Image Processing Reference
In-Depth Information
f
T
→
f
−T/2
T/2
←
T
→∞
→
Fig. 6.1.
The graph drawn
solid
illustrates
f
T
.As
T
grows beyond every bound,
f
T
will
include the
dashed graph
and will equal
f
Exercise 6.1.
Show that (i) complex exponentials are orthogonal under any scalar
product taken over an interval with the length of the basic period. (ii) The norms of
the complex exponentials are not affected by the shift of the interval over which the
scalar product is taken. (iii) Show the results in (i) and (ii) for 2D functions, i.e.,
C
exp(
iω
x
x
+
iω
y
y
)
.
Remembering that
ω
m−
1
=
m
2
π
1)
2
π
T
=
2
π
T
ω
m
−
T
−
(
m
−
(6.3)
we restate the synthesis formula of Eq. (5.14) as
f
T
(
t
)=
m
F
(
ω
m
) exp(
iω
m
t
)(
ω
m
− ω
m−
1
)
(6.4)
and the analysis formula of Eq. (5.13) as
T/
2
1
2
π
F
(
ω
m
)=
f
T
(
t
) exp(
−
iω
m
t
)
dt
(6.5)
−T/
2
ω
m−
1
=
2
T
, which is the
distance between two subsequent elements of the equidistant
discrete grid
:
Passing to the limit with
T
→∞
, we observe that
ω
m
−
···
ω
−
2
,ω
−
1
,ω
0
,ω
1
,ω
2
···
,
(6.6)
approaches to zero,
2
π
T
=
ω
m
−
ω
m−
1
=
Δω
→
0
(6.7)
