Digital Signal Processing Reference
In-Depth Information
A popular lattice implementation of LPC analysis is that developed by
Burg [3]. Burg derived the
k
i
parameters by minimizing the sum of the mean
squared forward and backward prediction errors, i.e.
(e
(i)
(m))
2
(b
(i)
(m))
2
N
−
1
E
(i)
=
+
(4.51)
m
=
0
E
(i)
is differentiated with respect to
k
i
and then is set to zero to give,
N
−
1
e
(i
−
1
)
(m)b
(i
−
1
)
(m
2
−
1
)
m
=
0
k
i
=
(4.52)
N
−
1
N
−
1
[
e
(i
−
1
)
(m)
]
2
[
b
(i
−
1
)
(m
1
)
]
2
+
−
m
=
0
m
=
0
It can also be shown [8] that the above equation results in
k
i
parameters,
−
1
≤
k
i
≤
1. Burg's algorithm operates as follows [3]:
1. Set
e
(
0
)
(m)
b
(
0
)
(m)
=
s(m)
=
α
(
1
)
2. Compute
k
1
=
1
3. Determine
e
(
1
)
(m)
and
b
(
1
)
(m)
using (4.47) and (4.48)
4. Set
i
=
2
α
(i)
=
5. Find
k
i
using (4.52)
i
6. Find
α
(i)
j
1 using (4.27)
7. find
e
(i)
(m)
and
b
(i)
(m)
using (4.47) and (4.48)
8. Set
i
for
j
=
1
,
2
, ... ,i
−
=
i
+
1
9. If
i
≤
p
go to step 5
10. End
4.3.3 Practical Implementationof theLPCAnalysis
In the practical implementation of the LPC analysis, several important groups
of factors need to be addressed. The first group comprises the performance,
efficiency and stability factors, which are not too dissimilar for all three
methods, although the LM is preferred in real-time systems where guaranteed
stability is very important. However, with careful choice of windowing and
fine precision arithmetic, the AM and CM are equivalent to the LM for
stability. As quantization is usually applied to the coefficients, stability can
always be maintained to some extent. The second group involves the choice of
the filter order,
p
, and the analysis frame size,
N
. Speech is usually sampled at
8 kHz, thus giving a 4 kHz spectrum for analysis. Within the 4 kHz spectrum,
the maximum number of formants displayed is usually four, thus indicating
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