Triangle wave
Template:Infobox mathematical function
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).
Definitions
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Definition
[edit]A triangle wave of period p that spans the range [0, 1] is defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t) = 2 \left| \frac{t}{p} - \left\lfloor \frac{t}{p} + \frac{1}{2} \right\rfloor \right|,} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lfloor\ \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 [−1, 1] the expression becomes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t)= 2 \left | 2 \left( \frac{t}{p} - \left\lfloor \frac{t}{p} + \frac{1}{2} \right\rfloor \right) \right| - 1.}
A more general equation for a triangle wave with amplitude Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} and period Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} using the modulo operation and absolute value is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) = \frac{4a}{p} \left| \left( \left(x - \frac{p}{4}\right) \bmod p \right) - \frac{p}{2} \right| - a.}
For example, for a triangle wave with amplitude 5 and period 4: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) = 5 \left| \bigl( (x - 1) \bmod 4 \bigr) - 2 \right| - 5.}
A phase shift can be obtained by altering the value of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -p/4} term, and the vertical offset can be adjusted by altering the value of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -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.
Relation to the square wave
[edit]The triangle wave can also be expressed as the integral of the square wave: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x(t) = \int_0^t \sgn\left(\sin\frac{2 \pi u}{p}\right)\,du.}
Expression in trigonometric functions
[edit]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): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) = \frac{2a}{\pi} \arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right).} The identity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \cos{x} = \sin\left(\frac{p}{4}-x\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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) = a - \frac{2a}{\pi} \arccos\left(\cos\left(\frac{2\pi}{p}x\right)\right).}
Expressed as alternating linear functions
[edit]Another definition of the triangle wave, with range from −1 to 1 and period p, is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.}
Harmonics
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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 π) and multiplying the amplitude of the harmonics by one over the square of their mode number, n (which is equivalent to one over the square of their relative frequency to the fundamental).
The above can be summarised mathematically as follows: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f_0} is the fundamental frequency, and i is the harmonic label which is related to its mode number by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 2i + 1} .
This infinite Fourier series converges quickly to the triangle wave as N tends to infinity, as shown in the animation.
Arc length
[edit]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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s = \sqrt{(4a)^2 + p^2}.}