Additive & Spectral: Fourier Series

Fourier Series

  • any periodic signal can be approximated by a sum of \(N_\mathit{part}\) sinusoids
  • with individual
    • amplitude \(a_i\)
    • frequency \(f_i\)
    • phase \(\varphi_i\)

\(\displaystyle y = \sum\limits_{i=1}^{N_{part}} a_i \ sin(2 \pi f_i \ t+ \varphi_i)\)

Certain basic waveforms can be generated with specific Fourier series.

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Triangular

\(X(t) = \frac{8}{\pi^2} \sum\limits_{i=0}^{N} (-1)^{(i)} \frac{\sin(2 \pi (2i +1) f\ t)}{(2i +1)^2}\)

Pitch (Hz):

Number of Harmonics:

Output Gain:

Time Domain:

Frequency Domain:


Sawtooth

Pitch (Hz):

Number of Harmonics:

Output Gain:

Time Domain:

Frequency Domain:



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