View source for Triangle wave

{{Short description|Non-sinusoidal waveform}}
{{Infobox mathematical function
| name = Triangle wave
| image = triangle-td and fd.png
| imagesize = 400px
| imagealt = A bandlimited triangle wave pictured in the time domain and frequency domain.
| caption = A [[Bandlimiting|bandlimited]] triangle wave<ref name="bandlimited-synthesis">{{cite conference |title=LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-filtered Waveforms |last1=Kraft |first1=Sebastian |last2=Zölzer |first2=Udo |date=5 September 2017 |book-title=Proceedings of the 20th [[International Conference on Digital Audio Effects]] (DAFx-17) |pages=255–259 |location=Edinburgh |conference=20th International Conference on Digital Audio Effects (DAFx-17) |conference-url=http://www.dafx17.eca.ed.ac.uk/}}</ref> pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220&nbsp;Hz (A<sub>3</sub>).
| general_definition = <math>x(t) = 4 \left\vert t - \left\lfloor t + 3/4 \right\rfloor + 1/4 \right\vert - 1</math>
| fields_of_application = Electronics, synthesizers
| domain = <math>\mathbb{R}</math>
| codomain = <math>\left[ -1, 1 \right]</math>
| parity = Odd
| period = 1
| root = <math>\left\{ \tfrac{n}{2} \right\}, n \in \mathbb{Z}</math>
| derivative = [[Square wave]]
| fourier_series = <math>x(t) = -\frac{8}{{\pi}^{2}}\sum_{k=1}^{\infty} \frac{{\left( -1 \right)}^{k}}{\left( 2 k - 1 \right)^{2}} \sin \left(2 \pi \left( 2 k - 1 \right) t\right)</math>
}}

{{Listen|filename=220 Hz anti-aliased triangle wave.ogg|title=Triangle wave sound sample|description=5 seconds of triangle wave at 220 Hz|format=[[Software:Ogg|Ogg]]}}
{{Listen|filename=Additive_220Hz_Triangle_Wave.wav|title=Additive Triangle wave sound sample|description=After each second, a harmonic is added to a sine wave creating a triangle 220 Hz wave|format=[[Software:Ogg|Ogg]]}}

A '''triangular wave''' or '''triangle wave''' is a [[Physics:Non-sinusoidal waveform|non-sinusoidal waveform]] named for its [[Triangle|triangular]] shape.  It is a [[Periodic function|periodic]], [[Piecewise linear function|piecewise linear]], [[Continuous function|continuous]] [[Function of a real variable|real function]].

Like a [[Square wave|square wave]], the triangle wave contains only odd [[Physics:Harmonic|harmonic]]s.  However, the higher harmonics [[Physics:Roll-off|roll off]] much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).

==Definitions==
[[Image:Waveforms.svg|thumb|400px|[[Physics:Sine wave|Sine]], [[Square wave|square]], triangle, and [[Sawtooth wave|sawtooth]] waveforms]]

=== Definition ===
A triangle wave of period ''p'' that spans the range [0,1] is defined as:
<math display="block">x(t)= 2 \left| \frac{t}{p} - \left \lfloor \frac{t}{p} + \frac{1}{2} \right \rfloor \right|</math>
where <math>\lfloor\,\ \rfloor</math> is the [[Floor and ceiling functions|floor function]]. This can be seen to be the absolute value of a shifted [[Sawtooth wave|sawtooth wave]].

For a triangle wave spanning the range {{closed-closed|−1,1}} the expression becomes:
<math display="block"> x(t)= 2 \left | 2 \left ( \frac{t}{p} - \left \lfloor {t \over p} + {1 \over 2} \right \rfloor \right) \right | - 1. </math>

A more general equation for a triangle wave with amplitude <math>a</math> and period <math>p</math> using the [[Modulo operation|modulo operation]] and [[Absolute value|absolute value]] is:

[[File:Triangle wave with amplitude=5, period=4.png|right|thumb|Triangle wave with amplitude=5, period=4]]

<math display="block">y(x) = \frac{4a}{p} \left| \left( \left(x - \frac{p}{4}\right) \bmod p \right) - \frac{p}{2} \right| - a.</math>

For example, for a triangle wave with amplitude 5 and period 4:
<math display="block">y(x) = 5 \left | \bigl( (x - 1) \bmod 4 \bigr) - 2 \right | - 5.</math>

A phase shift can be obtained by altering the value of the <math>- p/4</math> term, and the vertical offset can be adjusted by altering the value of the <math>- a</math> term.

As this only uses the modulo operation and absolute value, it can be used to simply implement a triangle wave on hardware electronics.

Note that in many programming languages, the <code>%</code> operator is a remainder operator (with result the same sign as the dividend), not a [[Modulo operation#In programming languages|modulo operator]]; the modulo operation can be obtained by using <code>((x % p) + p) % p</code> in place of <code>x % p</code>. In e.g. JavaScript, this results in an equation of the form <code>4*a/p * Math.abs((((x-p/4)%p)+p)%p - p/2) - a</code>.

=== Relation to the square wave ===
The triangle wave can also be expressed as the [[Integral|integral]] of the [[Square wave|square wave]]:
<math display="block">x(t) = \int_0^t \sgn\left(\sin\frac{u}{p}\right)\,du.</math>

=== Expression in trigonometric functions ===
A triangle wave with period ''p'' and amplitude ''a'' can be expressed in terms of [[Sine|sine]] and arcsine (whose value ranges from −''π''/2 to ''π''/2):
<math display="block">y(x) = \frac{2a}{\pi} \arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right).</math>
The identity <math display="inline">\cos{x} = \sin\left(\frac{p}{4}-x\right)</math> can be used to convert from a triangle "sine" wave to a triangular "cosine" wave. This phase-shifted triangle wave can also be expressed with [[Cosine|cosine]] and arccosine:
<math display="block">y(x) = a - \frac{2a}{\pi} \arccos\left(\cos\left(\frac{2\pi}{p}x\right)\right).</math>

=== Expressed as alternating linear functions ===
Another definition of the triangle wave, with range from −1 to 1 and period ''p'', is:
<math display="block">x(t) = \frac{4}{p} \left (t-\frac{p}{2} \left \lfloor\frac{2 t}{p}+\frac{1}{2} \right \rfloor \right )(-1)^\left \lfloor\frac{2 t}{p} + \frac{1}{2} \right \rfloor</math>

===Harmonics===
[[Image:Synthesis triangle.gif|thumb|upright=1.6|right|Animation of the additive synthesis of a triangle wave with an increasing number of harmonics. See Fourier Analysis for a mathematical description. ]]

It is possible to approximate a triangle wave with additive synthesis by summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by {{pi}}) and multiplying the amplitude of the harmonics by one over the square of their mode number, {{math|''n''}} (which is equivalent to one over the square of their relative frequency to the [[Physics:Fundamental frequency|fundamental]]).

The above can be summarised mathematically as follows:
<math display="block">\begin{align}
x_\mathrm{triangle}(t) & {} = \frac8{\pi^2}\sum_{i=0}^{N - 1} (-1)^i n^{-2} \sin\left(2\pi f_0 n t\right)
\end{align}</math>
where {{math|''N''}} is the number of harmonics to include in the approximation, {{mvar|t}} is the independent variable (e.g. time for sound waves), <math> f_0 </math> is the fundamental frequency, and {{math|''i''}} is the harmonic label which is related to its mode number by <math>n = 2i + 1</math>.

This infinite [[Fourier series]] converges quickly to the triangle wave as {{math|''N''}} tends to infinity, as shown in the animation.

==Arc length==
The [[Arc length|arc length]] per period for a triangle wave, denoted by ''s'', is given in terms of the amplitude ''a'' and period length ''p'' by
<math display="block">s = \sqrt{(4a)^2 + p^2}.</math>

==See also==
* [[List of periodic functions]]
* [[Physics:Sine wave|Sine wave]]
* [[Square wave]]
* [[Sawtooth wave]]
* [[Pulse wave]]
* [[Physics:Sound|Sound]]
* Triangle function
* [[Wave]]
* Zigzag

==References==
{{Reflist}}
*{{MathWorld|id=FourierSeriesTriangleWave|title=Fourier Series - Triangle Wave}}

{{Waveforms}}

{{DEFAULTSORT:Triangle Wave}}
[[Category:Fourier series]]
[[Category:Waveforms]]

{{Sourceattribution|Triangle wave}}