Signals and Systems - Electrical Engineering

732 CHAPTER 12: Applications of Discrete-Time Signals and Systems

nExample 12.7

Suppose you have a binary signal 01001001, with a duration of 8 units of time, and wish to

represent it using rectangular pulses and sinc functions. Consider the bandwidth of each of the

representations.

Solution

Using pulsesφ(t), the digital signal can be expressed as

s(t)=

∑^7

n= 0

bnφ(t−nτs)

wherebnare the binary digits of the digital signal (i.e.,b 0 =0,b 1 =1,b 2 =0,b 3 =0,b 4 =1,

b 5 =0,b 6 =0,b 7 =1, andτs=1). Thus,

s(t)=φ(t− 1 )+φ(t− 4 )+φ(t− 7 )

and the spectrum ofs(t)is

S()=φ()(e−j+e−j^4 +e−j^7 )

=φ()e−j^4 (ej^3 + 1 +e−j^3 )

=φ()e−j^4 ( 1 +2 cos( 3 ))

so that

|S()|=|φ()|| 1 +2 cos( 3 )|

If the pulses are rectangular,

φ(t)=u(t)−u(t− 1 )

the PCM signal will have an infinite-support spectrum because the pulse is of finite support. On

the other hand, if we use sinc functions,

φ(t)=

sin(πt/τs)

πt

its time support is infinite but its frequency support is finite (i.e., the sinc function is band limited).

In which case, the spectrum of the PCM signal is also of finite support.

If this digital signal is transmitted and received without any distortion, at the receiver we can use

the orthogonality of theφ(t)signals or sample the received signal atnTsto obtain thebn. Clearly,

each of these pulses has disadvantages—the advantage of having a finite support in the time or in

the frequency becomes a disadvantage in the other domain. n