Image Processing Reference
In-Depth Information
5
Finite Extension or Periodic Functions—Fourier
Coefficients
In this section we will make use of Hilbert spaces being a vector space in which a
scalar product is defined. We will do this for a class of functions that are powerful
enough to include physically realizable signals. The scalar product for this space will
be made precise and utilized along with an orthogonal basis to represent the member
signals of the Hilbert space by means of a discrete set of coefficients.
5.1 The Finite Extension Functions Versus Periodic Functions
We define the
finite extension functions
and the
periodic functions
as follows:
Definition 5.1.
A function
f
is a finite extension FE function if there exists a finite
real constant
T
such that
T
2
,
T
t/
∈
[
−
2
]
⇒
f
(
t
)=0
(5.1)
A function
f
is a periodic function if there is a positive constant
T
, called a period,
such that
f
(
x
)=
f
(
x
+
nT
)
for all integers
n
.
The definitions for higher dimensions are analogous. Examples of finite extension
signals include (i) a sound recording in which there is a time when the signal starts
and another when it stops; (ii) a video recording which has, additionally, a space lim-
itation representing the screen size; (iii) an (analog) photograph, which has a bound-
ary outside of which there is no image. A periodic signal is a signal, that repeats
copies of itself in a basic interval that is given by a fixed interval
T
. Because of this
all real signals that are
finite extension signals
can be considered to be equivalent to
periodic signals
as well. The FE signals can be repeated to yield a periodic function,
and periodic functions can be truncated such that all periods but one are set to zero.
Accordingly, a study of periodic functions is also a study of FE functions, and vice
versa. In this chapter we will use
f
to mean either finite extension or periodic.
Arguments of periodic functions are often measured or labelled in angles because
after one period the function is back at the same point where it started, like in the
