Signals and Systems - Electrical Engineering

12.4 Application to Digital Communications 729

After replacing Ts=1 it becomes v[n]=v[n−1]+e[n], so the transfer function of the

integrator is

P(z)=

V(z)

E(z)

=

1

1 −z−^1

The transfer function of the DAC with ZOH and the plant is

Gˆ(s)=(^1 −e

−sTs)G(s)

s

=

( 1 −e−s)

s(s+ 1 )

If we letD(s)= 1 /s(s+ 1 ), thengˆ(t)=d(t)−d(t− 1 ). ExpandingD(s)as

D(s)=

1

s(s+ 1 )

=

A

s

+

B

s+ 1

=

1

s

−

1

s+ 1

so thatd(t)=u(t)−e−tu(t)sampled, gives

d(nTs)=u(nTs)−e−nTsu(nTs)

which has a Z-transform

D(z)=

1

1 −z−^1

−

1

1 −e−^1 z−^1

⇒ Gˆ(z)=( 1 −z−^1 )D(z)

The transfer function is then

Y(z)

C(z)

=

P(z)( 1 −z−^1 )D(z)

1 +P(z)( 1 −z−^1 )D(z)

=

D(z)

1 +D(z)

sinceP(z)( 1 −z−^1 )=1. n

12.4 Application to Digital Communications

Although over the years the principles of communications have remained the same, their implemen-
tation has changed considerably. Analog communications transitioned into digital communications,
while telephony and radio have coalesced into wireless communications. The scarcity of radio spec-
trum changed the original focus on bandwidth and energy efficiency into more efficient utilization
of the available spectrum by sharing it, and by transmitting different types of data together. Wireless
communications has allowed the growth of cellular telephony, personal communication systems,
and wireless local area networks.

Modern digital communications was initiated with the concept of pulse code modulation, which
allowed the transmission of binary signals. PCM is a practical implementation of sampling, quantiza-
tion, and coding, or analog-to-digital conversion, of an analog message into a digital message. Using
the sample representation of a message, the idea of mixing several messages—possibly of different