This preview shows pages 1–3. Sign up to view the full content.
9.3 / Nonuniform Quantization
230
TABLE 9.2.1
Optimum Step Sizes for Uniform
Quantization of a Gaussian PDF
Number of
Step Size
Minimum
Mean
Levels
(L)
(AopJ
Squared
Error
(D)
SQNR(dB)
4
0.9957
0.1188
9.25
8
0.5860
0.03744
14.27
16
0.3352
0.01154
19.38
Source:
J. Max, "Quantizing for Minirilum Distortion,"
IRE Trans.In/.
Theory,
©
1960 IEEE.
9.3
Nonuniform Quantization
The most important nonuniform quantization method
to
date has been the
logarithmic quantization used in the telephone network for speech digitization
for over
20
years. The general idea behind this type
of
quantization is
that
for
a fixed, uniform quantizer,
an
input signal with
an
amplitude less than
full
load
will have a lower
SQNR
than
a signal whose amplitude occupies the
full
dy
namic range of the quantizer (but without overload). This fact is illustrated by
Example
9.2.2.
Such a variation in performance (SQNR) as a function
of
quan.
.
tizer input signal amplitude is particularly detrimental for speech, since low
amplitude signals can be very important perceptually. There
is
the additional
consideration for speech signals that low amplitudes are more probable
than
larger amplitudes, since speech
is
generally stated to have a gamma
or
Laplacian
probability density, which is highly peaked about zero.
Therefore, for speech signals a type of nonlinear quantization was invented
called
logarithmic companding.
Initially, this scheme was implemented by pass
ing the analog speech signal through a characteristic of the form
In
[1
+
IllslJ
( )
F
( )
=
 1 ::;;
s
::;;
1,
(9.3.1)
II
S
In
[1
+
IlJ
sgn
s,
where
s
is
the normalized speech signal
and
11
is
a parameter, usually selected
to be
11
=
100,
or
more recently,
11
=
255.
The function
Fis)
is
shown in Fig.
9.3.1.
Notice that
F
is)
tends to amplify small amplitudes more than larger
amplitudes whenever
11
>
O.
The output of
F
is)
then served as input to a uni
form, nbit quantizer. To resynthesize the speech signal, the quantizer
output
s
was passed through the inverse function
of
Eq.
(9.3.1)
given by
(9.3.2)
where, of course,  1
<
s
~
1.
The performance in
SQNR
of this system for
Il
=
100 and
n
=
7 bits is shown
in Fig.
9.3.2.
It
is evident from this figure that the
SQNR is
relatively flat over
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document240
Pulse Code Modulation / 9
1.0
o
o
Input signal
1.0
FIGURE 9.3.1
Logarithmic compression characteristics.
a wide dynamic range of input signal power (amplitudes), and hence low
amplitude signals are reproduced almost as well as higheramplitude signals.
In particular, we
see
from Eqs. (9.2.8) and (9.2.9) that for linear quantization, if
we
decrease the input signal power
by 12
dB
(from a peak value of
V
/2
to
V /8),
the output SQNR decreases by 12
dB.
However, as we move along the "sine
wave" curve in Fig. 9.3.2 from 0
to
12
dB
on the input power axis, the
SQNR
decreases only
by
about 2
dB.
The companding clearly improves the
SQNR
for lowamplitude signals.
40
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 Staff

Click to edit the document details