Image Processing Reference
In-Depth Information
the dIRAC deltA FunCtIon
This function is very convenient and widely used and best understood in
terms of the following short derivation. If
+∞
+∞
+∞
∫ ∫
fx
()
=
Fk e
()
ik x
(
)
d
k
=
fxe
()
ik x
(
)
d
x
e
− (
ik x
) d
k
(A.4)
x
x
x
x
x
x
−∞
−∞
−∞
then this gives us the opportunity to define the delta function by writing
+∞
+∞
+∞
∫ ∫
fx
()
=
fx
()(
δ
xxx
′ −
)
d
′ =
f xe
()
ik x
x
(
)
d
xe
ik x
(
) d
k
(A.5)
x
x
−∞
−∞
−∞
+∞
δ(
xx
′ −=
−∞
)
e
ik xkx
(
′ −
) d
k
(A.6)
x
x
x
Fx
{( )}
δ
δ
()
xe
ix
ω
d
xe
=
i
ω
0
=
1
(A.7)
If f
=
e it
−ω 0 then F
()
ω
=
e
it
ω
×
et
it
ω
d
=
e
−−
i
(
ωω
)
t
d
t
=
δ ωω
(
)
0
0
0
If f
then
= cosω 0
t
1
2
F
()
ω
=
(
e
it
ω
+
eet
it
ω
)
it
ω
d
(A.8)
0
0
−∞
1
2
1
2
=
e
−−
i
(
ωω
)
t
d
t
+
e
−+
i
(
ωω
)
t
d
t
(A.9)
0
0
−∞
−∞
1
2
1
2
(A.10)
=
δω ω
(
)
+
δωω
(
+
)
0
0
examples of Some notations
Using the similarity or scaling theorem, that is,
1
u
a

-
() F.T ¨→
gax
¨¨
G
(A.11)
|
a
|
F.T.
¨¨ ( ) which has equidistant
real zeros at every u / a = m π, where the rect( x ) function is defined in Figure
A.1 and the Fourier transform of this function is illustrated in Figure A.2.
Fourier transform symmetries can be found in Figure A.3.
then, rect()is
ax
1
/
a
units wide
¨→
1
/
a
sinc
ua
/

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