CHAPTER 6 Application to Control and Communications
6.2 System Connections and Block Diagrams
Control and communication systems consist of interconnection of several subsystems. As we indicated in Chapter 2, there are three important connections of LTI systems:
n Cascade
n Parallel
n Feedback
Cascade and parallel result from properties of the convolution integral, while the feedback con- nection relates the output of the overall system to its input. With the background of the Laplace transform we present now a transform characterization of these connections that can be related to the time-domain characterizations given in Chapter 2.
The connection of two LTI continuous-time systems with transfer functionsH1(s)andH2(s)(and correspond- ing impulse responsesh1(t)andh2(t)) can be done in:
n Cascade (Figure 6.1(a)): Provided that the two systems are isolated, the transfer function of the overall system is
H(s)=H1(s)H2(s) (6.1)
n Parallel (Figure 6.1(b)): The transfer function of the overall system is
H(s)=H1(s)+H2(s) (6.2)
n Negative feedback (Figure 6.4): The transfer function of the overall system is H(s)= H1(s)
1+H2(s)H1(s) (6.3)
n Open-loop transfer function:Ho`(s)=H1(s).
n Closed-loop transfer function:Hc`(s)=H(s).
6.2 System Connections and Block Diagrams 361
Cascading of LTI Systems
Given two LTI systems with transfer functionsH1(s)=L[h1(t)] andH2(s)=L[h2(t)] whereh1(t)and h2(t)are the corresponding impulse responses of the systems, thecascadingof these systems gives a new system with transfer function
H(s)=H1(s)H2(s)=H2(s)H1(s)
provided that these systems are isolated from each other (i.e., they do not load each other). A graph- ical representation of the cascading of two systems is obtained by representing each of the systems with blocks with their corresponding transfer function (see Figure 6.1(a)). Although cascading of systems is a simple procedure, it has some disadvantages:
n It requires isolation of the systems.
n It causes delay as it processes the input signal, possibly compounding any errors in the processing.
Remarks
n Loading, or lack of system isolation, needs to be considered when cascading two systems. Loading does not allow the overall transfer function to be the product of the transfer functions of the connected systems.
Consider the cascade connection of two resistive voltage dividers (Figure 6.2), each with a simple transfer function Hi(s)=1/2,i=1, 2. The cascade in Figure 6.2(b) clearly will not have as transfer function H(s)=H1(s)H2(s)=(1/2)(1/2)unless we include a buffer (such as an operational amplifier voltage
(a) (b)
X(s) Y(s)
y(t) H1(s) H2(s)
x(t)
Y(s)
y(t) H2(s)
H1(s)
X(s) x(t)
+
FIGURE 6.1
(a) Cascade and (b) parallel connections of systems with transfer functionH1(s)andH2(s). The input and output are given in the time or in the frequency domains.
(a) (b)
V2(s) + V0(s) −
+
− 1Ω
1Ω
1Ω
1Ω
V0(s) V1(s)
+ +
− +
− −
1Ω 1Ω
1Ω 1Ω
FIGURE 6.2
Cascading of two voltage dividers: (a) using a voltage follower givesV1(s)/V0(s)=(1/2)(1/2)with no loading effect, and (b) using no voltage followerV2(s)/V0(s)=1/56=V1(s)/V0(s)due to loading.
FIGURE 6.3
Cascading of (a) an LTV and (b) an LTI system.
The outputs are different,y1(t)6=y2(t).
Modulator
x(t) y2(t)
d dt
Modulator
x(t) y1(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 H1(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
y2(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
y1(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, the parallel connection 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)=H1(s)+H2(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.