0195136047.pdf

848 APPENDIX G

c 0 =

a 0

2

(8)

c ̄n=

1

2

(an−jbn), n= 1 , 2 , ... (9)

c ̄−n=

1

2

(an+jbn)= ̄c∗n,n= 1 , 2 , ... (10)

where the asterisk represents complex conjugation.

The periodic functionf(t)of Equation (1) can also be represented as

f(t)=

a 0

2

+

∑∞

n= 1

2 c ̄ncos(nωt+φn) (11)

where the coefficients are related by

an= 2 c ̄ncosφn (12)

bn=− 2 c ̄nsinφn (13)

c ̄n=

1

2

√

an^2 +b^2 n (14)

φn=arctan(−bn/an) (15)

The periodic waveform of Equation (1) can also be expressed as

f(t)=

a 0

2

+

∑∞

n= 1

2 c ̄nsin(nωt+φn) (16)

where the coefficients are related by

an= 2 c ̄nsinφn (17)

bn= 2 c ̄ncosφn (18)

c ̄n=

1

2

√

a^2 n+b^2 n (19)

φn=arctan(an/bn) (20)

note that the quadrant ofφnis to be chosen so as to make the formulae foran,bn, andc ̄nhold.

PROPERTIES OF FOURIER SERIES

Existence: If a bounded single-valued periodic functionf(t)of periodThas at most a finite

number of maxima, minima, and jump discontinuities in any one period, thenf(t)can be

represented over a complete period by a Fourier series of simple harmonic functions, the

frequencies of which are integral multiples of the fundamental frequency. This series will converge

tof(t)at all points wheref(t)is continuous and to the average of the right- and left-hand limits

off(t)at each point wheref(t)is discontinuous.

Delay:If a periodic functionf(t)is delayed by any multiple of its periodT, the waveform is

unchanged. That is to say

f(t−nT )=f(t) , n=± 1 ,± 2 ,± 3 , ... (21)

Symmetry:A periodic waveformf(t)with even symmetry such thatf(−t)=f(t)will have a

Fourier series with no sine terms; that is to say, all coefficientsbngo to zero. If, on the other hand,