Signals and Systems - Electrical Engineering

362 C H A P T E R 6: Application to Control and Communications

FIGURE 6.3

Cascading of (a) an LTV and (b) an LTI system.

The outputs are different,y 1 (t)6=y 2 (t).

Modulator

x(t) y 2 (t)

d

dt

Modulator

x(t) y 1 (t)

f(t) f(t)

d

× dt ×

(a) (b)

follower) in between (see Figure 6.2(a)). The cascading of the two voltage dividers without the voltage

follower gives a transfer function H 1 (s)= 1 / 5 , as can be easily shown by doing mesh analysis on the

circuit.

n The block diagrams of the cascade of two or more LTI systems can be interchanged with no effect on the

overall transfer function, provided the connection is done with no loading. That is not true if the systems

are not LTI. For instance, consider cascading a modulator (LTV system) and a differentiator (LTI) as

shown in Figure 6.3. If the modulator is first, Figure 6.3(a), the output of the overall system is

y 2 (t)=

dx(t)f(t)

dt

=f(t)

dx(t)

dt

+x(t)

df(t)

dt

while if we put the differentiator first, Figure 6.3(b), the output is

y 1 (t)=f(t)

dx(t)

dt

It is obvious that if f(t)is not a constant, the two responses are very different.

Parallel Connection of LTI Systems

According to the distributive property of the convolution integral, theparallelconnection of two

or more LTI systems has the same input and its output is the sum of the outputs of the systems

being connected (see Figure 6.1(b)). The parallel connection is better than the cascade, as it does

not require isolation between the systems, and reduces the delay in processing an input signal. The

transfer function of the parallel system is

H(s)=H 1 (s)+H 2 (s)

Remarks

n Although a communication system can be visualized as the cascading of three subsystems—the transmitter,

the channel, and the receiver—typically none of these subsystems is LTI. As we discussed in Chapter 5, the

low-frequency nature of the message signals requires us to use as the transmitter a system that can generate

a signal with much higher frequencies, and that is not possible with LTI systems (recall the eigenfunction

property). Transmitters are thus typically nonlinear or linear time varying. The receiver is also not LTI. A

wireless channel is typically time varying.

n Some communication systems use parallel connections (see quadrature amplitude modulation (QAM)

later in this chapter). To make it possible for several users to communicate over the same channel, a

combination of parallel and cascade connections are used (see frequency division multiplexing (FDM)

systems later in this chapter). But again, it should be emphasized that these subsystems are not LTI.