0195136047.pdf

14.2 MODULATION, SAMPLING, AND MULTIPLEXING 643

0 W 2 fc

(b)

f

y(t)

cos 2πfct

Oscillator

Synch

Low-pass

filter

(a)

xc(t)

z(t)

Figure 14.2.4(a)Product demodula-

tor.(b)Spectrum prior to low-pass fil-

tering.

Sampling and Pulse Modulation

In most analog circuits, signals are processed in their entirety. However, in many modern electric
systems, especially those that convert waveforms for processing by digital circuits, such as digital
computers, onlysample valuesof signals are utilized for processing. Sampling makes it possible
to convert an analog signal to discrete form, thereby permitting the use of discrete processing
methods. Also, it is possible to sample an electric signal, transmit only the sample values, and use
them to interpolate or reconstruct the entire waveform at the destination. Sampling of signals and
signal reconstruction from samples have widespread applications in communications and signal
processing.
One of the most important results in the analysis of signals is thesampling theorem, which
is formally presented later. Many modern signal-processing techniques and the whole family
of digital communication methods are based on the validity of this theorem and the insight it
provides. The idea leading to the sampling theorem is rather simple and quite intuitive. Let us
consider a relatively smooth signalx 1 (t), which varies slowly and has its main frequency content
at low frequencies, as well as a rapidly changing signalx 2 (t) due to the presence of high-frequency
components. Suppose we are to approximate these signals with samples taken at regular intervals,
so that linear interpolation of the sampled values can be used to obtain an approximation of the
original signals. It is obvious that the sampling interval for the signalx 1 (t) can be much larger
than the sampling interval necessary to reconstruct signalx 2 (t) with comparable distortion. This
is simply a direct consequence of the smoothness of the signalx 1 (t) compared tox 2 (t). Therefore,
the sampling interval for the signals of smaller bandwidths can be made larger, or the sampling
frequency can be made smaller. The sampling theorem is, in fact, a statement of this intuitive
reasoning.
Let us now look from another point of view by considering a simple switching sampler
and waveforms shown in Figure 14.2.5(a). Let the switch alternate between the two con-
tacts at thesampling frequencyfs = 1 /Ts. The lower contact of the switch is grounded.