Digital Signal Processing Reference
In-Depth Information
so that
R
xx
=
E
[
x
re
x
im
]
+
j
E
[
x
im
x
im
]
.
T
T
T
T
re
]+
E
[
x
im
x
re
]
−
E
[
x
re
x
(6
.
33)
Clearly the real part
P
is symmetric and the imaginary part
Q
is antisymmetric.
Note that the individual terms
re
]
,E
[
x
im
x
im
]
,E
[
x
re
x
im
]
E
[
x
re
x
(6
.
34)
cannot be obtained from a knowledge of the autocorrelation, so
E
[
xx
†
]onlyhas
limited information about correlations. Suppose the real and imaginary parts
are related such that
re
]=
E
[
x
im
x
im
]
E
[
x
re
x
(6
.
35)
and
im
]=
re
]
,
E
[
x
re
x
−
E
[
x
im
x
(6
.
36)
so that
re
]+2
jE
[
x
im
x
re
]
.
R
xx
=2
E
[
x
re
x
(6
.
37)
That is, the real and imaginary parts have identical autocorrelation, and fur-
thermore, the cross correlation between them is antisymmetric. We then say
that the complex random vector is
circularly symmetric
. Such complex random
vectors arise frequently in digital communication models. For such vectors,
R
xx
indeed contains all information about correlation components.
6.6.1 Properties of circularly symmetric random vectors
Equations (6.35) and (6.36) are the defining equations for circular symmetry of
a complex random vector. In what follows we will first show that this property
can be restated in a number of other ways.
1.
Symmetry in correlation.
Given the complex random vector
x
=
x
re
+
j
x
im
,
define the real vector
u
=
x
re
x
im
.
(6
.
38)
The autocorrelation of
u
is
⎡
⎤
re
]
T
E
[
x
re
x
E
[
x
re
x
im
]
⎣
⎦
R
uu
=
(6
.
39)
re
]
im
]
E
[
x
im
x
E
[
x
im
x
and contains all the individual information listed in Eq. (6.34). It is clear
that this reduces to the form
R
uu
=0
.
5
P
−
Q
QP
(6
.
40)
if and only if both Eq. (6.35) and (6.36) are true. So Eq. (6.40) is often
used as an equivalent definition for circular symmetry [Telatar, 1999].
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