Image Processing Reference
In-Depth Information
F, G
=
∞
−∞
F
(
ω
)
∗
G
(
ω
)
dω
(7.13)
We use the inverse FT to transfer the scalar product in the time domain to the
frequency domain:
F
−
1
(
F
)
,
F
−
1
(
G
)
f, g
=
=
[
F
(
ω
) exp(
iωt
)
dω
][
G
(
ω
) exp(
iω
t
)
dω
]
∗
dt
=
F
(
ω
) exp(
iωt
)[
G
(
ω
) exp(
iω
t
)]
∗
dωdω
dt
=
F
(
ω
)
G
(
ω
)
∗
[
iω
t
)
dt
]
dωdω
exp(
iωt
) exp(
−
=
F
(
ω
)
G
(
ω
)
∗
[
ω
)
t
)
dt
]
dωdω
exp(
−
i
(
ω
−
=
F
(
ω
)
G
(
ω
)
∗
2
πδ
(
ω
ω
)
dωdω
−
=2
π
F
(
ω
)
G
(
ω
)
dω
=2
π
F, G
(7.14)
This establishes the
Parseval-Plancherel theorem
for the FT.
Theorem 7.2 (Parseval-Plancherel).
The FT conserves the scalar products:
=
∞
−∞
f
(
t
)
∗
g
(
t
)
dt
=2
π
∞
−∞
F
(
ω
)
∗
G
(
ω
)
dω
=2
π
f, g
F, G
(7.15)
As a consequence of this theorem, we conclude that FT preserves the norms of
functions.
Exercise 7.2.
i) Show that the shifted sinc functions, see Eq. (6.26), that interpolate band-limited
signals constitute an orthonormal set under the scalar product, Eq. (6.29)
HINT: Apply the Parseval-Plancherel theorem to the scalar products.
ii) There are functions that are not band-limited. What does the projection of such
a signal on sinc functions correspond to?
HINT: Decompose the signal into a sum of two components corresponding to
portions of the signal inside and outside of
Ω
.
iii) Can the sinc functions of Eq. (6.26) be used as a basis for square integrable
functions?
HINT: Are the square integrable functions band-limited?
