Digital Signal Processing Reference
In-Depth Information
known time intervals enabling a unit variance nonuniform quantizer to be
used. These quantizers are called forward adaptive nonuniform quantizers.
3.3.4 LogarithmicQuantizer
As was discussed above, an optimum quantizer is advantageous if the
dynamic range (or variance) of the input signal is fixed to a small known
range. However, the performance of such a quantizer deteriorates rapidly
as the power of the signal moves away from the value that the quantizer
is designed for. Although, this can be controlled by normalizing the input
signal to unit variance, this process requires the transmission of the signal
variance at known time intervals for correct scaling of the de-quantized signal
amplitudes.
In order to cater for the wide dynamic range of the input speech signal,
Cattermole [2] suggested two companding laws called A-Law and
µ
-Law
Pulse Code Modulation (PCM). In both schemes, the signal to quantization
noise performance can be very close to that of a uniform quantizer, but their
performances do not change significantly with changing signal variance and
remain relatively constant over a wide range of input speech levels. When
compared with uniform quantizers, companded quantizers require fewer
bits per input sample for a specified signal dynamic range and signal to
quantization noise ratio. In a companding quantizer, quantizer levels are
closely spaced for small amplitudes which progressively increase as the
input signal range increases. This ensures that, when quantizing speech
signals where the probability density function is zero mean and maximum
at the origin, the frequently occurring small amplitudes are more accurately
quantized than the less frequent large amplitudes, achieving a significantly
better performance than a uniform quantizer.
The A-Law compression is defined by:
Ax
1
A
A
Law
(x)
=
for
0
≤
x
≤
(3.22)
1
+
log
10
(A)
1
+
log
10
(Ax)
for
1
A
Law
(x)
=
A
≤
x
≤
1
(3.23)
1
+
log
10
(A)
where
A
is the compression parameter with typical values of 86 for 7 bit (North
American) PCM and 87.56 for 8 bit (European) PCM speech quantizers.
The
µ
-Law compression on the other hand is defined by:
V
o
log
10
1
µ
|
x
|
+
V
o
=
µ
Law
(x)
sign(x)
(3.24)
log
10
[1
+
µ
]
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