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
=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
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