Digital Signal Processing Reference
In-Depth Information
Figure 7.8 Original spectral envelope and two quantized versions (6 bits).
As shown in [34]
A
(
z
,
n
) is guaranteed to be a minimum phase filter [and therefore the
synthesis filter
H
(
z
,
n
)
¼ s
(
n
)
/
A
(
z
,
n
) is guaranteed to be a stable filter] if the zeros of
P
(
z
,
n
)
¼
0 and
Q
(
z
,
n
)
¼
0 lie on the unit circle and if they are increasing monoto-
nously and alternating. In fact
P
(
z
,
n
) has a real zero at
z ¼
2
1 and
Q
(
z
,
n
)at
z ¼
1.
In Figure 7.8 two versions of an original spectral envelope (also depicted) quan-
tized with 6 bits are depicted. Representations using LPC coefficients as well as
LSF coefficients have been used. It is obvious that LSF coefficients are more robust
against quantization than LPC coefficients [35].
3
However, the benefit concerning
robustness gained by the usage of line spectral frequencies does not always justify
the increased computational complexity that results from the search for the zeroes.
7.4.3 Distance Measures
Distance measures play an important role in speech quality assessment, speech coding,
and—as we will see in the next sections—also in training and search procedures in
bandwidth extension schemes. Plenty of different distance measures for all kinds of
applications exist [36]. Since we are dealing with bandwidth extension and one
major task is the extension of the spectral envelope, we will focus here on distance
measures that are appropriate to evaluate distances between parametric representations
of spectral envelopes. Most of the spectral distance measures are
L
p
-norm-based
measures
2
3
1
p
ð
p
1
2
p
d
p
(
H
(
e
jV
,
n
),
H
(
e
jV
,
n
))
¼
4
jH
(
e
jV
,
n
)
H
(
e
jV
,
n
)
j
p
dV
5
:
(7
:
27)
p
3
The 6 bit quantization (see Fig. 7.8) leads in the LPC case to large overestimation errors around 3600 Hz
(fourth formand) as well as to large underestimation errors below 1300 Hz (first and second formand). The
quantized LSF envelope version exhibits a much smaller error.
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