MAE107 Homework #4
Prof. M’Closkey
Due Date
The homework is due by Friday, 5PM, May 14, 2010, to David Shatto in 38137 E4.
Problem 1
Consider the periodic signal
u
shown in Fig. 1. Answer the following questions:
1. Compute the Fourier series coeﬃcients,
c
k
, using the “complex” representation, i.e., ﬁnd a
formula for
c
k
in
u
(
t
) =
1
T
∞
X
k
=
∞
c
k
e
jkω
0
t
,
ω
0
=
2
π
T
,
c
k
=
Z
T/
2
T/
2
u
(
t
)
e

jkω
0
t
dt.
2. Use Matlab to plot an approximation of
u
using the following
truncated
Fourier series:
u
(
t
)
≈
1
T
10
X
k
=

10
c
k
e
jkω
0
t
.
In other words you will sum only the terms in the Fourier series corresponding to
k
=

10
,

9
,...,

1
,
0
,
1
,...,
9
,
10. Plot
u
and this approximation on the same ﬁgure. Your
plot axes must extend from 0 to 1 seconds and 0.2 to 1.2 for the “
y
” axis. Hand in your
code.
Problem 2
Consider the ﬁrst order linear system
˙
x
+ 3
x
= 3
u.
(1)
You will compute the periodic response of this system to the periodic input from Problem 1 in
two ways. The ﬁrst method will use timedomain calculations, and the second method will use a
Fourier series approach.
1.
Timedomain approach.
Imagine applying
u
from Problem 1 to this system since
t
=
∞
.
No matter what the initial condition was when the input was ﬁrst applied, its eﬀect on
x
via
a homogeneous solution of the ODE will have decayed to zero by the time we start taking our
measurements. The system is in “steadystate” response at this point. Here is a general fact
that we proved:
an asymptotically stable linear system when driven by a periodic
input, reaches a periodic (with same period as the input) steadystate once the
eﬀect of initial conditions has decayed
. Your objective is to compute the periodic steady
1
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Periodic function
seconds
Figure 1: A periodic signal –you will ﬁnd a closed form representation for the Fourier series coeﬃ
cients for this signal.
state of this system when
u
is the signal from Problem 1. Here is a big hint: if the system is
in periodic steadystate then
x
(
T
) =
x
(0), where
T
is the period of
u
. You can ﬁnd
x
(0) (or,
what is the same,
x
(
T
)) from
x
(
T
) =
e

3
T
x
(0) +
Z
T
0
h
(
T

τ
)
u
(
τ
)
dτ,
where you set
x
(
T
) =
x
(0) =
c
(ﬁnd
c
). Note that
h
is the impulse response of this system,
and the term
e

3
T
x
(0) represents the homogeneous solution. Another way to think about this
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 Spring '06
 TSAO
 Fourier Series, LTI system theory, Impulse response

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