known as the WienerKhinchine relation. It illustrates the fundamental equivalence of the PSD and
the autocorrelation function for characterizing widesense stationary random processes.
Even moment theorem for GRVs
There is a very useful theorem that applies to the even moments of Gaussian random variables. According to this
theorem, any 2
n
th moment of a set of GRVs can be expressed in terms of all the permutations of the
n
th moments.
This is best illustrated by example. Consider the fourth moment
(
x
1
x
2
x
3
x
4
)
. According to the theorem, we may write
(
x
1
x
2
x
3
x
4
)
≡
(
x
1
x
2
)(
x
3
x
4
)
+
(
x
1
x
3
)(
x
2
x
4
)
+
(
x
1
x
4
)(
x
2
x
3
)
25
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The theorem holds for complex GRVs such as are represented by the outputs of quadrature radio receivers and is
especially useful for predicting the mean and variance of quantities derived from them, like the signal power and
autocorrelation function.
In the next chapter, we begin the main material of the text, starting with the basic tenets of antenna theory. More
complicated aspects of the theory emerge in subsequent chapters. Material from this introductory chapter will mainly
be useful in the second half of the text, which covers noise, signals, and radar signal processing.
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 Spring '13
 HYSELL
 The Land, power density, Solid angle

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