Fourier series of the elementary waveforms

In general, given a repeating waveform , we can evaluate its Fourier
series coefficients by directly evaluating the Fourier transform:

but doing this directly for sawtooth and parabolic waves will require pages of algebra (somewhat less if we were willing resort to differential calculus). Instead, we rely on properties of the Fourier transform to relate the transform of a signal with its

In general, to evaluate the strength of the th harmonic, we'll make the assumption that is much larger than , or equivalently, that is negligible.

We start from the Time Shift Formula for Fourier Transforms
(Page ) setting the time shift to one sample:

Here we're using the assumption that, because is much larger than , is much smaller than unity and we can make approximations:

which are good to within a small error, on the order of . Now we plug this result in to evaluate:

- Sawtooth wave
- Parabolic wave
- Square and symmetric triangle waves
- General (non-symmetric) triangle wave