Digital Signal Processing Reference
In-Depth Information
Substitution of Equation (14.4) into Equation (14.2) gives
∞
x
(
t
)
=
(
c
i
+
c
1
h
i
)
φ
i
(
t
)
,
i
=
2
which means that it is possible to express the function
x
(
t
) by all other functions
of the corresponding system
. Hence this function,
when it linearly depends on the other functions of the system
Φ
without the function
{
φ
i
(
t
)
}
Φ
, is redundant.
It seems that no more than these two conditions have to be satisfied for a correct
application of the signal transforms discussed here. The discussion of when and
why the orthogonality of the basis functions is required follows.
14.2.2 Transforms by Means of a Finite Number of
Basis Functions
According to Equation (14.2), the signal
x
(
t
) is in general represented by an
infinite series. In practice, the number of terms of such a series is always finite.
Therefore, it is more appropriate to talk about approximating the signal. To do
this, the series
m
x
∗
(
t
)
=
φ
c
i
i
(
t
)
(14.5)
i
=
1
has to be constructed, so that
x
∗
(
t
) approximates
x
(
t
) sufficiently closely. The
least squares approximation error can be used as a criterion for evaluating this
closeness.
Now the approximation task both for analog and digital signals will be consid-
ered.
Analog Processing
The coefficients
for the series (14.5) can be determined by solving the fol-
lowing minimization task:
Θ
{
c
i
}
x
(
t
)
2
m
c
i
φ
i
(
t
)
−
d
t
=
min
.
(14.6)
0
i
=
1
To find the minimum of the integral (14.6), all the individual derivatives of
{
c
j
}
should be considered as being equal to zero. Then
2
Θ
0
x
(
t
)
m
c
1
φ
i
(
t
)
−
φ
j
(
t
)d
t
=
0
for
j
=
1
,
m
.
i
=
1
