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156
Baseband Pulse and Digital Signaling Chap. 3
60
50
(dB)
Q —
S
R out N
m-law companding
m = 255
40
30
Uniform quantization
(no companding)
20
10
–50
–40 –30 –20
Relative input level
20 log (x rms /V)
–10 –4.77
(dB)
0
Figure 3–10
Output SNR of 8-bit PCM systems with and without companding.
The voltage (or current) waveforms for digital signals can be expressed as an orthogonal
series with a finite number of terms N. That is, the waveform can be written as
N
w(t) = a w k w k (t), 0 6 t 6 T 0
k=1
(3–27)
where w k represents the digital data and wk(t), k = 1, 2, ..., N, are N orthogonal functions that
give the waveform its waveshape. (This will be illustrated by examples that follow in the sections
titled “Binary Waveforms” and “Multilevel Waveforms.”) N is the number of dimensions
required to describe the waveform. The term dimension arises from the geometric interpretation
as described in the next section on vector representation. The waveform w(t), as given by
Eq. (3–27) to represent a PCM word or any message of the M message digital source, is
assigned a unique set of digital data {w k }, k 1, 2, ..., N, to represent that message. For
example, for a binary source consisting of an ASCII computer keyboard, the letter X is
assigned the code word 0001101. In this case, w 1 0, w 2 0, w 3 0, w 4 1, w 5 1, w 6
0, w 7 1, and N = 7. This message (i.e., the letter X) is sent out over a time interval of T 0
seconds, and the voltage (or current waveform) representing the message, as described by