Image Processing Reference
In-Depth Information
6
Fourier Transform—Infinite Extension Functions
In the previous chapter we worked with finite extension functions and arrived at
useful conclusions for signal analysis. In this chapter we will lift the restriction on
finiteness on the functions allowed to be in the Hilbert space. As a result, we hope to
obtain similar tools for a wider category of signals.
6.1 The Fourier Transform (FT)
If
f
is not a finite extension signal, we still would like to express it in terms of the
complex exponentials basis via the synthesis Eq. (5.14), and the analysis Eq. (5.13)
formulas. To find a workaround, we will express a part of such a function in a finite
interval and then let this interval grow beyond all bounds. To be precise, we will
start with the finite interval [
−T/
2
,T/
2], and then let
T
go to infinity (Fig. 6.1)
Synthesizing the function
f
on a finite interval by complex exponentials amounts to
setting the function values to zero outside the interval, knowing that we will only
be able to reconstruct the signal inside of it. Outside of [
T/
2
,T/
2], the synthesis
delivers a repeated version of the (synthesized) function inside the interval. We call
the function restricted to the finite interval
f
T
, to put forward that it is constructed
from a subpart of
f
, and that it converges to
f
when
T
approaches,
−
∞
:
f
= lim
T→∞
f
T
(6.1)
The symmetric interval is a trick that will enable us to control both ends of the in-
tegration domain with a single variable
T
. By sending
T
to infinity both ends of the
integration domain will approach infinity while the function
f
T
will be replaced by
f
.
We first restate our scalar product, Eq. (5.3), for convenience:
=
T/
2
−T/
2
f
∗
g
f, g
(6.2)
The vector space of periodic functions or the space of FE functions having the ex-
tension
T
constitutes a Hilbert space, with the above definition of the scalar product.
