Image Processing Reference
In-Depth Information
T/ 2
μ ( ω )= 1
T
exp(
iωt ) dt
(6.16)
−T/ 2
Clearly F ( ω ) is reconstructed from its samples F ( ω m ) by using these as weight
functions for a series of a displaced versions of μ ,the interpolation function . We can
identify μ in Eq. (6.16) as the Fourier transform of a piecewise constant function
χ T ( t ), which is constant inside the interval where f is nonzero, and zero elsewhere.
χ T ( t )= 1 ,
T
2
, T
2
if t
[
];
(6.17)
0 ,
otherwise .
T
2
, T
2
We will refer to χ T as the characteristic function of the interval [
]. The rela-
tionship in Eq. (6.16) along with the definition in Eq. (6.17) reveals that
μ ( ω )= 1
T
iωt ) dt = 2 π
χ T ( t ) exp(
T F
( χ T )( ω )
(6.18)
−∞
Computing the integral in Eq. (6.16) in a straightforward manner yields
exp(
t = T/ 2
μ ( ω )= 1
T
iωt )
(6.19)
t = −T/ 2
1
T
ω 2 sin( ω T
1
=
2 )
(6.20)
In consequence, defining the sinc function as
sinc( ω )= sin ω
ω
(6.21)
yields the interpolator function
μ ( ω )=sinc T
2 ω
(6.22)
which, using Eq. (6.18), results in
2 π sinc T
2 ω
( χ T )( ω )= T
F
(6.23)
A characteristic function and its corresponding interpolation function is illustrated in
Fig. 6.3. Before we discuss the properties of the sinc functions further, we present a
few observations and elucidate the concept of functions with finite frequency exten-
sion, also known as band-limited functions.
ω m− 1 = 2 T , so that if the distance
between the samples defines the unit measurement, ω m
The sampling interval for F is given by ω m
ω m− 1 =1, then T 's
value is locked to T =2 π . If another measurement unit is used to quantify the
length of the interval that samples F , alternatively the length of the period of f ,
an ordinary scaling of the arguments will restore the correspondence. Because of
this, it is customary to assume that a finite extension 1D function is nonzero in
[
π, π ], and the sampling interval of its FT is equal to unit length.
 
Search WWH ::




Custom Search