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Baseband Pulse and Digital Signaling Chap. 3
The resulting detected signal-to-noise ratio can also be calculated. It has been determined
experimentally that the spectrum of the granular noise is uniformly distributed over the
frequency band |f| … f . It can also be shown that the total granular quantizing noise is d 2 s
3
(See Sec. 7–7, where d2 of PCM is replaced by d for DM.) Thus, the PSD for the noise is
n (f) = d 2 >(6f s ). The granular noise power in the analog signal band, |f| … B, is
B
N = 8n 2 9 = n (f)df = d 2B
(3–85)
L 3f s
or, from Eq. (3–84), with equality,
-B
N = 4p2 A 2 f a 2 B
3f s
3
The signal power is
S = 8w 2 (t)9 = A2
2
(3–86)
The resulting average signal-to-quantizing noise ratio out of a DM system with a sine-wave
test signal is
a S (3–87)
N b 3
=
f s 3
out 8p 2 f 2 a B
where f s is the DM sampling frequency, f a is the frequency of the sinusoidal input, and B is the
bandwidth of the receiving system. Recall that Eq. (3–87) was shown to be valid only for
sinusoidal-type signals.
For voice-frequency (VF) audio signals, it has been shown that Eq. (3–84) is too
restrictive if f a = 4 kHz and that slope overload is negligible if [deJager, 1952]
d Ú 2p800W p
f s
(3–88)
where W p is the peak value of the input audio waveform w(t). (This is due to the fact that the
midrange frequencies around 800 Hz dominate in the VF signal.) Combining Eqs. (3–88) and
(3–85), we obtain the SN for the DM system with a VF-type signal, viz.,
a S N b out
= Hw2 (t)I
N
=
3f s
3
(1,600p) 2 B a8w2 (t)9
W p
2
b
(3–89)
where B is the audio bandwidth and (8w 2 (t)9>W 2 ) is the average-audio-power to peak-audiopower
ratio.