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Material Type: Notes; Professor: Karam; Class: Multidimension Signal Process; Subject: Electrical Engineering; University: Arizona State University - Tempe; Term: Spring 2004;
Typology: Study notes
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-^
A 2D LSI system is separable if its impulse response is a separablesequencesequence
h
( n
n
h
n
h
n
-^
If LSI system is separable, we can implement it more efficiently
y
p
p
y
−^
−
1 0
1 0
2 1 2 2 1 1 2 1
N k
N k
=^
= 0
0
1
2
k^
k
If
h
( n
, 1 n
) is NxN, we need 2
N
2
multiplications and
N
2 -1 adds for each
computed output sample
y
( n
, 1 n
) for a general 2D LSI system. 2
Copyright 2004 by Prof. Lina Karam
EEE 507 - Lecture 2
1
¾
If
h
( n
, 1 n
) separable: 2
∑∑
− =
− =
1 0
1 0
2 2 1 1 2 2 1 1 2 1
1
2
N k
N k
−^
−
1
1
N^
N
∑
∑
=^
=
0
2 1
1
0
2 2 2 2 1 1 1 1 1
2
k^
k
n k
n g
k h k n k n x k h 4 4 4 4
-^
1D
convolution in
n
2
direction
∑
− =
1 0
2 1 1 1 1 1
N k
n k n g k h
-^
1D
convolution in
n
1
direction
¾
Then
) ( ) , ( ) ,
(^
2 2 2
2
1 0
1
2 1
2
k h k n n x n n g
N k^
−
=
∑
− =
∑
−
−
=
1
1 1 2 1 1 2 1 ) ( ) , ( ) , (
N
k h n k n g n n y ¾
Then
Copyright 2004 by Prof. Lina Karam
2
∑
=^01 k
Separable SystemsSeparable
Systems
Theorem:
then ),
( ) ( ) , (
and ) ( ) ( ) , (
If
2 2 1 1 2 1 2 2 1 1 2 1 n h n h n n h n x n x n n x
=
=
) , ( ) , ( ) , ( 2 1 2 1 2 1 n n h n n x n n y
∗ ∗
=
)
(
)
(
2
2
1
1
n y n y = ) , ( ) , ( ) , (
2 1 2 1 2 1 n n h n n x n n y
.)
(
)
(
)
(
and ) ( ) ( ) (
where
2 2 2 2 2 1 1 1 1 1 n h n x n y n h n x n y
∗
=
∗
=
)
(
)
(
2
2
1
1
n
y
n
y^ Copyright 2004 by Prof. Lina Karam
4
Computational SavingsComputational
Savings
Assume:
x: M×Mh: N×N
(M>N)
h: N×N
(M>N)
x
(M-point) with 1-D
h
(N-point)
Multiplies (Mults)
ith 2 D
h
2
M lt
x
with 2-D
h
2
Mults
x
with separable
h
Mults
2-D separable
x
with separable
h
2
Mults
Numerical Illustration: M=8; N=
576 Mults
312 Mults => 46% less
58 Mults
Copyright 2004 by Prof. Lina Karam
5
2D Discrete
Domain Systems
2D
Discrete-Domain Systems
-^
Auxiliary Definitions:
S pport ¾
Support
A sequence
x(n
,n 1
has support on R if
x(n
,n 1
for
(n
,n 1
A system has support on
if its impulse response has support on
Copyright 2004 by Prof. Lina Karam
EEE 507 - Lecture 2
7
Frequency Response of 2
D Systems
Frequency
Response of 2-D Systems
)
(^
n
n
j
ω
ω
)
(^
2 2
1 1
n
n
j e
ω
ω
)
,
(^
2
1
n
n h
= )
, (^
2
1
n
n y Copyright 2004 by Prof. Lina Karam
8
Frequency Response of 2-D Systems
q
y
p
y
-^
Example
n^2
h(n
,n 1
) 2
n^1
Question: Is this a separable system?Note: If a sequence is separable each row is a multiple of everyNote: If a sequence is separable, each row is a multiple of everyother row and each column is a multiple of every other column.
Copyright 2004 by Prof. Lina Karam
10
Frequency Response of 2
D Systems
Frequency
Response of 2-D Systems
-^
Example
n^2
h(n
,n 1
) 2
n^1
Determine the system frequency response:
∑
∑
j
j
∑
∑
−
−
=
1
2
2 2
1 1
)
,
(
)
,
(^
2
1
2
1
n^
n
jn
jn
n n h e e H
ω
ω
?
)
,
(^
2
1
=
n
n h
)
,
(^
2
1
Copyright 2004 by Prof. Lina Karam
11
Frequency Response of 2
D Systems
Frequency
Response of 2-D Systems
-^
ExampleDetermine the system frequency response:
Note:
∑
∑
δ
∑
δ
Copyright 2004 by Prof. Lina Karam
13
2
1
2 1
1
2
n^
n
∑
∑
δ
Frequency Response of 2
D Systems
Frequency
Response of 2-D Systems
-^
ExampleDetermine the system frequency response:
Copyright 2004 by Prof. Lina Karam
14
The 2-D Fourier Transform•^
The Fourier Transform of a sequence x(n
,n 1
) is defined as: 2
) ( 2 1 2 1 2 2 1 1
1
2
)
,
(
)
,
(^
n
n j
n^
n
e
n
n x
X
ω
ω
−
Note: The 2D Fourier Transform is periodic with period 2
π
in
both
ω
1
and
ω
. 2 -^
The 2D inverse Fourier Transform is given by:
g
y
2 1 ) ( 2 1 2 2 1
ω
ω
ω
ω
π
ω
ω
π^
π
n
n j^
∫ ∫
-^
Note:
The frequency response is the Fourier transform of the
impulse response.
π
π^
π
−^
−
Copyright 2004 by Prof. Lina Karam
16
2-D Fourier Transform
⎧
≤
≤
b
1
Example:
⎧ ⎨ ⎩
<
≤
<
≤
=
otherwise
b
a
H
, 0
,
, 1
)
,
(^
2
1
2
1
π ω π ω ω ω
ω
2
b
π
ω
1
a
π
-^
π
-a
-b
-
π
Copyright 2004 by Prof. Lina Karam
17
2
D Fourier Transform
2
D
Fourier Transform
⎧ ⎨
<
≤
<
≤
,
, 1
)
(^
2
1
π ω π ω ω ω
b
a
H Example:
otherwise
, 0
)
, (^
2
1
ω
ω H
ω
2
b
π
ω
1
a
π
-^
π
-a
-b
-
π
Copyright 2004 by Prof. Lina Karam
19
M
D Fourier Transform
M
D
Fourier Transform
-^
M-D Fourier Transform of a sequence x(n
,n 1
,…,n 2
M
):
) ( 2 1 2 1
1 1
1
2
) , , , ( ) , , , (
M M
M
n
n j
n^
n
M
n
M
e n n n x X
ω
ω
ω
ω
ω
K
K
K
K
−
∑
∑
∑
=
-^
Inverse M-D Fourier Transform:
1
2
M
π^
π
1
M
n
n j
M
egral
M M
M
d d d e X n n n x
M
ω ω ω ω ω ω π
ω
ω
ω
π^
π
K
3 2 1
K
K
K
2 1 ) ( 2 1
int
2 1
2 2 1 1
)
,...,
,
(
) (^2) (
1
)
,
, , (^
−^
−
∫^
∫
=
Copyright 2004 by Prof. Lina Karam
20#
20
