ECE 301, Homework #6, due date: 10/6/2010
http://cobweb.ecn.purdue.edu/
∼
chihw/10ECE301F/10ECE301F.html
Question 1:
[Basic] Review of linear algebra: Consider row vectors of dimension 3. Let
x
1
= (
√
2
/
2
,

√
2
/
2
,
0),
x
2
= (
√
3
/
3
,
√
3
/
3
,
√
3
/
3), and
x
3
= (
√
6
/
6
,
√
6
/
6
,

2
√
6
/
6))
•
Show that
{
x
1
, x
2
, x
3
}
is an
orthonormal
basis. Namely, show that

x
i

2
= 1 for all
i
= 1
,
2
,
3, and show that the inner product
x
i
·
x
j
= 0 for
i
6
=
j
.
•
If we know that
x
= 0
.
7
x
1
+ 0
.
3
x
2
+ 0
.
4
x
3
, find
x
.
•
If we know that
x
0
= (0
.
7
,
0
.
3
,
0
.
4), find
α
1
,
α
2
,
α
3
such that
x
0
=
α
1
x
1
+
α
2
x
2
+
α
3
x
3
.
•
Why are we interested in rewriting
x
0
=
α
1
x
1
+
α
2
x
2
+
α
3
x
3
?
Note: There is a simple formula of solving
α
1
,
α
2
,
α
3
when
x
1
,
x
2
, and
x
3
being
orthonor
mal
. Please refer to any linear algebra textbook or website, or come to the office hours if
you are not familiar with that formula. It might take too much time for you to rederive
existing results.
Question 2:
[Basic] Consider a LTI system with impulse response
h
(
t
) = 3

t
U
(
t
). What
is the output
y
(
t
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 Fall '06
 V."Ragu"Balakrishnan
 Fourier Series, Fourier series representation, Fourier series coefficients

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