Triangle wave

Triangle wave

A triangle wave is a 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.

One simple definition of a triangle wave is

:egin{align}x_mathrm{triangle}(t) = arcsin(sin(t))end{align}

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 pi), 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:

:egin{align}x_mathrm{triangle}(t) & {} = frac {8}{pi^2} sum_{k=1}^infty sin left(frac {kpi}{2} ight)frac{ sin (2pi kft)}{k^2} \& {} = frac{8}{pi^2} left( sin (2pi ft)-{1 over 9} sin (6 pi ft)+{1 over 25} sin (10 pi ft) + cdots ight)end{align}

It is also possible to approximate a triangle wave with abs() and floor():

:egin{align}x_mathrm{triangle}(t) = 2 * mbox{abs}(t - 2 * mbox{Floor}(t/2) - 1) - 1end{align}

Or with modulo:

:egin{align}x_mathrm{triangle}(t) = 4 (t%1)^2+2(t%2)-4(t%1) (t%2)-1end{align}

See also

*Triangle function
*Sine wave
*Sawtooth wave
*Square wave

Wikimedia Foundation. 2010.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • Triangle (disambiguation) — A triangle is a geometric shape that has three straight sides. Triangle can also refer to: Mathematics * Spherical triangle * Sierpinski triangle * Pascal s triangle * Triangle wave * Triangle inequality Objects * Triangle (instrument), a musical …   Wikipedia

  • Triangle Sun — Основная информация …   Википедия

  • Sine wave — Sinusoid redirects here. For the blood vessel, see Sinusoid (blood vessel). The graphs of the sine and cosine functions are sinusoids of different phases. The sine wave or sinusoid is a mathematical function that describes a smooth repetitive… …   Wikipedia

  • List of wave topics — This is a list of wave topics.0 ndash;9*21 cm lineA*Abbe prism *absorption spectrum *acoustics *Airy disc *Airy wave theory *Alfvén wave *Alpha waves *amphidromic point *amplitude *amplitude modulation *analog sound vs. digital sound *animal… …   Wikipedia

  • Continuous-wave radar — is a type of radar system where a known stable frequency continuous wave radio energy is transmitted and then received from any reflecting objects.[1]Continuous wave (CW) radar uses Doppler, which renders the radar immune to interference from… …   Wikipedia

  • Sawtooth wave — The sawtooth wave (or saw wave) is a kind of non sinusoidal waveform. It is named a sawtooth based on its resemblance to the teeth on the blade of a saw.The usual convention is that a sawtooth wave ramps upward as time goes by and then sharply… …   Wikipedia

  • Square wave — A square wave is a kind of non sinusoidal waveform, most typically encountered in electronics and signal processing. An ideal square wave alternates regularly and instantaneously between two levels. Origins and uses Square waves are universally… …   Wikipedia

  • Memories Off 6: T-wave — One of the two PlayStation 2 cover artworks. Developer(s) 5pb. Series Memories Off …   Wikipedia

  • Pulse wave — A pulse wave or pulse train is a kind of non sinusoidal waveform that is similar to a square wave, but does not have the symmetrical shape associated with a perfect square wave. It is a term common to synthesizer programming, and is a typical… …   Wikipedia

  • Belgian UFO wave — The Belgian UFO wave refers to a series of sightings of triangular UFOs in Belgium, which lasted from November 29, 1989 to April 1990. The sightings The Belgian UFO wave peaked with the events of the night of 30/31 March 1990. On that night… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”