Editing Waveform (section)

== Common periodic waveforms ==

Simple examples of periodic waveforms include the following, where <math>t</math> is [[time]], <math>\lambda</math> is [[wavelength]], <math>a</math> is [[amplitude]] and <math>\phi</math> is [[Phase (waves)|phase]]:

*[[Sine wave]]: <math display="inline">(t, \lambda, a, \phi) = a\sin \frac{2\pi t - \phi}{\lambda}.</math> The amplitude of the waveform follows a [[trigonometric]] sine function with respect to time.
*[[Square wave (waveform)|Square wave]]: <math display="inline">(t, \lambda, a, \phi) = \begin{cases}
a, (t-\phi) \bmod \lambda < \text{duty} \\
-a, \text{otherwise}
\end{cases}.</math> This waveform is commonly used to represent digital information. A square wave of constant [[frequency|period]] contains odd [[harmonic]]s that decrease at −6&nbsp;dB/octave.
*[[Triangle wave]]: <math display="inline">(t, \lambda, a, \phi) = \frac{2a}{\pi} \arcsin \sin \frac{2\pi t - \phi}{\lambda}.</math> It contains odd [[harmonic]]s that decrease at −12&nbsp;dB/octave.
*[[Sawtooth wave]]: <math display="inline">(t,\lambda, a, \phi) = \frac{2a}{\pi} \arctan \tan \frac{2\pi t - \phi}{2\lambda}.</math> This looks like the teeth of a saw. Found often in time bases for display scanning. It is used as the starting point for [[subtractive synthesis]], as a sawtooth wave of constant [[frequency|period]] contains odd and even [[harmonic]]s that decrease at −6 [[decibel|dB]]/octave.

The [[Fourier series]] describes the decomposition of periodic waveforms, such that any periodic waveform can be formed by the sum of a (possibly infinite) set of fundamental and harmonic components. Finite-energy non-periodic waveforms can be analyzed into sinusoids by the [[Fourier transform]].

Other periodic waveforms are often called composite waveforms and can often be described as a combination of a number of sinusoidal waves or other [[basis functions]] added together.