Signals and Systems - Electrical Engineering

1.4 Representation Using Basic Signals 101

signalx(t)by a train of very narrow pulses of the sampling periodTs. For simplicity, considering that
the width of the pulses is much smaller thanTs, the train of pulses can be approximated by a train of
impulses that is periodic of periodTs—that is, thesampling signalδTs(t)is

δTs(t)=

∑∞

n=−∞

δ(t−nTs) (1.33)

The sampled signalxs(t)is then

xs(t)=x(t)δTs(t)

=

∑∞

n=−∞

x(nTs)δ(t−nTs) (1.34)

or a sequence of uniformly shifted impulses with amplitude the value of the signalx(t)at the time
when the impulse occurs.

A fundamental result in sampling theory is the recovery of the original signal, under certain con-
strains, by means of an interpolation usingsinc signals. Moreover, we will see that the sinc is connected
with ideal low-pass filters. The sinc function is defined as

S(t)=

sinπt

πt

−∞<t<∞ (1.35)

This signal has the following characteristics:

n The time support of this signal is infinite.
n It is an even function oft, as

S(−t)=

sin(−πt)

−πt

=

−sin(πt)

−πt

=S(t) (1.36)

n Att=0 the numerator and the denominator of the sinc are zero; thus the limit ast→0 is found
using L’Hopital’s rule—that is,ˆ

lim

t→ 0

S(t)=lim

t→ 0

dsin(πt)/dt

dπt/dt

=lim

t→ 0

πcos(πt)

π

= 1 (1.37)

n S(t)is bounded—that is, since− 1 ≤sin(πt)≤1, then fort≥0,

− 1

πt

≤

sin(πt)

πt

=S(t)≤

1

πt

(1.38)

and given thatS(t)is even, it is equally bounded fort<0. Ast→±∞,S(t)→0.
n The zero-crossing time ofS(t)are found by letting the numerator equal zero—that is, when
sin(πt)=0, the zero-crossing times are such thatπt=kπ, ort=kfor a nonzero integerkor
k=±1,±2,....