Image Processing Reference
In-Depth Information
This equation, together with Eq. (6.39), establishes
f
and
F
as a discrete transform
pair when sampled on the grids of
t
m
and
ω
n
, respectively. We state the result as a
lemma.
Lemma 6.2.
Let
f
(
t
)
be a finite extension function with the extension
T
and have
N
nonzero FCs. The samples of
f
N−
1
2
π
T
t
n
=
n
T
f
(
t
n
)=
F
(
ω
m
) exp(
iω
m
t
n
)
,
with
N
,
(6.45)
m
=0
and the samples of its FT,
N−
1
F
(
ω
m
)=(
N
2
π
T
ω
m
=
m
2
π
T
)
−
1
f
(
t
n
) exp(
−iω
m
t
n
)
,
with
,
(6.46)
n
=0
constitute a discrete transform pair.
Using an analogous reasoning and theorem 6.2 we can restate this result for band-
limited functions.
Lemma 6.3.
Let
f
(
t
)
be a band-limited function, i.e.,
F
(
ω
)
has the finite extension
Ω
, that has at most
N
nonzero samples
f
(
t
n
)
with
t
n
=
n
2
Ω
. The samples of
f
N−
1
2
π
TN
F
(
ω
m
) exp(
iω
m
t
n
)
,
with
t
n
=
n
2
π
f
(
t
n
)=
Ω
,
(6.47)
m
=0
and the samples of its FT,
N−
1
F
(
ω
m
)=(
2
π
T
ω
m
=
m
Ω
)
−
1
f
(
t
n
) exp(
−
iω
m
t
n
)
,
with
N
,
(6.48)
n
=0
constitute a discrete transform pair.
The two lemmas can be simplified further because we are free to define
T
=
2
π
,or
Ω
=2
π
. In either case, the transform pair can be interpreted as a finite
discrete sequence, even without reference to the sampling distance between
t
and
ω
variables. Accordingly, both lemmas can be reduced to a dimensionless form that
only differs with where we place the transform constant 1
/N
. We choose to reduce
the first lemma and give it as a theorem.
Theorem 6.4 (DFT).
The discrete Fourier transform (DFT) for arrays with
N
ele-
ments, defined as
f
(
m
)exp
N−
1
F
(
n
)=
1
N
imn
2
π
N
−
(6.49)
m
=0
