McGill University
Fall 2009
ECSE 303: Signals and Systems 1
FINAL EXAMINATION
PROBLEM 1
Consider a Linear TimeInvariant (LTI) system with impulse response
h
(
t
). Let
x
(
t
) be the input signal and
y
(
t
) the corresponding output signal. In the following, ’prime’ (
0
) will
denote the derivative of a signal, i.e.,
x
0
(
t
) is the derivative of
x
(
t
),
h
0
(
t
) is the derivative of
h
(
t
),
etc.
(a)
Show that
y
0
(
t
) =
x
(
t
)
*
h
0
(
t
) =
x
0
(
t
)
*
h
(
t
)
,
where ’*’ denotes convolution.
(b)
For this part only, assume that the response of the system to
x
(
t
) =
e

5
t
u
(
t
), where
u
(
t
) =
1
,
t
≥
0
0
,
t <
0
is the unit step function, is sin(
ω
0
t
) for some
ω
0
>
0. Find the impulse response
of the system
h
(
t
).
(c)
Let
s
(
t
) be the unit step response of the system, i.e., the output of the system when the input
is the unit step function
u
(
t
). Show that the response
y
(
t
) of the system to an arbitrary input
x
(
t
) is
y
(
t
) =
x
0
(
t
)
*
s
(
t
)
.
(d)
For this part only, assume that the unit step response of the system is
s
(
t
) =
e

2
t
u
(
t
). Find
the response of the system to the input
x
(
t
) =
1
t
≥
0
e
t
t <
0
(e)
Is the statement below correct? Justify your answer.
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 Winter '09
 Champagne
 unit step response, unit step, Hinv

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