1
EE 211A
Digital Image Processing I
Fall Quarter, 2011
Handout 10
Instructor: John Villasenor
Homework 2 Solutions
Problems:
1.
Textbook Problem 2.4a, b, and f. Let
)
,
(
1
n
m
y
and
)
,
(
2
n
m
y
be the outputs
corresponding to the inputs
)
,
(
1
n
m
x
and
).
,
(
2
n
m
x
(a)
9
)
,
(
3
)
,
(
+
=
n
m
x
n
m
y
)
,
(
9
)
,
(
3
)]
,
(
[
1
1
1
n
m
y
n
m
x
n
m
x
T
=
+
=
)
,
(
9
)
,
(
3
)]
,
(
[
2
2
2
n
m
y
n
m
x
n
m
x
T
=
+
=
)
,
(
9
))
,
(
)
,
(
(
3
]
[
3
2
1
2
1
n
m
y
n
m
bx
n
m
ax
bx
ax
T
=
+
+
=
+
Since
2
1
3
by
ay
y
+
≠
, the system is non-linear.
)
,
(
9
)
,
(
3
)]
,
(
[
4
1
1
n
m
y
n
n
m
m
x
n
n
m
m
x
T
=
+
ʹ′
−
ʹ′
−
=
ʹ′
−
ʹ′
−
Since
)
,
(
)
,
(
1
4
n
n
m
m
y
n
m
y
ʹ′
−
ʹ′
−
=
, the system is shift-invariant.
IR =
T
9
)
,
(
3
)]
,
(
[
+
−
−
=
−
−
b
n
a
m
b
n
a
m
δ
δ
The system is IIR, since the IR is non-zero for all
m
,
n
.
System (a) is non-linear, shift-invariant, and IIR.
(b)
).
,
(
)
,
(
2
2
n
m
x
n
m
n
m
y
=
)
,
(
)
,
(
)]
,
(
[
1
1
2
2
1
n
m
y
n
m
x
n
m
n
m
x
T
=
=
)
,
(
)
,
(
)]
,
(
[
2
2
2
2
2
n
m
y
n
m
x
n
m
n
m
x
T
=
=
)
,
(
))
,
(
)
,
(
(
]
[
3
2
1
2
2
2
1
n
m
y
n
m
bx
n
m
ax
n
m
bx
ax
T
=
+
=
+
Since
2
1
3
by
ay
y
+
=
, the system is linear.
)
,
(
)
,
(
)]
,
(
[
4
1
2
2
1
n
m
y
n
n
m
m
x
n
m
n
n
m
m
x
T
=
ʹ′
−
ʹ′
−
=
ʹ′
−
ʹ′
−
Since
)
,
(
)
,
(
1
4
n
n
m
m
y
n
m
y
ʹ′