Triangle wave

A triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

A bandlimited triangle wave pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).

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).

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:

$\begin{align} x_\mathrm{triangle}(t) & {} = \frac {8}{\pi^2} \sum_{k=0}^\infty (-1)^k \, \frac{ \sin \left( (2k+1)\omega t \right)}{(2k+1)^2} \\ & {} = \frac{8}{\pi^2} \left( \sin (\omega t)-{1 \over 9} \sin (3\omega t)+{1 \over 25} \sin (5\omega t) - \cdots \right) \end{align}$

where $\scriptstyle \omega$ is the angular frequency.

Animation of the additive synthesis of a triangle wave with an increasing number of harmonics. See Fourier Analysis for a mathematical description.

Another definition of the triangle wave, with range from -1 to 1 and period 2a is:

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

where the symbol $\scriptstyle \lfloor n \rfloor$ represent the floor function of n.

Also, the triangle wave can be the absolute value of the sawtooth wave:

$x(t)= \left | 2 \left ( {t \over a} - \left \lfloor {t \over a} + {1 \over 2} \right \rfloor \right) \right |$

The triangle wave can also be expressed as the integral of the square wave:
$\int\sgn(\sin(x))\,dx\,$

See also

Sine, square, triangle, and sawtooth waveforms

• Triangle function
• Sine wave
• Sawtooth wave
• Square wave
• Waves
• Sound
• Zigzag

References

• Weisstein, Eric W., "Fourier Series - Triangle Wave" from MathWorld.