Basis Functions and Signal Representation - Signal Processing of Random Physiological Signals - page 24

For the exponential Fourier Series, the basis function is defined as given by (4.19).

e ± jnw 0 t =

n ( t )

, where n =± (0 , 1 , 2 ,...., ∞ ) , and (4.19)

T (the fundamental frequency is in radians).
To prove that the functions are orthogonal over the interval start with (4.20).

0 2 where: ω

t 1 + T

e jn ω 0 t e − jn ω 0 t dt ∗ = 0 when n = k

(4.20) t 1

= T when n = k

The coefficients for the series can be expressed as (4.21).

t 1 + T

1
T

x ( t ) e − jnw 0 t dt

a n = (4.21)

t 1

The coefficients are usually complex and can be expressed as a − n = α n * (* means con-
jugate). The signal x ( t ) is expressed as (4.22):

∞

n e jnw 0 t x ( t ) = α (4.22)

n =−∞

The accuracy of an approximate representation using a finite number of terms is obtained
from the energy ratio given by (4.23) and (4.24).

M

error
signal

energy
energy =

1
E

0 λ n a n

η m = 1 − (4.23)

=

n

where

t 2

x 2 ( t ) dt E = (4.24)

t 1

2 2 .

For the Fourier series, λ n = T , since α − n = α n *. Then, | α n | =| α − n |