Homework 6 (ECE220  Fall 2007)
•
Due:
Wednesday, October 24 by Noon in the dropbox outside 219 Phillips Hall.
•
Reasoning and work must be shown to gain full/partial credit.
•
WRITE YOUR NAME AND NET ID ON ALL PAGES HANDED IN!
1.
(40 points) Fourier Series for analog signal
An analog image is a 2D signal which has
limited support in the independent variables
I
(
x, y
). Let
I
(
x, y
) be defined over a square region

x

<
L
2
and

y

<
L
2
.
For any given point y we can find the Fourier series of
I
(
x, y
) pretending that the image
periodically repeats itself outside the period. We can do the same fixing x and pretending the
image repeats itself periodically.
(a) (5 points) Using this argument extend the concept of Fourier series in 2D an indicate
under what conditions we show that an image can be represented as:
I
(
x, y
) =
∞
X
k
1
=
∞
∞
X
k
2
=
∞
a
k
1
,k
2
e
j
2
π
L
k
1
x
e
j
2
π
L
k
2
y
Holds in the sense that the error:
e
N
(
x, y
) =
I
(
x, y
)

N
X
k
1
=

N
N
X
k
2
=

N
a
k
1
,k
2
e
j
2
π
L
k
1
x
e
j
2
π
L
k
2
y
is such that:
lim
N
→∞
Z
L
2

L
2
Z
L
2

L
2

e
N
(
x, y
)

2
dxdy
= 0
(b) (10 points) Suppose that the white level is 1 and the black level is 0. Calculate the Fourier
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 Fall '05
 JOHNSON
 Fourier Series, Gibbs, 2 L, 1D, 8 bits, 2 L

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