0195136047.pdf

FOURIER SERIES 849

f(t)has odd symmetry such thatf(−t)=−f(t), its Fourier series will have no cosine terms;

that is to say, allancoefficients become zero.

Decomposition:An arbitrary periodic waveformf(t)can be expressed as

f(t)=fe(t)+fo(t) (22)

wherefe(t)represents a part with even symmetry, andfo(t)represents another part with odd

symmetry. These parts may be evaluated from the original signal by

fe(t)=

1

2

[f(t)+f(−t)] (23)

fo(t)=

1

2

[f(t)−f(−t)] (24)

Integration:The integral of a periodic signal that has a valid Fourier series can be found by

termwise integration of the Fourier series of the signal.

Differentiation:If a periodic functionf(t)is continuous everywhere and its derivative has a

valid Fourier series, then wherever it exists, the derivative off(t)can be found by termwise

differentiation of the Fourier series off(t).

SOME USEFUL AUXILIARY FORMULAE FOR FOURIER SERIES

sin

nπ

2

=

(j )n+^1

2

[(− 1 )n−1] (25)

cos

nπ

2

=

(j )n

2

[(− 1 )n+1] (26)

The following table of trigonometric functions will be helpful for developing Fourier series:

nn n n/ 2 n/ 2

Function Any integer Even Odd Odd Even

sinnπ 00 0 00

cosnπ (− 1 )n 1 − 111

sin

nπ

2 0 (−^1 )

(n− 1 )/ (^200)
cos
nπ
2
(− 1 )n/^20 − 11
sinnπ
4
√
2
2
(− 1 )(n^2 +^4 n+^11 )/^8 (− 1 )(n−^2 )/^40
1 =
4
π
[ sin
πt
k

• 1
  3
  sin
  3 πt
  k


• 
  

  1
  5
  sin
  5 πt
  k
  +...] ( 0 <t<k) (27)
  t=
  2 k
  π
  [ sin
  πt
  k
  −
  1
  2
  sin
  2 πt
  k

  
  


• 
  

  1
  3
  sin
  3 πt
  k
  +...] (−k<t<k) (28)
  t=
  k
  2
  −
  4 k
  π^2
  [ cos
  πt
  k

  
  


• 
  

  1
  32
  cos
  3 πt
  k

  
  


• 
  

  1
  52
  cos
  5 πt
  k
  +...] ( 0 <t<k) (29)
  t^2 =
  2 k^2
  π^3
  [
  (
  π^2
  1
  −
  4
  1
  )sin
  πt
  k
  −
  π^2
  2
  sin
  2 πt
  k
  +(
  π^2
  3
  −
  4
  33
  )sin
  3 πt
  k