Digital Signal Processing Reference
In-Depth Information
For the initialization step, we obtain the initial APES estimates of
H
(
ω
,ω
2
)
1
α
ω
,ω
2
)from the available data
γ
in the following way. Let
S
and
(
be the set of
1
,
snapshot indices (
l
1
l
2
) such that the elements of the corresponding initial data
M
2
M
1
snapshot indices
{
(
l
1
,
l
2
)
,...,
(
l
1
,
l
2
+
−
1)
,...,
(
l
+
−
1
,
l
2
)
,...,
(
l
1
+
M
1
M
2
M
1
M
2
−
1
,
l
2
+
−
1)
}∈
G.
Define
the
set
of
×
1vectors
{
y
l
1
,
l
2
:
(
l
1
,
l
2
)
∈
S
}
, which contain only the available data samples, and let
|
S
|
be the
number of vectors in
S
.Furthermore, define the initial sample covariance matrix
1
|
S
|
ˆ
R
y
l
1
,
l
2
y
l
1
,
l
2
.
=
(3.33)
(
l
1
,
l
2
)
∈
S
M
0
2
must be chosen such that the
R
calculated in (3.33) has full rank. Similarly, the initial Fourier transform of the data
snapshots is given by
The size of the initial filter matrix
M
1
×
1
|
S
|
y
l
1
,
l
2
e
−
j
(
ω
1
l
1
+
ω
2
l
2
)
g
(
ω
,ω
2
)
=
.
(3.34)
1
(
l
1
,
l
2
)
∈
S
So the initial estimates of
H
(
ω
,ω
2
) and
α
(
ω
,ω
2
)can be calculated by (3.29)-
1
1
ˆ
R
and
g
(
(3.31) but by using the
2
) given above.
Next, we introduce some additional notation that will be used later for the step
of interpolating the missing samples. Let the
L
1
L
2
ω
,ω
1
×
−
+
(
L
2
N
1
M
1
1) matrix
T
be defined by
I
L
1
0
L
1
,
M
1
−
1
I
L
1
0
L
1
,
M
1
−
1
T
=
.
(3.35)
.
.
.
I
L
1
Hereafter,
0
K
1
,
K
2
denotes a
K
1
×
K
2
matrix of zeros only and
I
K
stands for the
K
×
K
identity matrix. Furthermore, let
G
be the following (
L
2
N
1
−
M
1
+
1)
×
N
1
N
2
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