Digital Signal Processing Reference
In-Depth Information
Using (4.42) and (4.45),
z
−
j
i
−
1
α
(i
−
1
)
E
(i)
(z)
=
A
(i
−
1
)
(z)S(z)
−
z
−
i
S(z)
k
i
−
(4.46)
i
−
j
=
j
1
1
)
th
The first term in equation (4.46) represents the prediction error for an
(i
−
order predictor employing the
s(m
−
1
),s(m
−
2
), ... ,s(m
−
i
+
1
)
samples to
predict
s(m)
. Since the output of the filter
i
−
1
1
α
(i
−
1
)
j
z
−
j
operating on
S(z)
is
α
i
−
1
s
m
−
1
+
α
i
−
2
s
m
−
2
+
...
+
α
1
s
m
−
i
+
1
the second term of equation (4.46)
represents the backward prediction error of the same predictor attempting to
predict
s(m
j
=
i
−
−
i)
from the
i
samples
s(m
−
i
+
j),j
=
1
,
2
,
3
, ... ,i
that follow
i)
. The prediction error sequence
e
(i
m
can therefore be expressed in terms
of the forward and backward error sequences as,
s(m
−
e
(i)
(m)
e
(i
−
1
)
(m)
k
i
b
(i
−
1
)
(m
=
−
−
1
)
(4.47)
It can also be shown that the
i
th
stage backward prediction error
b
(i)
(m)
can
be expressed as,
b
(i)
(m)
b
(i
−
1
)
(m
k
i
e
(i
−
1
)
(m)
=
−
1
)
−
(4.48)
Equations (4.47) and (4.48) provide the forward and backward prediction
error sequences for an
i
th
order filter, in terms of the corresponding errors of
a
(i
1
)
th
order filter. Note that
−
e
(
0
)
(m)
n
(
0
)
(m)
=
=
s(m)
(4.49)
i.e, the zero order filter error equals the original input. Furthermore, the
k
i
parameters can be directly computed from the forward and backward
prediction errors as [8],
N
−
1
e
(i
−
1
)
(m)b
(i
−
1
)
(m
−
1
)
m
=
0
k
i
=
(4.50)
N
−
1
N
−
1
[
e
(i
−
1
)
(m)
]
2
[
b
(i
−
1
)
(m
1
)
]
2
×
−
m
=
0
m
=
0
without using the prediction coefficients
α
j
. From the above expression, it
is clear that the
k
i
parameters represent the normalized cross-correlation
function between the forward and backward error sequences. It is for this
reason the
k
i
parameters are known as the partial correlation (PARCOR)
coefficients [8].
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