Digital Signal Processing Reference
In-Depth Information
the corresponding system of functions is described as orthonormalized. In the
case of digitized signals, conditions (14.17) become
α
N
>
0
for
i
=
j
,
ii
1
φ
φ
=
i
(
t
k
)
j
(
t
k
)d
t
(14.18)
0
for
i
=
j
.
k
=
Now consider solving the equation system (14.18) in the case when the basis
functions are orthogonal. Then
α
2
=
φ
i
(
t
)
>
0
for
j
=
1
,
ii
α
=
(14.19)
ji
0
for
j
=
i
,
and the equation system in question is reduced to the following system of equal-
ities:
1
φ
1
(
t
)
c
1
=
2
c
1
1
φ
2
(
t
)
c
2
=
2
c
2
(14.20)
.......................
c
m
=
1
2
c
m
.
φ
m
(
t
)
Thus we come to the conclusion that the orthogonal basis functions are a particular
case of linearly independent basis functions and their application considerably
simplifies the corresponding signal transforms. The coefficients
calculated in
the course of the orthogonal transforms can be obtained directly without solving
the equation system under consideration.
{
c
i
}
14.3.1 Analog Processing
It follows from Equations (14.10) and (14.20) that the coefficients
{
c
i
}
in the case
of the orthogonal transforms are given as
Θ
1
c
i
=
x
(
t
)
φ
i
(
t
)d
t
,
(14.21)
2
φ
i
(
t
)
0
where the norm
Θ
i
(
t
)
1
/
2
2
φ
i
(
t
)
=
φ
d
t
,
i
=
1
,
m
.
0
