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6.003 Homework 3
Due at the beginning of recitation on
Wednesday, September 30, 2009
.
Problems
1. Complex numbers
a.
Evaluate the real and imaginary parts of
j
j
.
b.
Evaluate the real and imaginary parts of (1
−
j
√
3)
12
.
c.
Express the real part of
e
5
jθ
in terms of sin
θ
and cos
θ
.
2. Characterizing Systems
Consider the system represented by the following diﬀerence equation:
y
[
n
] = 3
y
[
n
−
1]
−
2
y
[
n
−
2] +
x
[
n
] +
x
[
n
−
1]
.
a.
Write a program to tabulate the ﬁrst 20 values of the unitsample response.
b.
Find the system functional.
c.
Expand the system function in the previous part using partial fractions.
d.
Express the unitsample response of the system as a weighted sum of fundamental
modes.
e.
Repeat parts ad for the following diﬀerence equation:
y
[
n
] = 3
y
[
n
−
1]
−
2
y
[
n
−
2] +
x
[
n
]
Point out important diﬀerences with the previous parts.
3. Black’s Equation
Consider the general form of a feedback problem:
+
F
G
−
X
Y
Notice the minus sign on the adder: it indicates that the lower input is subtracted rather
than added.
a.
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 Spring '11
 DennisM.Freeman

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