Triangle wave

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).

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

A more general equation for a triangle wave with amplitude ${\displaystyle a}$ and period ${\displaystyle p}$ using the modulo operation and absolute value is $${\displaystyle y(x)={\frac {4a}{p}}\left|\left(\left(x-{\frac {p}{4}}\right){\bmod {p}}\right)-{\frac {p}{2}}\right|-a.}$$

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.