Lpc - LPC.PPT 5.1 LPC.PPT 5.2 Deller 266 Lecture 5 Linear...

Page 5.1
Linear Prediction
E.4.14 – Speech Processing
LPC.PPT(4/15/2002)
5.1
Lecture 5
Linear Prediction (LPC)
•
Aims of Linear Prediction
•
Derivation of Linear Prediction Equations
•
Autocorrelation method of LPC
•
Interpretation of LPC filter as a spectral whitener
LPC.PPT(4/15/2002)
5.2
Deller: 266++
Notes: •
We will neglect the pure delay term
z
–½
p
in the numerator of
V(z)
.
•
50% of the world puts a + sign in the denominator of
V(z)
(this is
almost essential when using MATLAB).
V(z)
R(z)
u (n)
u
l
(n)
s(n)
1
½
1
½
1
)
(
)
(
1
)
(
)
(
)
(
)
(
−
−
=
−
−
−
=
=
−
= = = =
∑
z
z
R
z
A
Gz
z
a
Gz
z
V
n
s
n
u
n
u
p
p
j
j
j
p
l
microphone
the
at
pressure
lips
the
at
flow
volume
two
the
of
mixture
or
noise
waveform,
glottal
a time-varying all-pole filter
Linear Prediction: Analysis & Coding (LPC)
The aim of Linear Predication Analysis (LPC) is to estimate
V(z)
from the speech signal
s(n)
.

Page 5.2
Linear Prediction
E.4.14 – Speech Processing
LPC.PPT(4/15/2002)
5.3
Deller: 266++
If the vocal tract resonances have high gain, the second term will dominate:
V(z)
R(z)
u(n)
u
l
(n)
s(n)
∑
=
−
+
′
=
p
j
j
j
n
s
a
n
u
G
n
s
1
)
(
)
(
)
(
s n
a s n
j
j
j
p
( )
(
)
≈
−
=
∑
1
Prediction Error
We can reverse the order of
V
(
z
)
and
R
(
z
)
since both are
linear and
V
(
z
)
doesn’t change substantially during the
impulse response of
R
(
z
)
or vice-versa:
The right hand side of this expression is a
prediction
of s(n)
as a
linear sum
of past speech samples. Define the
prediction error
at sample
n
as
e n
s n
a s n
j
s n
a s n
a s n
a s n
p
E z
S z A z
j
j
p
p
( )
( )
(
)
( )
(
)
(
)
(
)
( )
( )
( )
=
−
−
=
−
−
−
−
−
−
−
−
=
=
∑
1
1
2
1
2
…
or in terms of z
transforms: R(z)
V(z)/G=
1
/A(z)
× G
u(n)
u'(n)
s(n)
LPC.PPT(4/15/2002)
5.4
Given a frame of speech
{
F
}
, we would like to find the
values
a
i
that minimize:
e n s n
i
i
p
s n s n
i
a s n
j s n
i
i
p
a
s n
j s n
i
s n s n
i
a
where
s n
i s n
j
n
F
j
j
p
n
F
j
n
F
j
p
n
F
ij
j
i
ij
n
F
( ) (
)
,
,
( ) (
)
(
) (
)
,
,
(
) (
)
( ) (
)
(
) (
)
{
}
{
}
{
}
{
}
{
}
−
=
=
⇒
−
−
−
−
  
  
=
=
⇒
−
−
=
−
⇒
=
=
−
−
∈
=
∈
∈
=
∈
∈
∑
∑
∑
∑
∑
∑
∑
0
1
0
1
1
1
0
for
for
…
…
φ
φ
φ
j
p =
∑
1
∑ ∈
=
}
{
2
)
(
F
n
E
n
e
Q
(
)
∑
∑
∑
∈
∈
∈
−
−
=
∂
∂
=
∂
∂
=
∂
∂
}
{
}
{
}
{
2
)
(
)
(
2
)
(
)
(
2
)
(
F
n
F
n
i
F
n
i
i E
i
n
s
n
e
a
n
e
n
e
a
n
e
a
Q
The optimum values of
a
i
must satisfy
p
equations:
To do so, we differentiate w.r.t each
a
i
:
or in matrix form:
Φ
Φ
Φ
a
c
a
c
=
⇒
=
−
−
1
1
providing
exists
the matrix
Φ
is symmetric and positive semi-definite.

Page 5.3
Linear Prediction
E.4.14 – Speech Processing
LPC.PPT(4/15/2002)
5.5
Matrices with Special Properties
– Symmetric: –
Positive Definite:
–
Positive Semi-Definite: as above but with
≥
.
–
Toeplitz: Constant diagonals:
Inverting Matrices
Any special properties possessed by a matrix can be used when inverting it to:
–
reduce the computation time
–
improve the accuracy
φ
φ
ji
ij
T
=
⇔
=
Φ
Φ
x
x
i
ij
j
i j
T
φ
,
∑
>
⇔
>
≠
0
0
0
x
x
x
Φ
for any
(
)
φ
φ
i
j
ij
f i
j
+
+
=
=
−
1
1
,
Matrix
(
p×p
)
Computation
General
∝
p
3
Symmetric, +ve definite
∝
½
p
3
Toeplitz, Symmetric,
+ve definite
∝
p
2
LPC.PPT(4/15/2002)
5.6
Autocorrelation LPC
We start with a frame of windowed speech (typ 20-30 ms): We take {
F