0195136047.pdf

644 SIGNAL PROCESSING

x(t) xs(t)

xs(t)

xs(t)

x(t)

x(t)

s(t)

s(t)

fs

(a)

(b)

t

−Ts Ts

D 2 T

s^3 Ts

0

t

−Ts Ts

D

0 2 Ts 3 Ts

1

Figure 14.2.5(a)Switching sampler.(b)Model using switching functions(t).

While the switch is in touch with the upper contact for a short interval of timeD<<Ts,

and obtains a sample piece of the input signalx(t) everyTsseconds, the output sampled

waveformxs(t) will look like a train of pulses with their tops carrying the sample values of

x(t), as shown in the waveform in Figure 14.2.5(a). This process can be modeled by using

aswitching function s(t), shown in the waveform of Figure 14.2.5(b), and a multiplier in

the form

xs(t)=x(t)s(t) (14.2.5)

shown in Figure 14.2.5(b). The periodic switching functions(t) is simply a rectangular pulse train

of unit height, whose Fourier expansion is given by

s(t)=a 0 +a 1 cos 2πfst+a 2 cos 2π( 2 fs)t+... (14.2.6)

witha 0 =D/Tsandan=( 2 /π n)sin (π Dn/Ts) forn=1, 2,... [see Figure 14.1.4(a)]. Using

Equation (14.2.6) in Equation (14.2.5), we get

xs(t)=a 0 x(t)+a 1 x(t)cos 2πfst+a 2 x(t)cos 2π( 2 fs)t+... (14.2.7)

By employing the frequency-domain methods, one can gain insight for signal analysis and easily

interpret the results. Supposing thatx(t) has a low-pass amplitude spectrum, as shown in Figure

14.2.6(a), the corresponding spectrum of the sampled signalxs(t) is depicted in Figure 14.2.6(b).

Taking Equation (14.2.7) term by term the first term will have the same spectrum asx(t) scaled by

the factora 0 ; the second term corresponds to product modulation with a scale factora 1 and carrier

frequencyfs, so that it will have a DSB spectrum over the rangefs−W≤f ≤fs+W; the

third and all other terms will have the same DSB interpretation with progressively higher carrier

frequencies 2fs,3fs,....

Note that provided the sampling frequency satisfies the condition