6.003 Homework 4
The following problems are intended to help you prepare for the upcoming exam. Please do
them by
Wednesday, October 7, 2009
. You need not submit your answers: they will
NOT be graded. Solutions will be posted.
Problems
1. Avoiding excitation
Consider the system described by the following diﬀerence equation:
y
[
n
] =
x
[
n
] +
5
2
y
[
n
−
1]
−
y
[
n
−
2]
.
Find an input
x
[
n
] such that the output
y
[
n
] is proportional to
(
1
2
)
n
for large values of
n
. Try to minimize the number of nonzero samples in
x
[
n
].
2. Periodic system
Consider this variant of the Fibonacci system:
y
[
n
] =
y
[
n
−
1]
−
y
[
n
−
2] +
x
[
n
]
where
x
[
n
] represents the input and
y
[
n
] represents the output.
a.
Compute the unitsample response and show that it is periodic. What is the period?
b.
Determine the poles of the system.
c.
Decompose the system functional into partial fractions, and use the result to deter
mine a closedform expression for
h
[
n
], the unitsample response.
3. Closedloop poles
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 Spring '11
 DennisM.Freeman
 following block diagram

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