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                                   ɢ
                                                                         +K
      ¥ ¡          2          1       1                                   ¤
       4 £                    3       4                                   ¦

								
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