Biomedical Engineering Reference
In-Depth Information
CHAPTER
4
Basis Functions and Signal
Representation
4.1 INTRODUCTION TO BASIS FUNCTIONS
To obtain a quantitative description of a signal, it is necessary to represent the signal in
terms of explicit time functions. Mathematical convenience dictates that the signal can
be represented as a linear combination of a set of elementary time functions called
Basis
Functions
. Equation (4.1) is the general equation representing a signal,
x
(
t
), in terms
of a set of basis functions designated as
0
(
t
),
1
(
t
),
2
(
t
), ...,
N
(
t
).
N
x
(
t
)
=
a
n
n
(
t
)
(4.1)
n
=
0
4.2 DESIRABLE PROPERTIES OF BASIS FUNCTIONS
One property that is desirable for a set of basis functions is the
finality of coefficients
, which
allows the determination of a coefficient without needing to know any other coefficients.
This means that more terms can be added without making changes or recalculating the
earlier or previous coefficient. To achieve “finality of coefficients,” the basis functions
must be orthogonal over the time interval of interest.
The condition of orthogonality requires that the integral of two functions satisfies
two conditions. The first condition is when the two basis functions are not equal, then
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