Editing Square wave (waveform) (section)

== Definitions ==

The square wave in mathematics has many definitions, which are equivalent except at the discontinuities:

It can be defined as simply the [[sign function]] of a sinusoid:
<math display="block">\begin{align}
   x(t) &= \sgn\left(\sin \frac{2\pi t}{T}\right) = \sgn(\sin 2\pi ft) \\
   v(t) &= \sgn\left(\cos \frac{2\pi t}{T}\right) = \sgn(\cos 2\pi ft),
 \end{align}</math>
which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities.  Here, ''T'' is the [[Period (physics)|period]] of the square wave and ''f'' is its frequency, which are related by the equation ''f'' = 1/''T''.

A square wave can also be defined with respect to the [[Heaviside step function]] ''u''(''t'') or the [[rectangular function]] Π(''t''):
<math display="block">\begin{align}
 x(t) &= 2\left[\sum_{n=-\infty}^\infty \Pi\left(\frac{2(t - nT)}{T} - \frac{1}{2}\right)\right] - 1 \\
   &= 2\sum_{n=-\infty}^\infty \left[u \left(\frac{t}{T} - n\right) - u \left(\frac{t}{T} - n - \frac{1}{2} \right) \right] - 1.
 \end{align}</math>

A square wave can also be generated using the [[floor function]] directly:
<math display="block">x(t) = 2\left(2\lfloor ft\rfloor - \lfloor 2 ft\rfloor \right) + 1</math>
and indirectly:
<math display="block">x(t) = \left(-1\right)^{\lfloor 2ft \rfloor}.</math>

Using the fourier series (below) one can show that the floor function may be written in trigonometric form<ref>{{cite web |title=Partial sum formula |url=https://www.wolframalpha.com/input?i=%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Csin%5Cleft%28%5Cleft%282n-1%5Cright%29x%5Cright%29%7D%7B2n-1%7D |website=www.wolframalpha.com |access-date=9 July 2023 |archive-url=https://web.archive.org/web/20230122100923/https://www.wolframalpha.com/input?i=%5Csum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D%5Cfrac%7B%5Csin%5Cleft%28%5Cleft%282n-1%5Cright%29x%5Cright%29%7D%7B2n-1%7D |archive-date=January 22, 2023 |language=en |url-status=live}}</ref>
<math display="block">\frac{2}{\pi}\arctan\left(\tan\left(\frac{\pi ft}{2}\right)\right)+\frac{2}{\pi}\arctan\left(\cot\left(\frac{\pi ft}{2}\right)\right)</math>