Signals and Systems - Electrical Engineering

6.2 System Connections and Block Diagrams 361

Cascading of LTI Systems
Given two LTI systems with transfer functionsH 1 (s)=L[h 1 (t)] andH 2 (s)=L[h 2 (t)] whereh 1 (t)and
h 2 (t)are the corresponding impulse responses of the systems, thecascadingof these systems gives a
new system with transfer function

H(s)=H 1 (s)H 2 (s)=H 2 (s)H 1 (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)=H 1 (s)H 2 (s)=( 1 / 2 )( 1 / 2 )unless we include a buffer (such as an operational amplifier voltage

(a) (b)

X(s) Y(s)

y(t)

H 1 (s) H 2 (s)

x(t)

Y(s)

y(t)

H 2 (s)

H 1 (s)

X(s)

x(t)

+

FIGURE 6.1
(a) Cascade and (b) parallel connections of systems with transfer functionH 1 (s)andH 2 (s). The input and output
are given in the time or in the frequency domains.

(a) (b)

V 2 (s)

+

V 0 (s) −

+

−

1 Ω

1 Ω

1 Ω

V 0 (s) V 1 (s) 1 Ω

+ +−

+

−

−

1 Ω 1 Ω

1 Ω 1 Ω

FIGURE 6.2
Cascading of two voltage dividers: (a) using a voltage follower givesV 1 (s)/V 0 (s)=( 1 / 2 )( 1 / 2 )with no loading
effect, and (b) using no voltage followerV 2 (s)/V 0 (s)= 1 / 5 6=V 1 (s)/V 0 (s)due to loading.