563489578934

Sec. 3–3 Pulse Code Modulation 149
As shown in Chapter 7, under certain assumptions, the ratio of the recovered analog
peak signal power to the total average noise power is given by †
a S (3–16a)
N b 3M 2
=
pk out 1 + 4(M 2 - 1)P e
and the ratio of the average signal power to the average noise power is
a S (3–16b)
N b =
out 1 + 4(M 2 - 1)P e
where M is the number of quantized levels used in the PCM system and P e is the probability of
bit error in the recovered binary PCM signal at the receiver DAC before it is converted back into
an analog signal.
M 2
Example 3–3 AVERAGE SIGNAL-TO-NOISE RATIO FOR A RECOVERED
ANALOG SIGNAL
Using Eq. (3–16b), calculate the average SNR dB of the analog signal that is recovered from a
PCM signal that has error bits with a probability of error of P e . Plot the SNR dB for P e over a range
from 10 -7 to 10 -1 . See Example3_03.m for the solution.
In Chapter 7, P e is evaluated for many different types of digital transmission systems. In
Chapter 1, it was shown how channel coding could be used to correct some of the bit errors
and, consequently, reduce P e . Therefore, in many practical systems, P e is negligible. If we
assume that there are no bit errors resulting from channel noise (i.e., P e 0) and no ISI, then,
from Eq. (3–16a), the peak SNR resulting from only quantizing errors is
a S N b = 3M 2
pk out
(3–17a)
and from Eq. (3–16b), the average SNR due only to quantizing errors is
a S N b out
= M 2
(3–17b)
Numerical values for these SNRs are given in Table 3–2.
To realize these SNRs, one critical assumption is that the peak-to-peak level of the analog
waveform at the input to the PCM encoder is set to the design level of the quantizer. For
example, referring to Fig. 3–8a, this corresponds to the input traversing the range -V to +V
volts, where V 8 volts is the design level of the quantizer. Equations (3–16) and (3–17) were
derived for waveforms with equally likely values, such as a triangle waveshape, that have a
peak-to-peak value of 2V and an RMS value of V> 23, where V is the design peak level of the
quantizer.
† This derivation is postponed until Chapter 7 because a knowledge of statistics is needed to carry it out.