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Sec. 3–2 Pulse Amplitude Modulation 137
Analog multiplier
(four-quadrant multiplier)
w s (t)
PAM (natural sampling)
Low-pass filter,
H(f)
Cw(t)
cos(nv s t)
H(f)
Oscillator
wo = nv s
– f co f co
f
where B < f co < f s – B
Figure 3–4
Demodulation of a PAM signal (naturally sampled).
At the receiver, the original analog waveform, w(t), can be recovered from the PAM signal,
w s (t), by passing the PAM signal through a low-pass filter where the cutoff frequency is
B 6 f cutoff 6 f s - B. This is seen by comparing Fig. 3–3b with Fig. 3–3a. Because the spectrum out
of the low-pass filter would have the same shape as that of the original analog signal shown in
Fig. 3–3a, the waveshape out of the low-pass filter would be identical to that of the original analog
signal, except for a gain factor of d (which could be compensated for by using an amplifier). From
Fig. 3–3b, the spectrum out of the low-pass filter will have the same shape as the spectrum of the original
analog signal only when f s Ú 2B, because otherwise spectral components in harmonic bands (of f s )
would overlap. This is another illustration of the Nyquist sampling rate requirement. If the analog
signal is undersampled (f s 6 2B), the effect of spectral overlapping is called aliasing. This results in
a recovered analog signal that is distorted compared to the original waveform. In practice, physical
signals are usually considered to be time limited, so that (as we found in Chapter 2) they cannot be
absolutely bandlimited. Consequently, there will be some aliasing in a PAM signal. Usually, we
prefilter the analog signal before it is introduced to the PAM circuit so we do not have to worry about
this problem; however, the effect of aliasing noise has been studied [Spilker, 1977].
It can also be shown (see Prob. 3–5) that the analog waveform may be recovered from
the PAM signal by using product detection, as shown in Fig. 3–4. Here the PAM signal is multiplied
with a sinusoidal signal of frequency v o nv s . This shifts the frequency band of the
PAM signal that was centered about nf s to baseband (i.e., f 0) at the multiplier output. We
will study the product detector in Chapter 4. For n 0, this is identical to low-pass filtering,
just discussed. Of course, you might ask, Why do you need to go to the trouble of using a
product detector when a simple low-pass filter will work? The answer is, because of noise on
the PAM signal. Noise due to power supply hum or noise due to mechanical circuit vibration
might fall in the band corresponding to n 0, and other bands might be relatively noise free.
In this case, a product detector might be used to get around the noise problem.
Instantaneous Sampling (Flat-Top PAM)
Analog waveforms may also be converted to pulse signaling by the use of flat-top signaling
with instantaneous sampling, as shown in Fig. 3–5. This is another generalization of the
impulse train sampling technique that was studied in Sec. 2–7.