Signals and Systems - Electrical Engineering

2.3 LTI Continuous-Time Systems 149

Parallel Connection

If we connect inparalleltwo LTI systems with impulse responsesh 1 (t)andh 2 (t), the impulse response of the

overall system is

h(t)=h 1 (t)+h 2 (t)

In fact, the output of the parallel combination is

y(t)=[x∗h 1 ](t)+[x∗h 2 ](t)

=[x∗(h 1 +h 2 )](t)

which is thedistributive propertyof convolution.

Feedback Connection
In these connections the output of the system is fed back and compared with the input of the system.
The fedback output is either added to the input giving apositive feedbacksystem or subtracted from the
input giving anegative feedbacksystem. In most cases, especially in control systems, negative feedback
is used. Figure 2.13(c) illustrates the negative feedback connection.

Given two LTI systems with impulse responsesh 1 (t)andh 2 (t), a negative feedback connection (Figure 2.13(c))

is such that the output is

y(t)=[h 1 ∗e](t)

where the error signal is

e(t)=x(t)−[y∗h 2 ](t)

The overall impulse responseh(t), or the impulse response of theclosed-loopsystem, is given by the implicit

expression

h(t)=[h 1 −h∗h 1 ∗h 2 ](t)

Ifh 2 (t)= 0 (i.e., there is no feedback) the system is called anopen-loopsystem andh(t)=h 1 (t).

Using the Laplace transform we will obtain later an explicit expression for the Laplace transform of
h(t). To obtain the above result we consider the output of the system as the overall impulse response
y(t)=h(t)due to an inputx(t)=δ(t). Thene(t)=δ(t)−h∗h 2 , and so when replaced in the
expression for the output

h(t)=[e∗h 1 ](t)=[(δ−h∗h 2 )∗h 1 ](t)=[h 1 −h∗h 1 ∗h 2 ](t)

the implicit expression is as given above. When there is no feedback,h 2 (t)=0, thenh(t)=h 1 (t).