Image Processing Reference
In-Depth Information
nΩ
+
Ω
2
f
(
t
m
)=
n
F
(
ω
) exp(
it
m
(
ω
−
nΩ
))
dω
Ω
2
nΩ−
nΩ
+
Ω
2
it
m
nΩ
)
n
=exp(
−
F
(
ω
) exp(
it
m
ω
)
dω
Ω
2
nΩ−
imn
2
π
)
∞
−∞
=exp(
−
F
(
ω
) exp(
it
m
ω
)
dω
=
f
(
t
m
)
(7.41)
Thus,
F
(0) according to the formula in Eq. (7.40) yields
Ω
m
1
f
(
m
2
π
F
(0) =
Ω
)
(7.42)
On the other hand, using the definition of
F
(
ω
) in Eq. (7.39) we obtain the same
F
(0) as
F
(0) =
n
F
(
nΩ
)
(7.43)
so that we can write the
Poisson summation
as a theorem.
Theorem 7.5 (Poisson summation).
If
f
and
F
are a FT pair, then their sample
sums fulfill
Ω
m
F
(
nΩ
)=
1
f
(
m
2
π
Ω
)
(7.44)
n
The power of this formula is appreciable because we did not require
F
to have finite
frequencies nor
f
to have a finite extension. It says that if we know an FT pair, then
we can deduce their discrete sums from one another.
The
Comb distribution
is defined as a train of
δ
distributions:
Comb
T
(
t
)=
n
δ
(
t
−
nT
)
(7.45)
It is a convenient analytic tool when sampling physical functions, as it relates sam-
pling in one domain with the periodization in the other domain, elegantly. However,
we need to derive its behavior under the FT before exploiting it.
The left side of Eq. (7.44) is
F
(
nΩ
)=
δ
(
ω
−
nΩ
)
,F
(7.46)
n
n
whereas the right side of Eq. (7.44) yields
