Image Processing Reference
In-Depth Information
Table 7.1. Frequently referenced Fourier transform pairs
Function
FT of function
f
(
t
)
F
(
ω
)
f
(
t
+
t
)
exp(
−it
ω
)
F
(
ω
)
d
f
(
x
)
d
x
iωF
(
ω
)
(
f ∗ g
)(
x
)
2
πF
(
ω
)
G
(
ω
)
1
2
π
δ
(
t
)
P
m
δ
(
t − mT
)
P
n
δ
(
ω − n
2
T
)
2
π
T
(
1
,
if
t ∈
[
−
2
,
2
];
T
2
ω
)
sin(
χ
T
(
t
)=
T
2
π
T
2
ω
)
0
,
otherwise
.
(
√
2
πσ
2
exp(
−
t
2
exp(
−
ω
2
1
2
σ
2
)
2(
σ
)
2
)
The same essential properties hold true for FTs, FCs and DFTs. For this reason
and to keep the clutter of variables and indices to a minimum, it is natural to work
with FTs, at least at the design stage of applications. We will summarize the impor-
tant aspects of FT pair interrelationship by using only the forward and inverse FT
integrals. In table 7.1, we list some commonly used properties of the transform, and
useful FT pairs in signal analysis studies. Here, we recall that lemma 6.1 is conve-
nient to use to obtain some other entries for the table.
Exercise 7.7.
What is the FT of
δ
(
t
−
t
0
)
?
t
)
.
HINT: Study the FT of
f
(
t
−
Exercise 7.8.
What is the FT of
cos(
t
)
?
HINT: Use
cos(
t
)=
2
(exp(
it
)+exp(
−it
))
.
By interpreting the symbols according to the
FT correspondence table
(Table
7.2), the FT pair can represent both variants of the FC transform and the DFT. How-
ever, it should be noted that the table can not be used for translation of FT pairs. For
example, the FT of a Gaussian is a Gaussian, but the FC and DFTs of a Gaussian are
strictly speaking not sampled Gaussians, although for many purposes this is a good
approximation.
Exercise 7.9.
What are the FT, FC and DFT of
cos(
t
)
on the appropriate parts of
the real axis?
HINT: Apply FT, FC, and DFT to translated versions of
δ
while interpreting the
latter as Dirac and Kronecker-
δ
s.
Exercise 7.10.
Plot the FT, FC and DFT of a Gaussian by using a numerical soft-
ware package. How should we proceed to approximate a true Gaussian by the results
better and better?
