Digital Signal Processing Reference
In-Depth Information
TABLE 6.1
2-d DTFT Theorems
Property
f
(
n
1
,
n
2
)
F
(
ω
1
,
ω
2
)
Convolution
f
(
n
1
,
n
2
) **
h
(
n
1
,
n
2
)
F
(
ω
1
,
ω
2
)
H
(
ω
1
,
ω
2
)
Spatial shift
f
(
n
1
-
m
1
,
n
2
-
m
2
)
F
(
ω
1
,
ω
2
)
e
-j
ω
1
m
1
e
-j
ω
2
m
2
Spatial reflection
f
(-
n
1
,
n
2
)
F
(-
ω
1
,
ω
2
)
Some of the other important 2-d DTFT properties are given in Table 6.1.
2-d Discrete Fourier Transform (2-d DFT)
In Section 3.1.2, we discussed the one-dimensional Discrete Fourier Trans-
form (DFT) as the practical extension of the 1-d DTFT. Similarly, we can
extend the DFT concept to the 2-d DTFT,
F
(
ω
1
,
ω
)
,
of the 2-d spatial signal
2
f
(
n
1
, n
). Because the 2-d DTFT
F
(
ω
1
,
ω
)
is periodic in both variables
ω
and
2
2
1
ω
with period 2
π
radians, this property is used to the divide the frequency
2
interval (0, 2
) equally spaced points. This
discretization yields the 2-d Discrete Fourier Transform (2-d DFT) of the 2-d
spatial signal
f
π
) into
N
(for
ω
) and
N
(for
ω
1
1
2
2
(
n
1
, n
) as follows:
2
N
−
1
N
−
1
1
2
∑
∑
−
j
2
π
nkN
−
j
2
π
nkN
Fk k
(, )
=
F
( , )
ωω
ωπ
ωπ
=
f n n
(,
)
e
e
(6.10)
11
1
2 2
2
=
=
2
2
kN
kN
12
1 2
12
1
1
1
2
2
2
n
=
0
n
=
0
1
2
for
k
=
012
,,
…−
N
1
;
1
1
k
=
012
,,
…
N
−
1
;
2
2
Similarly the inverse 2-d DFT is defined as follows:
N
−
1
N
−
1
1
2
1
∑
∑
(
)
=
(
)
j
2
π
nkN j
2
π
nkN
fnn
,
Fk k e
,
e
(6.11)
11
1
2 2
2
12
12
NN
12
k
=
0
k
=
0
1
2
for
n
=
012
,,
…
N
−
1
;
1
1
n
=
012
,,
…
N
−
1
;
2
2
. Additionally, as seen
from Equation 6.10 and Equation 6.11, the 2-d DFT and 2-d IDFT are both
finite sums, making it very convenient to program in computers and micro-
processors.
Both
f
(
n
1
, n
)
and
F
(
k
1
, k
)
are matrices of size
N
×
N
2
2
1
2
