Editing Triangle wave (section)

==Definitions==
[[Image:Waveforms.svg|thumb|400px|[[sine wave|Sine]], [[Square wave (waveform)|square]], triangle, and [[sawtooth wave|sawtooth]] waveforms]]

=== Definition ===
A triangle wave of period ''p'' that spans the range [0,&nbsp;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]].

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

[[File:Triangle wave with amplitude=5, period=4.png|right|thumb|Triangle wave with amplitude&nbsp;=&nbsp;5, period&nbsp;=&nbsp;4]]
A more general equation for a triangle wave with amplitude <math>a</math> and period <math>p</math> using the [[modulo operation]] and [[absolute value]] is
<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]] of the [[Square wave (waveform)|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]] 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]] 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{2t}{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 Transform|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 [[fundamental frequency|fundamental]]).

The above can be summarised mathematically as follows:
<math display="block">
x_\text{triangle}(t) = \frac8{\pi^2} \sum_{i=0}^{N - 1} \frac{(-1)^i}{n^2} \sin(2\pi f_0 n t),
</math>
where {{mvar|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 {{mvar|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 {{mvar|N}} tends to infinity, as shown in the animation.