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Pulse-Code Modulation 609
called a µ-law compressor, has an input-output magnitude characteristic
of the form
ln (1 + µ|s|)
y = sgn(s); |s| ≤1, |y| ≤1 (12.5)
ln(1 + µ)
where s is the normalized input, y is the normalized output, sgn (·) isthe
sign function, and µ is a parameter that is selected to give the desired
compression characteristic.
In the encoding of speech waveforms the value of µ = 255 has been
adopted as a standard in the U.S. and Canada. This value results in about
a24dBreduction in the quantization noise power relative to uniform
quantization. Consequently, an 8-bit quantizer used in conjunction with
a µ = 255 logarithmic compressor produces the same quality speech as a
12-bit uniform quantizer with no compression. Thus the compressed PCM
speech signal has a bit rate of 64,000 bps.
The logarithmic compressor standard used in European telecommunication
systems is called A-law and is defined as
⎧
1+ln(A|s|)
⎪⎨
1+lnA sgn(s), 1
A
≤|s| ≤1
y =
(12.6)
⎪⎩
A|s|
1+lnA sgn(s), 0 ≤|s| ≤ 1 A
where A is chosen as 87.56. Although (12.5) and (12.6) are different
nonlinear functions, the two compression characteristics are very similar.
Figure 12.1 illustrates these two compression functions. Note their strong
similarity.
FIGURE 12.1
Comparison of µ-law and A-law nonlinearities
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