0195136047.pdf

APPENDIX

G

Fourier Series

A periodic waveformf(t)=f(t+T), which is said to be periodic with a periodT, fundamental

frequencyf= 1 /T, and fundamental radian frequencyω= 2 π/T, can be expressed in terms of

an infinite series (known as theFourier series) of sinusoidal signals. Expressed mathematically,

we have

f(t)=

a 0

2

+

∑∞

n= 1

(ancosnωt+bnsinnωt) (1)

where the coefficientsanandbnare given by

an=

2

T

∫T/^2

−T/ 2

f(t)cosnωt dt=

2

T

∫T

0

f(t)cosnωtdt, n= 0 , 1 , 2 , 3 , ... (2)

bn=

2

T

∫T/^2

−T/ 2

f(t)sinnωt dt=

2

T

∫T

0

f(t)sinnωtdt, n= 1 , 2 , 3 , ... (3)

The average value off(t), which is also known as the dc component, is given by

a 0

2

=

1

T

∫T

0

f(t) dt

COMPLEX (EXPONENTIAL) FOURIER SERIES

By making use of the trigonometric identities

cost=

1

2

(ejt+e−jt) (4)

sint=

1

2 j

(ejt−e−jt) (5)

wherej=

√

−1, one can express Equation (1) as follows in terms ofcomplex Fourier series:

f(t)=

∑∞

n=−∞

c ̄nejnωt (6)

where the complex coefficientsc ̄nare given by

c ̄n=

1

T

∫T/^2

−T/ 2

f(t)e−jnωtdt , n= 0 ,± 1 ,± 2 , ... (7)

The set of coefficients{ ̄cn}is often referred to as the Fourier spectrum. These coefficients are

related to the coefficientsanandbnof Equations (2) and (3) by

847