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ECEN 314
Solution to Quiz # 1
Georghiades
Note:
Quiz is openbook, 15 minutes long.
1. A linear, timeinvariant system, has the following unitstep response:
y
(
t
) =
‡
e

t

e

2
t
·
u
(
t
)
.
Find the following:
(a) The impulseresponse of the system.
(b) The response of the system when the input is:
1
1
t
1
2
w(t)
Solution
(a) The unitimpulse function is the derivative of the unitstep function and therefore the
impulse response is the derivative of the unitstep response. If we let the impulse response
be
h
(
t
), we have:
h
(
t
) =
dy
(
t
)
dt
=
‡
e

t

e

2
t
·

{z
}
0
at
t
=0
δ
(
t
) +
‡

e

t
+ 2
e

2
t
·
u
(
t
)
=
‡
2
e

2
t

e

t
·
u
(
t
)
.
The above solution is the fastest for this problem, but not the most general. Since we
now know Laplace transforms, there is a much more general way of solving this problem,
which works in ﬁnding the impulse response when one knows
any
inputoutput pair of
signals,
x
(
t
) and
y
(
t
). The approach is as follows:
•
Take the Laplace transform of the input
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This note was uploaded on 01/01/2012 for the course ECEN 314 taught by Professor Halverson during the Spring '08 term at Texas A&M.
 Spring '08
 HALVERSON

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