Basis Functions and Signal Representation - Signal Processing of Random Physiological Signals - page 27

4.4.2 Legendre Functions
Other signal representation includes the Legendre functions, which are an orthonormal
set of basis functions in the time interval from

− 1 to 1. The general basis function is

given by (4.38) and (4.39).

2 n

+ 1

n ( t ) = P n ( t ) for − 1 ≤ t ≤ 1 . (4.38)

2

where P n ( t ) is the Legendre polynomials, given by (4.39) and (4.40).

d n

1
2 n n ! dt n ( t 2

1) n

P n ( t )

= − for = 0 , 1 , 2 ,... for − 1 ≤ ≤ 1 (4.39)

n t

1
√ 2 0 ( t ) =

3
2 t

1 ( t ) =

(4.40)

5
2

3

1
2 2 t 2

2 ( t )

= −

7
2

5

2 t

3

2 t 3

3 ( t )

= −

These basis functions may be convenient when the signals have a predominant linear or
quadratic term.

4.4.3 Laguerre Functions
When the time interval of the representation is from 0 to ∞ , the Laguerre functions

form a complete orthonormal set defined by (4.41):

1

t
2

n ! e −

n ( t )

= ln( t ) for 0 ≤ < ∞ (4.41)

t

ln is not natural logarithm, but rather as defined by (4.42):

e t d n

dt n ( t n e − t )

ln ( t ) = (4.42)