Image Processing Reference
In-Depth Information
function remains 1, and this maximum is attained at the origin for all
T
s. However,
when
T
increases, the values of the functions
{
B
T
(
ω
)
}
T
at
ω
=0increase with
T
,
since sinc(
T
0) = 1. As a result,
B
T
(
ω
) approaches infinity at
ω
=0, whereas the
effective width of the
B
T
functions shrink to zero. Thus, the
B
T
(
ω
) approaches to
something which is zero everywhere except at the origin and yet has an area which
is always 1!
That something is the so-called
Dirac distribution
represented by
δ
·
1
2
π
T
sinc(
ω
T
lim
T→∞
B
T
(
ω
) = lim
T→∞
·
2
)=
δ
(
ω
)
(7.6)
which is shown in Fig. 7.1. Other names for the Dirac distribution are the
delta
function
,
unit impulse
, and the
Dirac function
, although,
δ
(
ω
) is not very useful when
considered strictly as a function. This is because the
δ
(
ω
) interpreted as a function,
will only deliver the value zero, except at the origin, where it will not even deliver a
finite value, but delivers the value “infinity”.
In mathematics, the limit of the
B
T
function is not defined in terms of what it
does to the points
ω
, its argument, but instead what it does to a class of other func-
tions under integration. That is also where its utility arises, namely, it consistently
delivers a value as a result of the integration. Such objects are called generalized
functions, or distributions. To obtain a hint on how we should define the limit of
B
T
, we should thus study the behavior of
B
T
when it is integrated with arbitrary
(integrable) functions
f
:
=
B
T
,f
B
T
f
(
ω
)
dω
=
(
1
2
π
χ
T
(
t
) exp(
−
iωt
)
dt
)
f
(
ω
)
dω
=
(
1
2
π
exp(
−
iωt
)
f
(
ω
)
dω
)
χ
T
(
t
)
dt
=
F
(
t
)
χ
T
(
t
)
dt
Thus, in the limit we will obtain
=
T→∞
lim
B
T
,f
F
(
t
)
dt
=
f
(0)
(7.7)
because
χ
T
approaches to the constant function 1 with increasing
T
. Also, changing
the order of the integration and the limit operator is not a problem for physically
realizable functions. This shows that as
T
grows beyond any bound,
B
T
will consis-
tently “kick out” the value of its fellow integrand
f
at origin, no matter the choice of
f
. Therefore, we give the following precision to
δ
, the Dirac distribution.
Definition 7.1.
The Dirac-
δ
distribution is defined as
=
δ, f
f
(
ω
)
δ
(
ω
)
dω
=
f
(0)
(7.8)
