Biomedical Engineering Reference
In-Depth Information
4.4.4 Hermite Functions
Hermite functions
(4.43) are orthonormal over the range from
−∞
to
∞
.
(2
n
n
!
√
π
t
2
2
)
−1
/
2
e
−
n
(
t
)
=
H
n
(
t
)
1)
n
e
t
2
d
n
dt
n
(
e
−
t
2
)
H
n
(
t
)
=
(
−
(4.43)
Cardinal functions
are given by (4.44).
sin
π
(2
wt
−
n
)
n
(
t
)
=
for
n
=
0
,
±
1
,
±
2
,...
(4.44)
π
(2
wt
−
n
)
Another property of orthogonal functions has to do with the accuracy of the
representation when not all of the terms can be used for an exact representation. In most
cases,
N
is infinity for exact representation, but the coefficient,
a
n
, becomes smaller as
n
increases.
The approximation or estimation of
x
(
t
) is expressed as
x
(
t
) as in (4.45).
M
x
(
t
)
a
n
=
n
(
t
)
,
where
M
is finite
.
(4.45)
n
=
0
A measure of the closeness or goodness of approximation is the integral squared error
(4.46).
t
2
x
(
t
)]
2
dt
0 when
x
(
t
)
I
=
[
x
(
t
)
−
and
I
=
=
x
(
t
)
(4.46)
t
1
The smaller the value of
I
, the better the approximation. The integral squared
error is applicable
only
when
x
(
t
) is an energy signal or when
x
(
t
) is periodic. When
x
(
t
)
is an energy signal, the limits of integration may range from
−∞
to
+∞
. When
x
(
t
)is
periodic, then
t
2
t
1
is equal to the period
T
.
To find the optimum values of the estimator for the basis coefficient, we use the
integral squared error or sum of squared error (SSE) as shown in (4.47).
−
x
(
t
)
x
(
t
)
2
t
2
t
1
=
−
(4.47)
I
dt
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