Triangle wave

Image:Waveforms.svg, square, triangle, and sawtooth waveforms]]

A triangle wave of period p that spans the range [0, 1] is defined as

$$x(t)\; =\; 2\; \backslash left|\; \backslash frac\{t\}\{p\}\; -\; \backslash left\backslash lfloor\; \backslash frac\{t\}\{p\}\; +\; \backslash frac\{1\}\{2\}\; \backslash right\backslash rfloor\; \backslash right|,$$

where $\backslash lfloor\backslash \; \backslash rfloor$ 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 {{closed-closed|−1, 1}} the expression becomes

$$x(t)=\; 2\; \backslash left\; |\; 2\; \backslash left(\; \backslash frac\{t\}\{p\}\; -\; \backslash left\backslash lfloor\; \backslash frac\{t\}\{p\}\; +\; \backslash frac\{1\}\{2\}\; \backslash right\backslash rfloor\; \backslash right)\; \backslash right|\; -\; 1.$$

File:Triangle wave with amplitude=5, period=4.png

A more general equation for a triangle wave with amplitude $a$ and period $p$ using the modulo operation and absolute value is

$$y(x)\; =\; \backslash frac\{4a\}\{p\}\; \backslash left|\; \backslash left(\; \backslash left(x\; -\; \backslash frac\{p\}\{4\}\backslash right)\; \backslash bmod\; p\; \backslash right)\; -\; \backslash frac\{p\}\{2\}\; \backslash right|\; -\; a.$$

For example, for a triangle wave with amplitude 5 and period 4:

$$y(x)\; =\; 5\; \backslash left|\; \backslash bigl(\; (x\; -\; 1)\; \backslash bmod\; 4\; \backslash bigr)\; -\; 2\; \backslash right|\; -\; 5.$$

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

The triangle wave can also be expressed as the integral of the square wave:

$$x(t)\; =\; \backslash int\_0^t\; \backslash sgn\backslash left(\backslash sin\backslash frac\{u\}\{p\}\backslash right)\backslash ,du.$$

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):

$$y(x)\; =\; \backslash frac\{2a\}\{\backslash pi\}\; \backslash arcsin\backslash left(\backslash sin\backslash left(\backslash frac\{2\backslash pi\}\{p\}x\backslash right)\backslash right).$$

The identity $\backslash cos\{x\}\; =\; \backslash sin\backslash left(\backslash frac\{p\}\{4\}-x\backslash right)$ 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:

$$y(x)\; =\; a\; -\; \backslash frac\{2a\}\{\backslash pi\}\; \backslash arccos\backslash left(\backslash cos\backslash left(\backslash frac\{2\backslash pi\}\{p\}x\backslash right)\backslash right).$$

Another definition of the triangle wave, with range from −1 to 1 and period p, is

$$x(t)\; =\; \backslash frac\{4\}\{p\}\; \backslash left(t\; -\; \backslash frac\{p\}\{2\}\; \backslash left\backslash lfloor\backslash frac\{2t\}\{p\}\; +\; \backslash frac\{1\}\{2\}\; \backslash right\backslash rfloor\; \backslash right)(-1)^\backslash left\backslash lfloor\backslash frac\{2\; t\}\{p\}\; +\; \backslash frac\{1\}\{2\}\; \backslash right\backslash rfloor.$$

Image:Synthesis triangle.gif 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).

The above can be summarised mathematically as follows:

x_\text{triangle}(t) = \frac8{\pi^2} \sum_{i=0}^{N - 1} \frac{(-1)^i}{n^2} \sin(2\pi f_0 n t),

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), $f\_0$ is the fundamental frequency, and {{mvar|i}} is the harmonic label which is related to its mode number by $n\; =\; 2i\; +\; 1$.

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

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

$$s\; =\; \backslash sqrt\{(4a)^2\; +\; p^2\}.$$

• List of periodic functions
• Sine wave
• Square wave
• Sawtooth wave
• Pulse wave
• Sound
• Triangle function
• Wave
• Zigzag

{{Reflist}}

• {{MathWorld|id=FourierSeriesTriangleWave|title=Fourier Series - Triangle Wave}}

{{Waveforms}}

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Category:Fourier series

Category:Waveforms