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13.1
13.1 Trigonometric Fourier Series Fourier series of periodic functions T=period offunctionf(t)
f(~
il = 2; = basic angular frequency
Cosine - sine - form
-2T
Orthogonality
o
2T
T,n=k=O TI2, n=k>O
o
T
T
{~n*k
T
f coskiltcosniltdt=
f sin kilt sinniltdt=
-T
{TI2 k=n>O ' 0, k*n
T
f sin kilt cos nilt dt = 0
o
ao
f(t) = -2 +
L (an cos nilt + bn sin nilt) * 00
n=l
a +T
2 an=f f f(t)cosniltdt a
(n~O)
2 a +T bn=f f f(t)sinniltdt a
(n~l)
Special case. Period T= 21t a f(t) = 20 +
1
00
L (an cos nt + bn sin nt)
n=l
1" an = 1rJ/(t) cos nt dt
f(t) even ~ an =
4 TI2 f f(t) cos nilt dt,
f
f(t)odd~an=O,
o
bn =
1rj" f(t) sin nt dt 1"
bn = 0
4 TI2 bn=f ff(t)sinniltdt
o
* For bounded and piece-wise differentiable functions, the equality holds at points t where f(t) is continuous. At jumps the Fourier series equals (f(t + ) + f(t -))12. 310 L. Råde et al., Mathematics Handbook for Science and Engineering © Springer-Verlag Berlin Heidelberg 2004
13.1 Approximation in mean (cf. sec. 12.1) ao n sn(t)= -2 + L (ak cos kDt+ hk sinHU) k=l
Amplitude - phase form (f(t) real, £2= 2rrlT) f(t) =Ao +
L
n=l
An cos(n£2t+ an)' An;::: 0 for n;::: l.
Calculation of An' an' see below. Complex form (£2= 2rr1T) Orthogonality
Io eikQte-infltdt-_{T,k=n 0 k ' T
:#:n
f(t)=
=
L
n=-oo
cne inflt'
I
1 a+T . cn = T f(t)e-m!ltdt a
Relations between Fourier coefficients sin, cos form - amplitude, phase form (f(t) real) ~
I AO=2ao,
An=Ajan+b/i
_ _. _ {-arctan(bnlan), an>O > a,,-arg(an lb n),n_l 1t-arctan(bn la n), an1 ,n_
13.1
Parseval's identities [Below, an,bn,An' lXn,cn refer to!(t), a~, b~, A~, a~, c~, refer to g(t)]
1 a+T
J f(t) g(t) dt= ~ cnc~ a
T
J f(t- r)g( r)dr = L n
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