Digital Signal Processing Reference
In-Depth Information
The property (2.2) can be shown to hold if, and only if,
f
(
z
)
¼ f
(
x
,
y
) is a function of
jzj
2
) for some nonnegative function
g
(
) and normal-
izing constant
c
. Hence the regions of constant contours are circles in the complex
plane, thus justifying the name for this class of distributions. rva
z
is said to be
sym-
metric
,ortohavea
symmetric distribution
,if
z ¼
d
z
. Naturally, circular symmetry
implies symmetry.
Characteristics of a complex rva
z
can also be described via its moments, for
example, via its second-order moments. The variance
s
2
2
¼ x
2
þy
2
, that is,
f
(
z
)
¼ cg
(
jzj
¼ s
2
(
z
)
.
0of
z
is defined as
s
2
(
z
)
W
E
[
jzj
2
]
¼ E
[
x
2
]
þE
[
y
2
]
:
Note that variance does not bear any information about the correlation between the real
and the imaginary part of
z
, but this information can be retrieved from pseudo-variance
t
(
z
)
[ C
of
z
, defined as
t
(
z
)
W
E
[
z
2
]
¼ E
[
x
2
]
E
[
y
2
]
þ
2
jE
[
xy
]
:
Note that
E
[
xy
]
¼
Im[
t
(
z
)]
=
2. The complex covariance between complex rvas
z
and
w
is defined as
cov(
z
,
w
)
W
E
[
zw
]
:
Thus,
s
2
(
z
)
¼
cov(
z
,
z
) and
t
(
z
)
¼
cov(
z
,
z
). If
z
is circular, then
t
(
z
)
¼
0. Hence a
rva
z
with
t
(
z
)
¼
0 is called second order circular. Naturally if
z
or
w
are (or both
z
and
w
are) circular and
z
=
w
, then cov(
z
,
w
)
¼
0 as well.
Circularity quotient [41]
@¼ @
(
z
)
[ C
of a rva
z
(with finite variance) is defined as
the quotient between the pseudo-variance and the variance
cov(
z
,
z
)
s
2
(
z
)
s
2
(
z
)
t
(
z
)
s
2
(
z
)
:
@
(
z
)
W
p
¼
Thus we can describe
@
(
z
) as a measure of correlation between rva
z
and its conjugate
z
. The modulus
l
(
z
)
W
j@
(
z
)
j
[
[0, 1]
is referred to as the circularity coefficient [22, 41] of
z
. If the rva
z
is circular, then
t
(
z
)
¼
0, and consequently
l
(
z
)
¼
0. Circularity coefficient measures the “amount
of circularity” of zero mean rva
z ¼ x þ jy
in that
0,
if
x
and
y
are uncorrelated with equal varinaces
l
(
z
)
¼
(2
:
3)
1,
if
x
or
y
is zero, or
x
is a linear function of
y:
Note that
l
(
z
)
¼
1if
z
is purely real-valued such as a BPSKmodulated communication
signal, or, if the signal lie on a line in the scatter plot (also called constellation or I
/
Q
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