Biomedical Engineering Reference
In-Depth Information
the resulting integral must be zero as shown in (4.2).
t
2
n
(
t
)
k
(
t
)
dt
=
0;
k
=
n
(4.2)
t
1
The second condition is when the two basis functions are equal, then the resulting integral
will equal some value “Lambda,”
λ
k
, as shown in (4.3).
t
2
n
(
t
)
k
(
t
)
dt
=
λ
k
;
k
=
n
for all
k
and
n
(4.3)
t
1
If both sides of (4.3) are divided by
1) for all
k
,
then the basis function is called
orthonormal
. Note that the limits of integration can be
defined as finite interval or an infinite or semi-infinite interval.
λ
k
and lambda
λ
k
is made unity (
λ
k
=
4.3 EVALUATION OF COEFFICIENTS
Recall the general expression of a signal represented in terms of a basis function, as the
weighted summation of the elementary functions:
N
x
(
t
)
=
a
n
n
(
t
)
(4.4)
n
=
0
Hence, to evaluate the coefficients
a
n
of the basis function, the following steps are
necessary:
(1) Multiply both sides by
j
(
t
) as in (4.5).
N
j
(
t
)
x
(
t
)
=
(
t
)
a
n
n
(
t
)
(4.5)
n
=
0
(2) Integrate both sides of the equation over the specified interval
t
2
−
t
1
(4.6).
j
(
t
)
n
(
t
)
dt
t
2
t
2
t
2
N
N
j
(
t
)
x
(
t
)
dt
=
a
n
=
a
n
j
(
t
)
n
(
t
)
dt
(4.6)
n
=
0
n
=
0
t
1
t
1
t
1
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