Image Processing Reference
In-Depth Information
Ω
m
1
f
(
m
2
π
Ω
)=
1
m
2
π
Ω
δ
(
t
−
Ω
)
,f
m
δ
t − m
2
π
Ω
F
(
ω
) exp(
itω
)
dω
)
dt
1
Ω
=
m
F
(
ω
)
exp(
itω
)
dt
dω
δ
t
1
Ω
m
2
π
Ω
=
−
m
by changing the order of the integrations and the summation. Because
δ
(
t
)=
δ
(
−
t
),
we can perform a variable substitution in the inner integral,
F
(
ω
)
itω
)
dt
dω
δ
t
exp(
Ω
m
1
f
(
m
2
π
Ω
)=
1
m
2
π
Ω
−
−
Ω
m
(
ω
)
dω
δ
t
1
Ω
m
2
π
Ω
=
F
(
ω
)
F
−
m
δ
t − m
2
π
Ω
1
Ω
F
=
,F
(7.47)
m
and identify it formally as a FT of a Comb. In fact, because
δ
is a distribution and
F
was an arbitrary function, the object
nΩ
)
F
δ
(
ω
−
(7.48)
n
is also a distribution which, thanks to Eqs. (7.44), (7.46), and (7.47),
,F
δ
t
1
Ω
F
m
2
π
Ω
F
(
nΩ
)=
δ
(
ω
−
nΩ
)
,F
=
−
(7.49)
n
n
m
acts in a well-defined manner on arbitrary functions under integration. Accordingly,
the last equation delivers the FT of a Comb, which we state in the following theorem.
Theorem 7.6 (FT of a Comb).
The FT of a Comb distribution yields a Comb distri-
bution
=
Ω
n
δ
t
m
2
π
Ω
F
−
δ
(
ω
−
nΩ
)
(7.50)
m
Up to now, we did not impose any restriction on
F
(
ω
). We now assume that
F
is zero outside of
Ω
2
,
Ω
2
. Then, we can, by applying the right-hand of the theorem,
see that the periodized
F
is obtained as a convolution:
−
