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Sec. 3–4 Digital Signaling 155
where |w 1 (t)| … 1 and A is a positive constant. The A-law compression characteristic is shown
in Fig. 3–9c. The typical value for A is 87.6.
In practice, the A-law characteristic is implemented by using a segmenting technique
similar to that shown in Fig. 3–9d, except that, for Segment 1, there are 16 steps of width ¢ ; for
segment 2, 16 steps of width ¢ ; for Segment 3, 16 steps of width 2 ¢ ; for Segment 4, 16 steps of
width 4 ¢ ; etc.
When compression is used at the transmitter, expansion (i.e., decompression) must be
used at the receiver output to restore signal levels to their correct relative values. The
expandor characteristic is the inverse of the compression characteristic, and the combination
of a compressor and an expandor is called a compandor.
Once again, it can be shown that the output SNR follows the 6-dB law [Couch, 1993]
a S N b dB
= 6.02n + a
(3–25)
where
a = 4.77 - 20 log (V>x rms ) (uniform quantizing)
(3–26a)
or for sufficiently large input levels,
a L 4.77 - 20 log [ ln (1 + m)] (m-law companding)
(3–26b)
or [Jayant and Noll, 1984]
a L 4.77 - 20 log [1 + ln A] (A-law companding)
(3–26c)
and n is the number of bits used in the PCM word. Also, V is the peak design level of the
quantizer, and x rms is the rms value of the input analog signal. Notice that the output SNR
is a function of the input level for the case of uniform quantizing (no companding), but
is relatively insensitive to the input level for µ-law and A-law companding, as shown in
Fig. 3–10. The ratio Vx rms is called the loading factor. The input level is often set for a
loading factor of 4, which is 12 dB, to ensure that the overload quantizing noise will be
negligible. In practice, this gives a -7.3 for the case of uniform encoding, as compared
to a 0 that was obtained for the ideal conditions associated with Eq. (3–17b). All of
these results give a 6-dB increase in the signal-to-quantizing noise ratio for each bit added
to the PCM code word.
3–4 DIGITAL SIGNALING
In this section, we will answer the following questions: How do we mathematically represent
the waveform for a digital signal, such as the PCM signal of Figure 3–8d, and how do we
estimate the bandwidth of the waveform?