Signals and Systems - Electrical Engineering

12.4 Application to Digital Communications 731

FIGURE 12.9
PCM system: transmitter, channel,
and receiver.

LPF and holdSample and coder Quantizer

Repeater

Decoder LPF

PCM transmitter

Repeater Repeater

PCM receiver

Channel

m(t) ···1011100···

···1011100··· ···1011100···

···1011100··· m^(t)

its analysis is complicated. Figure 12.9 shows a transmitter, a channel, and a receiver of a PCM

system.

The main disadvantage of PCM is that its bandwidth is wider than that of the analog message

it represents. This is due to the rectangular pulses in the signal. If we represent the PCM signal

s(t)as

s(t)=

N∑− 1

n= 0

anφ(t−nτs)

whereφ(t)is a function andτsis the duration of a symbol, the spectrum ofs(t)will be

S()=

∑N

n= 0

anF[φ(t−nτs)]

=

∑N

n= 0

anφ()e−jnτs

Suppose that the function is a sinc function having an infinite time support so thats(t)also has an

infinite time support. If

φ(t)=

sin(πt/τs)

πt

it has a band-limited spectrum,

φ()=u(+π/τs)−u(−π/τs)

so that the spectrum of the signal is also band limited. On the other hand, by duality if we use

a rectangular pulse as the functionφ(t), its spectrum will spread over all frequencies, making the

spectrum of the signal of very large bandwidth.