Digital Signal Processing Reference
In-Depth Information
2-d discrete-
space input
f(n
1
,n
2
)
2-d discrete-
space system
2-d discrete-
space output
g(n
1
,n
2
)
FIGURE 6.2
Two-dimensional discrete space system.
(
)
=
(
)
⎣
⎦
(6.4)
gn n
,
T f n n
,
12
12
where
T
is the system operator.
If the system is
LSI
(
Linear Shift Invariant
), then we have the 2-d convolution
relation:
∑
∑
(
)
=
(
)
(
)
(6.5)
gn n
,
f k k hn
,
−
k n
,
−
k
12
12
1
12
2
k
k
1
2
or
(
)
=
(
)
(
)
(6.6)
gn n
,
f n n
,
* *
hn n
,
12
12
12
where the symbol ** represents the 2-d discrete convolution, and
h
(
n
1
, n
)
is
2
the 2-d impulse response of the system.
The 2-d impulse response is defined as the output of the system, when the
input
f
(
n
1
, n
) =
δ
(
n
1
, n
), the 2-d impulse function, defined in Equation 6.2.
2
2
Two-dimensional Discrete-Time Fourier Transform (2-d DTFT)
The 2-d DTFT of a 2-d discrete function
f
(
n
1
, n
) is defined as:
2
∑
∑
(
)
=
(
)
−
jn
ω
−
j n
ω
(6.7)
F
ωω
,
f
n
,
n
e
e
11
2 2
12
12
n
n
1
2
Likewise, the 2-d inverse DTFT is given by the following equation:
π
π
∫
1
(
)
=
(
)
−π
jnj n
ωω
(6.8)
fnn
,
F
ωω
,
e
e
d
ω ω
d
11
2 2
12
12
1
2
2
4
π
−
π
Note:
The 2-d Fourier Transform
F
(
ω
1
,
ω
2
) is periodic in both variables
ω
1
and
ω
2
with period 2
π
radians. This implies that
(
)
=
(
)
(6.9)
F
ωπ ω π
±
k
2
,
±
k
2
F
ωω
,
; for integer
k
and
k
2
.
1
1
2
2
1
2
1
