ECE 802-601 Homework 1
Spring 2008
Due January 29, 2008
1. Consider the space of square integrable functions on the interval ],[
π
π
−
, ],[
2
π
π
−L,
and the associated othonormal basis given by ,...2,1,
2
)sin(
,
2
)cos(
,
2
1=
⎭
⎬
⎫
⎩
⎨
⎧n
nxnx
πππ
Consider the following two subspaces, S: space of symmetric functions on ],[
π
π
− and
A: space of antisymmetric functions on ],[
π
π
−
.
a) Show how any function ],[)( 2
π
π
−
∈
Lxf can be written as )()()( xfxfxf as +
=
,
where AxfSxf as ∈∈ )(,)(.
b) Give orthonormal bases for S and A.
2. Let for and
n
ntty =)( K,2,1,0=n11
≤
≤
−
t.
a) Show that
{
is a linearly independent set in
}
K,,, 210 yyy ]1,1[
2
−
L.
b) Write a MATLAB function that implements the Gram-Schmidt orthogonalization
process. Apply it to
{
and plot the first four orthonormal vectors. Given a
finite or countably infinite set of linearly independent vectors , we can construct an
orthonormal set with the same span as follows:
}
K,,, 210 yyy
i
y
i
x
• Start with
0
0
0y
y
x= .
• Then, recursively set
0011
0011
1,
,
xxyy
xxyy
x><−
>
<
−
=
• Therefore, i
k
iikk
kk
kk
kxxyv
vy
vy
x><=
−
−
=∑
−
=
1
0
,,
3. Inner products are preserved by unitary transforms.
a) Prove the general Parseval’s theorem, for any
in a vector space and an orthonormal basis,
*
2121 ,,, >><<>=< ∑ii
i
xyxyyy
21 ,yy
{
}
K,,, 321 xxx .
b) Using the result in part (a), show that
c) For )/(
)/sin(
)( Tt
Tt
thT
π
π
=, prove that
{
}
Zn
TnTth ∈
−
)( is an orthogonal basis of the space
U of functions whose Fourier transforms are bandlimited in ]/,/[ TT
π
π
−
.
