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Every (physically sensible) periodic function
f(t) = f(t + T) with T = 1/n = 2p/w and n, w = frequency and
angular frequency, respectively, may be written as a Fourier series as follows |
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f(t) » |
a0/2 |
+ a1 · cos wt + a2
· cos 2 wt + ... + an · cos nwt
+ .. | | |
+ b1 · sin wt + b2
· sin 2wt + ... + bn · sin nwt
+ .. |
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and the Fourier coefficients ak and bk (with
the index k = 0, 1, 2, ...) are determined by |
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ak = 2/T · |
T ó õ 0 |
f(t) · cos kwt · dt |
bk = 2/T · |
T ó õ 0 |
f(t) · sin kw
t · dt |
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This can be written much more elegantly using complex
numbers and functions as |
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f(t) = |
+¥
S -¥ |
cn · ein
wt |
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The coefficients cn are obtained by |
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cn = |
T ó õ 0 |
f(t) · e inwt
· dt | = |
{ |
½(an – ibn )
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for n > 0 |
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½(a
n + ibn) |
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for n < 0 |
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The function f(t) is thus expressed as a sum of sin functions with
the harmonic frequencies or simply harmonics
n·w derived from the fundamental frequency w0
= 2p/T. |
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The coefficients cn define the spectrum
of the periodic function by giving the amplitudes of the harmonics that the function contains. |
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© H. Föll (Electronic Materials - Script)