Signals and Systems - Electrical Engineering

148 C H A P T E R 2: Continuous-Time Systems

FIGURE 2.13

Block diagrams for connecting two LTI systems

with impulse responsesh 1 (t)andh 2 (t)in (a)

cascade, (b) parallel, and (c) negative

feedback.

h 1 (t)

h 2 (t)

x(t) + y(t)

(b)

h 1 (t)

h 2 (t)

y(t)

x(t) e(t)

+ −

(c)

h 1 (t) h 2 (t) y(t)

x(t)

(a)

Two LTI systems with impulse responsesh 1 (t)andh 2 (t)connected incascadehave as an overall impulse

response

h(t)=[h 1 ∗h 2 ](t)=[h 2 ∗h 1 ](t)

whereh 1 (t)andh 2 (t)commute (i.e., they can be interchanged).

In fact, if the input to the cascade connection isx(t), the outputy(t)is found as

y(t)=[[x∗h 1 ]∗h 2 ](t)

=[x∗[h 1 ∗h 2 ]](t)

=[x∗[h 2 ∗h 1 ]](t)

where the last two equations show thecommutative propertyof convolution. The impulse response of

the cascade connection indicates that the order in which we connect LTI systems is not important—

that we can put the system with impulse responseh 1 (t)first, or the system with impulse responseh 2 (t)

first with no effect in the overall response of the system (we will see later that this is true provided

that the two systems do not load each other). When dealing with linear but time-varying systems,

however, the order in which we connect the systems in cascade is important.