Digital Signal Processing Reference
In-Depth Information
where
L
1. Then, according to the data model in (2.1), the
l
th data
snapshot
y
l
canbewritten as
=
N
−
M
+
e
j
ω
l
y
l
=
α
(
ω
)
a
(
ω
)
·
+
e
l
(
ω
)
,
(4.3)
where
a
(
ω
)isan
M
×
1vector given by (2.5) and
e
l
(
ω
)
=
[
e
l
(
ω
)
e
l
+
1
(
ω
)
···
)]
T
. The APES algorithm mimics a ML approach to estimate
e
l
+
M
−
1
(
ω
α
(
ω
)by
assuming that
e
l
(
1, are zero-mean circularly symmetric com-
plex Gaussian random vectors that are statistically independent of each other and
have the same unknown covariance matrix
ω
)
,
l
=
0
,
1
,...,
L
−
E
e
l
(
)
.
)
e
l
Q
(
ω
)
=
ω
(
ω
(4.4)
Then the covariance matrix of
y
l
canbewritten as
=
α
)
2
)
a
H
(
R
(
ω
a
(
ω
ω
)
+
Q
(
ω
)
.
(4.5)
L
−
1
Since the vectors
0
in our case are overlapping, they are not statistically
independent of each other. Consequently, APES is not an exact ML estimator.
Using the above assumptions, we get the normalized surrogate log-likelihood
function of the data snapshots
{
e
l
(
ω
)
}
l
=
{
y
l
}
as follows:
−
}
α
ln
Q
(
)
−
L
1
y
l
)
e
j
ω
l
H
1
L
ln
p
(
1
L
{
y
l
(
ω
)
,
Q
(
ω
))
=−
M
ln
π
−
ω
−
α
(
ω
)
a
(
ω
l
=
0
Q
−
1
(
)
e
j
ω
l
]
×
ω
)[
y
l
−
α
(
ω
)
a
(
ω
(4.6)
tr
Q
−
1
(
ln
Q
(
)
−
L
−
1
1
L
=−
π
−
ω
ω
M
ln
)
l
=
0
H
,
y
l
)
e
j
ω
l
y
l
)
e
j
ω
l
−
α
(
ω
)
a
(
ω
−
α
(
ω
)
a
(
ω
(4.7)
where tr
{
·
}
and
|·|
denote the trace and the determinant of a matrix, respectively.
For any given
α
(
ω
)
,
maximizing (4.7) with respect to
Q
(
ω
) gives
−
L
1
1
L
Q
)
e
j
ω
l
][
y
l
)
e
j
ω
l
]
H
(
ω
)
=
[
y
l
−
α
(
ω
)
a
(
ω
−
α
(
ω
)
a
(
ω
.
(4.8)
α
l
=
0
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