Introduction to Signal Processing
Hayder Radha
- 1 -
ECE 366
Homework Set #10
Wednesday, April 15, 2009 (In class)
ECE 366 – Introduction to Signal Processing
Spring 2009
Michigan State University
Department of Electrical and Computer Engineering
Please remember to follow the rules and policies outlined in the ECE 366 Syllabus
[1]
Problem 3.3-1
(a)
For :
[ ]
(
)
1
n
x n
= −
Using:
[ ]
2
1
lim
2
1
N
x
N
n
N
P
x
n
N
+
→∞
=−
=
+ +
∑
(
)
(
)
2
1
1
lim
1
lim
2
1
1
2
1
2
1
N
n
x
N
N
n
N
P
N
N
N
+
→∞
→∞
=−
=
−
=
+
=
+ +
+ +
∑
.
(d)
For :
[ ]
(
)
[ ]
1
n
x n
u n
= −
(
)
(
)
2
0
1
1
1
lim
1
lim
2
1
2
1
2
N
n
x
N
N
n
P
N
N
N
+
→∞
→∞
=
=
−
=
=
+ +
+ +
∑
.
(e)
For:
[ ]
cos
3
6
x n
n
π
π
⎡
⎤
=
+
⎢
⎥
⎣
⎦
This is a periodic discrete-time signal with period
0
N
:
0
2
6
3
N
π
π
=
=
⎛
⎞
⎜
⎟
⎝
⎠
.
In other words, when evaluating the power of the signal, we need to consider a set
of discrete-time samples that cover a complete period (i.e., six samples in this
case), and take the average over such set of samples:

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Introduction to Signal Processing
Hayder Radha
- 2 -
ECE 366
2
5
0
2
2
2
2
2
2
1
cos
6
3
6
1
cos
cos
cos
2
cos
3
cos
4
cos
5
6
6
3
6
3
6
3
6
3
6
3
6
x
n
P
n
π
π
π
π
π
π
π
π
π
π
π
π
π
=
⎛
⎞
⎡
⎤
=
+
⎜
⎟
⎢
⎥
⎣
⎦
⎝
⎠
⎛
⎞
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
=
+
+
+
+
+
+
+
+
+
+
⎜
⎟
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
⎣
⎦
⎝
⎠
∑
( )
2
2
2
2
2
2
1
3
3
3
3
1
0
0
3
0.5
6
2
2
2
2
6
x
P
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎛
⎞
⎜
⎟
=
+
+
−
+
−
+
+
=
=
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
⎝
⎠
.
[2]
Problem 3.3-4
(a)
This signal can be represented as a linear combination of time-shifted ramp
functions. There are a couple of ways of performing this representation.
Below is an example of such expression.
First, we have one increasing ramp that begins as a zero value at
3
n
= −
and it
grows linearly until it reaches a value of 3 at
0
n
=
. Such function can be
expressed as:
[ ]
(
)
[
]
[
]
(
)
1
3
3
1
x
n
n
u n
u n
=
+
+
−
−
.
(Test the signal at different values of
n
from
3
n
= −
to
0
n
=
.)
[ ]
(
)
[
]
[
]
(
)
1
3
3
1
x
n
n
u n
u n
=
+
+
−
−
n
3
−
2
−
1
−