Triangle wave
A 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, demonstrating odd symmetry. 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).
It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every (4n−1)th harmonic by −1 (or changing its phase by π), and rolling off the harmonics by the inverse square of their relative frequency to the fundamental.
This infinite Fourier series converges to the triangle wave with cycle frequency f over time t:
Another definition of the triangle wave, with range from -1 to 1 and period 2a is:
Also, the triangle wave can be the absolute value of the sawtooth wave:
or, for a range from -1 to +1:
The triangle wave can also be expressed as the integral of the square wave:
A simple equation with a period of 4, with . As this only uses the modulo operation and absolute value, this can be used to simply implement a triangle wave on hardware electronics with less CPU power:
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