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First we apply this to the sawtooth wave . For we have:
Ignoring the DC offset of , this gives an
impulse,
zero everywhere except one sample per cycle. The summation in
the Fourier transform only has one term, and we get:
We then apply the difference formula backward to get:
valid for integer values of , small compared to , but with .
(To get the second
form of the expression we plugged in
and .)
This analysis doesn't give us the DC component
,
because we would have had to divide by . Instead, we can evaluate the DC
term directly as the sum of all the points of the waveform: it's approximately
zero by symmetry.
To get a Fourier series in terms of familiar real-valued sine and cosine functions,
we combine corresponding terms for negative and positive values of . The
first harmonic () is:
and similarly the th harmonic is
so the entire Fourier series is:
Next: Parabolic wave
Up: Fourier series of the
Previous: Fourier series of the
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Miller Puckette
2006-09-24