Image Processing Reference
In-Depth Information
T=0.5
T=1.2
T=16
Fig. 7.1.
The sinc function sequence
B
T
given by Eq. (7.3) for some
T
values. In
blue
the
(capped) function indicates that
B
T
approaches the Dirac distribution with “
∞
value” at the
origin
2
π
sinc
ω
T
dω
=1
T
(7.2)
2
Obviously, the “area under the curve”,
2
π
sinc
ω
T
B
T
(
ω
)=
T
(7.3)
2
is 1, independent of
T
, as long as
T
is finite! In fact for every
T
we have a different
function
B
T
, so that we can fix a sequence of
T
s and study
B
T
. This is what we will
do next.
The
B
T
s were obtained by Fourier transforming a function that is constant in
a finite and symmetric interval [
−
T
2
,
T
2
]. What happens to sinc when
T
approaches
infinity? In other words, what is the Fourier transform of an “eternal” constant? We
restate Eq. (6.23) and study it when
T
increases.
sinc
T
ω
2
1
2
π
T
F
(
χ
T
)(
ω
)=
B
T
(
ω
)=
·
(7.4)
Any function contracts when we multiply its argument with a large constant
T
. The
function
sinc(
T
ω
2
)
(7.5)
is accordingly a contracted version of sinc(
ω
) as
T
increases.
The sinc function is a smooth function that is both continuous and has continuous
derivatives everywhere, including at the origin. When we increase
T
, the sinc func-
tion contracts in the horizontal direction only. In particular, the maximum of the sinc