MAE143 A  Signals and Systems  Winter 11
Midterm, February 2nd
Instructions
(i) This exam is open book. You may use whatever written materials you choose, including
your class notes and textbook. You may use a hand calculator with no communication
capabilities
(ii) You have 50 minutes
(iii) Do not forget to write your name, student number, and instructor
1.
Signals
Consider the following mathematical description of a continuoustime signal
x
(
t
) =
u
(
t

1)

(1

e

(
t

2)
)
u
(
t

2)

δ
(
t
+ 1)
.
Sketch the plot of the following derived signals:
(a) (2 points)
x
(
t
)
(b) (2 points)
x
(2

t
)
(c) (2 points)
x
(
t/
2)
Solution:
NOTE: In this question we are not being picky about the value of
u
(
t
)
at
t
= 0
!
(a) Start with the third summand, which is a negative impulse at time
t
=

1
. This is the
only nonzero value taken by the function for
t <
1
. At
t
= 1
, the first summand, a unit
step, which start adding
1
to the value of the function. Finally, the second summand is
zero for
t <
2
, so
x
(
t
)
≡
1
for
1
< t <
2
. For
t
≥
2
, this summand is negative, with value
0
at
t
= 2
, and then smoothly approaching

1
as
t
grows. A sketch of the plot is
(+ 2 points)
2
0
2
4
6
8
1
0.5
0
0.5
1
x(t)
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
(b) The function
x
(

t
)
has the same plot as the one above except that it is mirrored with
respect to the vertical axis at
0
.
For example,
x
(

t
)
at
t
= 1
is the impulse.
The final
function
x
(2

t
)
is then shifted in time by two seconds. For example,
x
(2

t
)
at
t
= 3
is
the impulse. A sketch of the plot is
(+ 2 points)
2
0
2
4
6
8
1
0.5
0
0.5
1
x(2t)
3
(c) The function
x
(
t/
2)
has its time streched by a factor of
2
. For example, the impulse
This is the end of the preview.
Sign up
to
access the rest of the document.
 Summer '08
 MIROSLAVKRSTIC
 Laplace

Click to edit the document details