Triangle wave

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A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).

DefinitionsEdit

File:Waveforms.svg

Sine, square, triangle, and sawtooth waveforms

DefinitionEdit

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 function. This can be seen to be the absolute value of a shifted sawtooth wave.

For a triangle wave spanning the range Template:Closed-closed 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

Triangle wave with amplitude = 5, period = 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 % operator is a remainder operator (with result the same sign as the dividend), not a modulo operator; the modulo operation can be obtained by using ((x % p) + p) % p in place of x % p. In e.g. JavaScript, this results in an equation of the form 4*a/p * Math.abs((((x - p/4) % p) + p) % p - p/2) - a.

Relation to the square waveEdit

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

Expression in trigonometric functionsEdit

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 functionsEdit

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>

HarmonicsEdit

File:Synthesis triangle.gif

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 Template:Pi) and multiplying the amplitude of the harmonics by one over the square of their mode number, Template:Math (which is equivalent to one over the square of their relative frequency to the 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 Template:Mvar is the number of harmonics to include in the approximation, Template:Mvar is the independent variable (e.g. time for sound waves), <math>f_0</math> is the fundamental frequency, and Template:Mvar 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 Template:Mvar tends to infinity, as shown in the animation.

Arc lengthEdit

The 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>

ReferencesEdit

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|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:FourierSeriesTriangleWave%7CFourierSeriesTriangleWave.html}} |title = Fourier Series - Triangle Wave |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

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