Digital Signal Processing Reference
In-Depth Information
6.2.3 The Modified Riccati Equation (MRE)
P(k
Given
|
k
−
1
)
at time step
k
−
1, one step recursion of MRE algorithm
P(k
+
P(
1
produces
1
|
k)
at time step
k
. The recursion is initialized with
|
0
)
0
)F
T
(
0
)
+
G(
0
)Q(
0
)G
T
(
0
)
. The algorithm consists of two main parts,
whose derivation details can be found in [
20
]:
F(
0
)P(
0
|
(i)
Covariance Update:
S(k)
H P(k
1
)H
T
=
|
k
−
+
R(k),
(6.5)
W(k)
=
P(k
1
)H
T
S
−
1
(k),
|
k
−
(6.6)
c
n
z
g
n
z
S(k)
1
/
2
,
V(k)
=
(6.7)
Q
LU
2
λ V(k),P
D
,
q
2
(k)
=
(6.8)
P(k
|
k)
=
P(k
|
k
−
1
)
−
q
2
(k) W(k)S(k) W
T
(k)
P
MRE
(k
|
k),
(6.9)
(ii)
Covariance Prediction:
P(k
F(k)P(k
k)F
T
(k)
G(k)Q(k)G
T
(k).
+
1
|
k)
=
|
+
(6.10)
Here, the output of the algorithm
P(k
P
MRE
(k
|
k)
|
k)
is a deterministic approx-
k)
of the PDAF, i.e.,
P
MRE
(k
imation to the filter calculated covariance
P(k
|
|
k)
≈
Z
k
−
1
E
. At each time step
k
,thevalueof
q
2
(k)
can be obtained from a two
dimensional (2D) LUT,
Q
LUT
2
[
P(k
|
k)
|
]
)
via interpolation where
Q
LUT
2
·
·
·
·
)
is prepared
offline and only once from the
Information Reduction Factor
(IRF) given in (
6.11
):
(
,
(
,
n
z
g
n
z
m
k
−
1
∞
e
−
λ V(k)
(λ V(k))
m
k
−
1
(m
k
−
q
2
λ V(k),P
D
c
n
z
(
2
π)
n
z
/
2
P
D
1
)
!
m
k
=
1
×
I
2
λ V(k),P
D
,m
k
(6.11)
with
g
0
···
g
r
1
)r
1
b(λ V(k),P
D
)
+
m
k
I
2
λ V(k),P
D
,m
k
exp
(
−
1
exp
(
−
r
j
/
2
)
×
(r
1
r
2
···
r
m
k
)
n
z
−
1
dr
1
dr
2
···
dr
m
k
,
0
j
=
(6.12)
(
2
π)
n
z
/
2
λ V(k)
c
n
z
g
n
z
b
λ V(k),P
D
(
1
−
P
D
P
G
)
P
D
(6.13)
π
n
z
/
2
/Γ(n
z
/
2
where
c
n
z
)
being the gamma function, is the volume
of
the
n
z
-dimensional unit hypersphere (
c
1
=
+
1
)
, with
Γ(
·
√
γ
G
is referred to as “number of sigmas” (standard deviations) of the gate and
2
,c
2
=
π,c
3
=
4
π/
3, etc.), and
g
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