2.3 LTI Continuous-Time Systems 147
Solution
In this case,x(t)=h(t)=u(t)−u(t− 1 ). Again we plotx(τ)andh(t−τ)both as functions ofτ,
for−∞<t<∞.
n It should be noticed that while computing the convolution integral fortincreasing from nega-
tive to positive values,h(t−τ)moves from left to right whilex(τ)remains stationary, and that
they only overlap on a finite support.
n Fort<0,h(t−τ)andx(τ)do not overlap, soy(t)=0 fort<0.
n h(t−τ)andx(τ)increasingly overlap for 0≤t<1 and decreasingly overlap for 1≤t<2. So
thaty(t)=tfor 0≤t<1, andy(t)= 2 −tfor 1≤t<2.
n Fort>2, there is no overlap and soy(t)=0 fort>2.
Thus, the complete response is
y(t)=r(t)− 2 r(t− 1 )+r(t− 2 )
wherer(t)=tu(t)is the ramp signal.
Notice in this example that:
n The result of the convolution of these two pulses,y(t), is smoother thanx(t)andh(t). This is
becausey(t)is the continuous average ofx(t), ash(t)is the impulse response of the averager in
example 2.12.
n The length of the support ofy(t)equals the sum of the lengths of the supports ofx(t)andh(t).
This is a general result that applies to any two signalsx(t)andh(t). n
The length of the support ofy(t)=[x∗h](t)is equal to the sum of the lengths of the supports ofx(t)andh(t).
2.3.8 Interconnection of Systems—Block Diagrams
Systems can be considered a connection of subsystems. In the case of LTI systems, to visualize the
interaction of the different subsystems each of the subsystems is represented by a block with the
corresponding impulse response, or equivalently by its Laplace transform as we will see in the next
chapter. The flow of the signals is indicated by arrows, and the addition of signals or multiplication
of a signal by a constant is indicated by means of circles.
Two possible connections, thecascadeand theparallelconnections, result from the properties of
the convolution integral, while thefeedbackconnection is found in many natural systems and has
been replicated in engineering, especially in control. The concept of feedback is one of the greatest
achievements of the 20th century. See Figure 2.13.
Cascade Connection
When connecting LTI systems in cascade the impulse response of the overall system can be found
using the convolution integral.
