Digital Signal Processing Reference
In-Depth Information
The variance of the quantization error is given by a simplified expression
below:
2
∆
σ
2
=
(4.6)
12
Practical quantizers
are used for high-quality music work at
SNR
values
around 90 dB.
Nonuniform quantizers,
such as
µ
-law and
A
-law quantizers,
are widely used around the world to improve the
SNR
value. A detailed
analysis of nonuniform quantizers is given below.
Nonuniform Quantizer
The most important nonuniform quantization technique is logarithmic quan-
tization (
in Europe, Africa,
Asia, South America, and Australia), which has been used very successfully
for speech digitization. This technique evolved from the fundamental prop-
erty of speech, which has a gamma or Laplacian probability density in
amplitude, highly peaked about zero value. Hence, even though low ampli-
tudes of speech are more probable than large amplitudes, a uniform quan-
tizer amplifies all signals equally.
The principle behind nonuniform quantization is to pre-process (com-
press) the sampled signal before it enters the uniform quantizer, such that
the processed signal occupies the full dynamic range of the quantizer. How-
ever, the output of the uniform quantizer has to be post-processed
(expanded) to extract the true quantized signal. The dual process is called
logarithmic companding
µ
-law
in the U.S., Japan, and Canada and
A-law
.
The nonuniform quantization process is explained below in a series of four
steps, which is also illustrated in Figure 4.4.
, which is a combination of
compression
and
expanding
Sampled
Signal
x(n)
Normalizer
Compressor
F
µ
(s)
Uniform
quantizer
Final quantized
Signal
ˆ
x(n)
De-
normalizer
Expander
F
µ
−1
(
ˆ
FIGURE 4.4
Nonuniform
(
µ
-law)quantizationprocess.
