1
1
Digital Speech Processing—
Lecture 14
Linear Predictive
Coding (LPC)Lattice
Methods, Applications
2
Prediction Error Signal
H(z)
Gu(n)
s(n)
1
1
1
1
1
1
1
1
1
1. Speech Production Model
( )
(
)
( )
( )
( )
( )
2. LPC Model:
( )
( )
( )
( )
(
)
( )
( )
( )
3. LPC Error Model:
( )
( )
( )
α
α
α
=
−
=
=
−
=
=
−
+
=
=
−
=
−
=
−
−
=
=
−
=
=
−
∑
∑
∑
∑
%
p
k
k
p
k
k
k
p
k
k
p
k
k
k
k
s n
a s n
k
Gu n
S z
G
H z
U z
a z
e n
s n
s n
s n
s n
k
E z
A z
z
S z
S z
A z
E z
1
1
( )
( )
(
)
α
−
=
=
=
+
−
∑
∑
p
k
k
p
k
k
z
s n
e n
s n
k
A(z)
s(n)
e(n)
1/A(z)
e(n)
s(n)
Perfect reconstruction even if
a
k
not equal to
α
k
3
Lattice Formulations of LP
both covariance and autocorrelation methods use
two step solutions
1. computation of a matrix of correlation values
2. efficient solution of a set of linear equations
another class of LP
•
•
methods, called lattice methods,
has evolved in which the two steps are combined into
a recursive algorithm for determining LP parameters
begin with Durbin algorithmat the
stage the set of
•
th
i
1 2
( )
coefficients {
,
,
,...,
} are coefficients of the
order
optimum LP
α
=
i
th
j
j
i
i
4
1
1
( )
( )
ˆ
define the system function of the
order inverse filter
(prediction error filter) as
( )
if the input to this filter is the input segment
ˆ
(
)
(
)
(
α
−
=
•
=
−
•
=
+
∑
th
i
i
i
k
k
k
n
i
A
z
z
s
m
s n
m w m
1
1
( )
( )
ˆ
( )
( )
( )
( )
ˆ
), with output
(
)
(
)
(
)
(
)
(
)
ˆ
where we have dropped subscript
 the absolute
location of the signal
the ztransform gives
( )
( )
( )
α
α
=
=
+
=
−
−
•
•
=
=
−
∑
i
i
n
i
i
i
k
k
i
i
k
e
m
e
n
m
e
m
s m
s m
k
n
E
z
A
z S z
1
( )
( )
−
=
⎛
⎞
⎜
⎟
⎜
⎟
⎝
⎠
∑
i
i
k
k
z
S z
Lattice Formulations of LP
5
Lattice Formulations of LP
1
1
1
1
( )
(  )
(  )
( )

( )
( )
(
)
(
using the steps of the Durbin recursion
(
=
, and
)
we can obtain a recurrence formula for
( )
of the form
( )
( )
α
α
α
α
−
−
−
•
−
=
=
−
±
i
i
i
i
i
i
j
j
i j
i
i
i
i
i
i
i
k
k
A
z
A
z
A
z
k z
A
1
1
1
1
1
1
1
)
( )
(
)
(
)
( )
(
)
(
)
(
)
giving for the error transform the expression
( )
( ) ( )
(
) ( )
(
)
(
)
(
)
−
−
−
−
−
−
−
•
=
−
=
−
−
i
i
i
i
i
i
i
i
i
z
E
z
A
z S z
k z
A
z
S z
e
m
e
m
k b
m
6
Lattice Formulations of LP
1
where we can interpret the first term as the ztransform
of the forward prediction error for an (
)
order predictor,
and the second term can be similarly interpreted based on
defining a ba
•
−
st
i
1
1
1
1
1
1
1
1
1
( )
( )
(
)
(
)
(
)
(
)
( )
( )
(
)
ckward prediction error
( )
(
) ( )
(
) ( )
( ) ( )
( )
( )
with inverse transform
(
)
(
)
(
)
(
α
−
−
−
−
−
−
−
−
−
−
=
=
=
−
=
−
•
=
−
−
+
−
=
−
∑
i
i
i
i
i
i
i
i
i
i
i
i
i
i
k
k
B
z
z
A
z
S z
z
A
z
S z
k A
z S z
z
B
z
k E
z
b
m
s m
i
s m
k
i
b
m
1
1
(
)
( )
)
(
)
with the interpretation that we are attempting to predict
(
) from the
samples of the input that follow
(
) => we are doing a
prediction and
(
) is called the
−
−
•
−
−
i
i
i
k e
m
s m
i
i
s m
i
b
m
backward
backward prediction error sequence
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