# fourier by NiceTime

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```									                    Fourier Analysis
Fourier series:
i
()
wt a        + sin(
=a 0_ 1 t          n+   )É ¸
=             i    i
i
where w(t) is the waveform, a0 is the dc-level, ai are the
amplitudes, Éi are the angular frequencies, t is the time and
¸i are the phase angles. Note, the angular frequency Éi
corresponds to frequency, fi, such that, Éi = 2Àfi.

Discrete Fourier Transform using Fourier Coefficients:
i
()
wt a       [ _ t cos( +
= b sin( ) c n)] t
+                                  É        É
= 0          1   i     i        i
i
Equations of Known Waveforms
Sine wave:                      () sin t=
wt a
É
Square wave:
3             ¢
()
wt a sint                =
sint       sin
É+
t              +
É       +
É
5K
¡       ¥           1             1             ¤
Triangle wave:
£                   3             5             ¦
8        13                                  ¢
wt ()      cos Àa=cos t É+cost É+ t                             É+K
¥2 ¡        9         1                                  ¤
5                   25
Sawtooth wave:
£                                                    ¦
2        12        3
wt ()     sin Àa =sint É_ t É+sin É_ t
sin      t                                   É¢
+K
¥ ¡          2          1       1                                   ¤
4 £                    3       4                                   ¦

```
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