Autocorrelation in terms of model parameters.
If we normalize the autocorrelations
in (4.2.31) by dividing throughout by
r(
0
)
, we obtain the following system of equations
Pa
= −
ρ
(4.2.33)
where
P
is the normalized autocorrelation matrix and
ρ
= [
ρ(
1
) ρ(
2
)
· · ·
ρ(P )
]
H
(4.2.34)
is the vector of normalized autocorrelations. This set of
P
equations relates the
P
model
coef
fi
cients with the
fi
rst
P
(normalized) autocorrelation values. If the poles of the all-pole
fi
lter are strictly inside the unit circle, the mapping between the
P
-dimensional vectors
a
and
ρ
is
unique
. If, in fact, we are given the vector
a
, then the normalized autocorrelation vector
ρ
can be computed from
a
by using the set of equations that can be deduced from (4.2.33)
A
ρ
= −
a
(4.2.35)
where
A
ij
=
a
i
−
j
+
a
i
+
j
, assuming
a
m
=
0 for
m <
0 and
m > P
(see Problem 4.6).
Given the set of coef
fi
cients in
a
,
ρ
can be obtained by solving (4.2.35). We will see that,
under the assumption of a stable
H(z)
, a solution always exists. Furthermore, there exists a
simple, recursive solution that is ef
fi
cient (see Section 7.5). If, in addition to
a
, we are given
d
0
, we can evaluate
r(
0
)
with (4.2.20) from
ρ
computed by (4.2.35).Autocorrelation values
r(l)
for lags
l > P
are found by using the recursion in (4.2.18) with
r(
0
), r(
1
), . . . , r(P )
.
EXAMPLE 4.2.3.
For the AP(3) model with real coef
fi
cients we have
r(
0
)
r(
1
)
r(
2
)
r(
1
)
r(
0
)
r(
1
)
r(
2
)
r(
1
)
r(
0
)
a
1
a
2
a
3
= −
r(
1
)
r(
2
)
r(
3
)
(4.2.36)
d
2
0
=
r(
0
)
+
a
1
r(
1
)
+
a
2
r(
2
)
+
a
3
r(
3
)
(4.2.37)
Therefore, given
r(
0
)
,
r(
1
), r(
2
), r(
3
)
, we can
fi
nd the parameters of the all-pole model by
solving (4.2.36) and then substituting into (4.2.37).
Suppose now that instead we are given the model parameters
d
0
, a
1
, a
2
, a
3
. If we divide
both sides of (4.2.36) by
r(
0
)
and solve for the normalized autocorrelations
ρ(
1
), ρ(
2
),
and
ρ(
3
)
,
we obtain
1
+
a
2
a
3
0
a
1
+
a
3
1
0
a
2
a
1
1
ρ(
1
)
ρ(
2
)
ρ(
3
)
= −
a
1
a
2
a
3
(4.2.38)
The value of
r(
0
)
is obtained from
r(
0
)
=
d
2
0
1
+
a
1
ρ(
1
)
+
a
2
ρ(
2
)
+
a
3
ρ(
3
)
(4.2.39)
If
r(
0
)
=
2
, r(
1
)
=
1
.
6
, r(
2
)
=
1
.
2
,
and
r(
3
)
=
1, the Toeplitz matrix in (4.2.36) is positive
de
fi
nite because it has positive eigenvalues. Solving the linear system gives
a
1
= −
0
.
9063
,
a
2
=
0
.
2500
,
and
a
3
= −
0
.
1563. Substituting these values in (4.2.37), we obtain
d
0
=
0
.
8329.
Using the last two relations, we can recover the autocorrelation from the model parameters.
Correlation matching.
All-pole models have the unique distinction that the model
parameters are completely speci
fi
ed by the
fi
rst
P
+
1 autocorrelation coef
fi
cients via a set
of linear equations. We can write
d
0
a
↔
r(
0
)
ρ
(4.2.40)
Manolakis, Dimitris, et al. Statistical and Adaptive Signal Processing : Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing, Artech House, 2005.
ProQuest Ebook Central, .