Triangle wave

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

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

where ${\displaystyle \lfloor \,\ \rfloor }$ 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:

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

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

Triangle wave with amplitude=5, period=4

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

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

$${\displaystyle y(x)=5\left|{\bigl (}(x-1){\bmod {4}}{\bigr )}-2\right|-5.}$$

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

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

$${\displaystyle x(t)=\int _{0}^{t}\operatorname {sgn} \left(\sin {\frac {u}{p}}\right)\,du.}$$

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):

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

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

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

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

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

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 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:

$${\displaystyle {\begin{aligned}x_{\mathrm {triangle} }(t)&{}={\frac {8}{\pi ^{2}}}\sum _{i=0}^{N-1}(-1)^{i}n^{-2}\sin \left(2\pi f_{0}nt\right)\end{aligned}}}$$

where N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), ${\displaystyle f_{0}}$ is the fundamental frequency, and i is the harmonic label which is related to its mode number by ${\displaystyle n=2i+1}$.

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