Digital Signal Processing Reference
In-Depth Information
6.2
Problem Solving
Exercise 1: Solve the following problems, briefly outlining the important
steps.
a.
Sample the following 2-d continuous functions
f
(
x,y
) (in the interval
given) to obtain 2-d discrete functions
f
(
n
1
,n
2
) in the form of 4
×
4
matrices. Sketch the sampled 2-d functions.
i.
f
(
x,y
) =
rect
(
x
/2,
y
/2), in the interval -1
≤
η≤
1, -1
≤
y
≤
1,
⎧
⎨
1
,
−≤ ≤
axa bxb
;
−≤ ≤
⎪
⎪
(
)
=
where
rect x a y b
,
0otherw se
,
ii.
f
(
x,y
) =
sin
(
π
x
/4)
sin
(
π
y
/4), in the interval 0
≤
x
≤
8, 0
≤
y
≤
8.
b.
Find the 2-d DTFT
F
(
ω
1
,
ω
2
) of the following 2-d discrete functions
f
(
n
1
,n
2
):
i.
f
(
n
1
,
n
2
) =
δ
(
n
1
- 2,
n
2
- 2) +
δ
(
n
1
- 1,
n
2
- 3) +
δ
(
n
1
- 3,
n
2
- 1)
ii.
(
n
1
- 1)
u
(
n
1
,
n
2
)
iii.
f
(
n
1
,
n
2
) =
e
-(
n
1
+
n
2
)
u
(
n
1
,
n
2
)
f
(
n
1
,
n
2
) =
δ
c.
The block diagram of an LSI system is given in Figure 6.4:
where
x
(
n
1
,
n
2
) = (0.5)
n
1
(0.25)
n
2
u
(
n
1
,
n
2
)
h
(
n
1
,
n
2
) =
δ
(
n
1
,
n
2
) +
δ
(
n
1
- 1,
n
2
) +
δ
(
n
1
,
n
2
- 1) +
δ
(
n
1
- 1,
n
2
- 1)
i.
Determine the 2-d Fourier transform
H
(
ω
1
,
ω
2
) of the system im-
pulse response
h
(
n
1
, n
2
).
ii.
Determine the 2-d Fourier transform
Y
(
ω
1
,
ω
2
), and hence, deter-
mine the output
y
(
n
1
, n
2
).
x(n
1
,n
2
)
h(n
1
,n
2
)
y(n
1
,n
2
)
FIGURE 6.4
Figure for problem (c).
