Triangle wave
From Free net encyclopedia
A triangle wave is a basic kind of non-sinusoidal waveform named for its triangular shape.
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), and so its sound is smoother than a square wave and is nearer to that of a sine wave.
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 <math>\pi</math>), 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:
- <math>x_\mathrm{triangle}(t) = \frac {8}{\pi^2} \sum_{k=1}^\infty \sin \left(\frac {k\pi}{2}\right)\frac{ \sin (kt)}{k^2}</math>
Image:Synthesis triangle.gif Template:Listen
