Digital Signal Processing Reference
In-Depth Information
Unless mentioned otherwise, the term circular symmetry in our discussions al-
ways means Eqs. (6.35) and (6.36), which corresponds to Definition 1.
6.6.5.A Zero mean not implied by Definition 1
For the special case of a scalar random variable, the definition of circular sym-
metry given in Eqs. (6.35) and (6.36) is equivalent to
E
[
x
re
]=
E
[
x
im
]
,
E
[
x
re
x
im
]=0
.
(6
.
71)
The condition
E
[
x
re
x
im
] = 0 is just orthogonality. So, given
any two
real or-
thogonal random variables
x
re
and
x
im
with identical mean square values, the
complex random variable
x
=
x
re
+
jx
im
is circularly symmetric. Since
x
re
and
x
im
can have arbitrary mean,
x
need not have zero mean.
Example 6.4: Circularly symmetric variables with nonzero mean
Let
x
1
and
x
2
be real random variables with
E
[
x
1
]=
E
[
x
2
]=1
,
E
[
x
1
]=
E
[
x
2
]=
m
=0
,
and
E
[
x
1
x
2
]=
ρ,
with 0
<ρ<
1
.
Define
x
ρ
y
=
x
1
−
so that
E
[
yx
1
]=0
,
and
E
[
y
]=
m
(1
− ρ
−
1
)
=0
.
Let
βx
2
ρ
x
re
=
βy
=
βx
1
−
and
x
im
=
x
1
,
where
β>
0 is a constant such that
E
[
x
re
]=1
.
Then
E
[
x
re
]=
E
[
x
im
]=1
,
and
E
[
x
re
x
im
]=0
,
by construction. Furthermore,
E
[
x
re
]and
E
[
x
im
] are both nonzero. Thus
x
=
x
re
+
jx
im
is circularly symmetric with nonzero mean.
Example 6.5: Circularly symmetric Gaussian with nonzero mean
In Ex. 6.4, note that, by construction,
x
re
x
im
=
β
x
x
2
.
−
β/ρ
(6
.
72)
1
0
Assume the original real random variables
x
1
and
x
2
from which we started
are jointly Gaussian. That is, let the real vector [
x
1
x
2
]
T
be Gaussian.
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