Digital Signal Processing Reference
In-Depth Information
Lattice Methods
As shown in the previous sections, the solution to the LPC equation involves
two basic steps: (i) computation of a matrix of correlation values,
φ
n
(i, j)
,
and (ii) solution of a set of linear equations. Although the two steps are
already very efficient, another class of autocorrelation based methods, called
Lattice Methods (LM), have been developed which combine the two steps
tocomputetheLPCparameters.ThebasicideabehindtheLMisthat
knowledge of the forward and backward prediction errors are incorporated
during the calculation of the intermediate stages of the predictor parameters.
A major incentive for using the LM is that the computed parameters are
guaranteed to form a stable filter, a feature which neither the AM nor the
CM possess.
Consider the
i
th
stage of Durbin's algorithm where the set of coefficients
α
(i)
1
,
2
, ... ,i
are the optimum linear prediction coefficients of an
i
th
order
filter. The inverse filter
A(z)
based on these
i
optimum coefficients will be,
,
j
=
j
i
α
(i)
j
A
(i)
(z)
=
z
−
j
1
−
(4.40)
j
=
1
and the prediction error
e
(i
n
(m)
(or for simplicity
e
(i)
(m))
will be,
i
α
(i)
e
(i)
(m)
=
s(m)
−
s(m
−
j)
(4.41)
j
j
=
1
In a z-transform notation, the above equation becomes,
E
(i)
(z)
A
(i)
(z)S(z)
=
(4.42)
By combining (4.27) and (4.40) we have
i
−
1
[
α
(i
−
1
)
j
k
i
α
(i
−
1
)
−
α
(i)
i
A
(i)
(z)
=
]
z
−
j
z
−
i
1
−
−
(4.43)
−
i
j
=
j
1
but
α
(i)
=
k
i
and hence,
i
i
−
1
i
−
1
α
(i
−
1
)
j
α
(i
−
1
)
i
A
(i)
(z)
=
z
−
j
z
−
j
k
i
z
−
i
1
−
+
k
i
−
(4.44)
−
j
=
=
j
1
j
1
z
−
i
z
−
j
i
−
1
α
(i
−
1
)
A
(i
−
1
)
(z)
A
(i)
(z)
=
−
k
i
−
(4.45)
i
−
j
j
=
1
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