# Fourier Series: Sawtooth

## Formula

The sawtooth is an asymmetric waveform with a sharp timbre. The related Fourier series is described by the following characteristics:

odd and even harmonics

alternating sign

slow decrease towards higher partials

\begin{equation*}
X(t) = \frac{2}{\pi} \sum\limits_{k=1}^{N} (-1)^i \frac{\sin(2 \pi i f\ t)}{i}
\end{equation*}

## Interactive Example

Pitch (Hz):

Number of Harmonics:

Output Gain:

Time Domain:

Frequency Domain:

In contrast to the triangular wave, the interactive example shows the occurrence of ripples at the steep edges of the waveform. The higher the number of partials, the denser the ripples. This is referred to as the Gibbs phenomenon.