Digital Signal Processing Reference
In-Depth Information
Similarly,
B
p
−
1
Q
(z)
B
0
z
p
B
1
z
1
=
+
+
...
+
+
B
0
(5.25)
1
z
p/
2
[
B
0
(z
p/
2
z
−
p/
2
)
B
1
(z
(p/
2
−
1
)
z
−
(p/
2
−
1
)
)
=
+
+
+
+
+
...
B
p/
2
]
As all roots are on the unit circle, we can evaluate equation (5.24) on the unit
circle only.
e
jω
then
z
1
z
−
1
Let
z
=
+
=
2cos
(ω)
(5.26)
2
e
jpω/
2
A
0
cos
p
A
1
cos
p
ω
2
A
p/
2
2
ω
−
2
1
P
(z)
=
+
+
...
+
(5.27)
2
2
e
jpω/
2
B
0
cos
p
B
1
cos
p
ω
2
B
p/
2
2
ω
−
2
1
Q
(z)
=
+
+
...
+
(5.28)
2
=
By making the substitution
x
cos
(ω)
, equations (5.27) and (5.28) can be
=
solved for
x
. For example, with
p
10, the following is obtained:
P
10
(x)
16
A
0
x
5
8
A
1
x
4
20
A
0
)x
3
8
A
1
)x
2
=
+
+
(
4
A
2
−
+
(
2
A
3
−
+
(
5
A
0
−
3
A
2
+
A
4
)x
+
(A
1
−
A
3
+
0
.
5
A
5
)
(5.29)
and similarly,
Q
10
(x)
16
B
0
x
5
8
B
1
x
4
20
B
0
)x
3
8
B
1
)x
2
=
+
+
(
4
B
2
−
+
(
2
B
3
−
+
(
5
B
0
−
3
B
2
+
B
4
)x
+
(B
1
−
B
3
+
0
.
5
B
5
)
(5.30)
The LSFs are then given by:
cos
−
1
(x
i
)
2
πT
LSF(i)
=
,
for 1
≤
i
≤
p
(5.31)
The distribution plots of LSFs for a 10
th
orderLPCfilterareshownin
Figure 5.5 and a typical LSF plot is shown in Figure 5.6, where the first half
is active speech and the second half is silence. Notice that during silent
regions the frequencies are evenly spread between 0 and
f
s
/
2where
f
s
is the
sampling frequency. This method is obviously considerably simpler than the
complex root method, but it still suffers from indeterministic computation
time. However, a faster root search can be accomplished by noting that the
change from one LSF vector to the next is not too drastic in most cases. Thus
by using the previous values as the starting estimates of the roots, the number
of iterations required per root is considerably reduced, e.g. typically from 5
to 10 iterations.
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