Image Processing Reference
In-Depth Information
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
−0.2
−0.2
−pi
−4pi/8
−pi/8
0
pi/8
4pi/8
−64
−56
−48
−40
−32
−24
−16
−8
0
8
16
24
32
40
48
56
Fig. 6.3. The graph shows the finite extension of one domain that matches a particular inter-
polation function in the other domain where the domains are, interchangeably, the time and
the frequency domains. Notice that the interpolator (
sinc function
) crosses zero only at every
eighth integer, while its total extension (in the other domain) is
2
8
•
The function
F
(
ω
) can be represented exactly by means of its samples on an
integer grid, if
f
(
t
) vanishes outside an interval having length
T
=2
π
.
Functions with Finite Frequencies or Band-limited Functions
By using the symmetry of the Fourier transform, we can switch the roles of
F
and
f
while retaining the above results. In other words, we assume
F
to have a limited
extension
Ω
, outside of which it will vanish (and therefore can be periodized). We
will refer to such functions as finite frequency (FF) functions, which are also known
as
band-limited functions
. Consequently, the continuous function
f
(
t
) can be syn-
thesized faithfully by means of its samples. This combination, limited extension
F
in the
ω
-domain and discrete
f
in the
t
-domain, is the most commonly encountered
case in signal analysis applications. Evidently, the roles of
μ
and
χ
are also switched
to the effect that
μ
now interpolates the samples of
f
whereas
χ
defines the extension
of
F
. As discussed previously, the sampling interval of
f
equals 1 if the extension of
F
is
Ω
=2
π
.
Properties of Sinc Functions
We note that the
sinc function
,
sinc(
t
)=
sin
t
t
(6.24)
is a continuous function, even at the origin, where it attains the maximum value
1. In fact, it is not only continuous at the origin, but it is also
analytic
there (and
everywhere), meaning that all orders of its derivatives exist continuously yielding a
“smooth” function. As has been observed already, a scaled version of sinc yields the
interpolation function
μ
