0195136047.pdf

14.2 MODULATION, SAMPLING, AND MULTIPLEXING 641

signal needs to besampledas part of the modulation process. Frequency translation and sampling

have extensive use in communication systems. Both of these lend tomultiplexing, which permits

a transmission system to handle two or more information-bearing signals simultaneously.

Frequency Translation and Product Modulation

The basic operation needed to build modulators is the multiplication of two signals. Whenever

sinusoids are multiplied, frequency translation takes place. Figure 14.2.1(a) shows aproduct

modulator, which multiplies the signalx(t) and a sinusoidal carrier wave at frequencyfcto yield

xc(t)=x(t)cos 2πfct (14.2.1)

Choosingx(t) to be a low-pass signal with bandwidthW<<fc, Figure 14.2.1(b) depicts the

relationship betweenxc(t) andx(t). The modulated wavexc(t) can now be seen to have abandpass

spectrum resulting from frequency translation, which will be explained later.

Ifx(t) contains a sinusoidal componentAmcos 2πfmt, multiplication by a sinusoidal carrier

wave cos 2πfctwithfc>> fmyields

(Amcos 2πfmt)×(cos 2πfct)=

Am

2

cos 2π(fc−fm)t+

Am

2

cos 2π(fc+fm)t (14.2.2)

Waveforms of the signal, the carrier wave, and the product, as well as their respective line

spectra, are shown in Figure 14.2.2. Notice that the low frequencyfmhas been translated to the

higher frequenciesfc±fm.

Next, let us consider an arbitrary low-pass signalx(t) with the typical amplitude spectrum

of Figure 14.2.3(a). The amplitude spectrum of the modulated wavexc(t) will now havetwo

sidebands(lower and upper sidebands), each of widthWon either side offc, as illustrated in

Figure 14.2.3(b). Thus, we have a signal that can be transmitted over abandpasssystem with a

minimum bandwidth of

B= 2 W (14.2.3)

which is twice the bandwidth of the modulating signal. This process is then known asdouble-

sideband modulation(DSB). Either the lower or the upper sideband may be removed by filtering so

as to obtain single-sideband modulation (SSB) withB=W, if the bandwidth needs to be conserved.

By choosing the carrier frequencyfcat a value where the system has favorable characteristics,

the frequency translation by product modulation helps in minimizing the distortion and other

problems in system design.

xc(t)

xc(t)

cos 2πfct

0 t

Ideal

multiplier

Sinusoidal

carrier

wave

Modulated

wave

Information-bearing

input signal

(a) (b)

x(t)

x(t)

1/fc

Figure 14.2.1(a)Product modulator.(b)Waveforms.