1 (47)

Prévia do material em texto

Problem 3.11 PP
Using the convolution integral, find the step response of the system whose impulse response is 
given below and shown in Fig;
0 , r 2 .
Figure Impulse response
-I -OS 0 OS 1 15 2 25 3
1106 (nc)
Step-by-step solution
step 1 of 6
Given impulse response o f the system is
^ ri, o ^ t ^ 2
|0 , £ 2 
The given impulse response is
*(/)
-1 -0.5 0 05 I t.5 2 2.5 3 
lunc(soc)
Step 2 of 6 ^
There are three cases to consider as shown in the following figure.
w
For the case £ ^ 0, the situation is illustrated in the below Sgure.
There is no overlc^ between the two functions u (£ - r ) and h ( r) and the output 
is zero, that is y i (i) = 0.
(b)
Step 3 of 6
For the case 0 £ £ ̂ 2, the situation is displayed in the below Bgure and shows 
partial overls^. The ou^u t ofthe system is
s£
(c)
Step 4 of 6 ^
For the case £ ^ 2, the situation is displayed in the below figure and shows 
total overU^. The output of the system is
^ W = f * A W « ( £ - T ) ^ T
s 2
Step 5 of 6
The figures to illustrate convolution are
u(t) b
bet)
i(sec) ( 2 t(stc)
uCt) b
ll(T)
T (sec) • 2
h(T)ii(l-t)
1 h(T)
t(sa :}
x(sec)
t 2 xfsecj
h(T)u(l-T)^
1
^ t - j )
b(i)
step 6 of 6
The output of the system is the con^osite o f the three segments computed above a 
shown in the following figure.
Thus, the required step response is obtained.