HOME
The Info List - Triangle Wave



--- Advertisement ---


(i) (i)

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).

CONTENTS

* 1 Harmonics * 2 Definitions * 3 Arc Length * 4 See also * 5 References

HARMONICS

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

It is possible to approximate a triangle wave with additive synthesis by adding odd harmonics of the fundamental, multiplying every (4nāˆ’1)th harmonic by āˆ’1 (or changing its phase by Ļ€), 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: x t r i a n g l e ( t ) = 8 2 k = 0 ( 1 ) k sin ( 2 ( 2 k + 1 ) f t ) ( 2 k + 1 ) 2 = 8 2 ( sin ( 2 f t ) 1 9 sin ( 6 f t ) + 1 25 sin ( 10 f t ) ) {displaystyle {begin{aligned}x_{mathrm {triangle} }(t)&{}={frac {8}{pi ^{2}}}sum _{k=0}^{infty }(-1)^{k},{frac {sin left(2pi (2k+1)ftright)}{(2k+1)^{2}}}\ width:67.27ex; height:13.509ex;" alt="{begin{aligned}x_{{mathrm {triangle}}}(t)&{}={frac {8}{pi ^{2}}}sum _{{k=0}}^{infty }(-1)^{k},{frac {sin left(2pi (2k+1)ftright)}{(2k+1)^{2}}}\"> Sine , square , triangle, and sawtooth waveforms

Another definition of the triangle wave, with range from -1 to 1 and period 2a is: x ( t ) = 2 a ( t a t a + 1 2 ) ( 1 ) t a + 1 2 {displaystyle x(t)={frac {2}{a}}left(t-aleftlfloor {frac {t}{a}}+{frac {1}{2}}rightrfloor right)(-1)^{leftlfloor {frac {t}{a}}+{frac {1}{2}}rightrfloor }} where the symbol n {displaystyle scriptstyle lfloor nrfloor } represent the floor function of n.

Also, the triangle wave can be the absolute value of the sawtooth wave : x ( t ) = 2 ( t a t a + 1 2 ) {displaystyle x(t)=left2left({t over a}-leftlfloor {t over a}+{1 over 2}rightrfloor right)right}

or, for a range from -1 to +1: x ( t ) = 2 2 ( t a t a + 1 2 ) 1 {displaystyle x(t)=2left2left({t over a}-leftlfloor {t over a}+{1 over 2}rightrfloor right)right-1}

The triangle wave can also be expressed as the integral of the square wave : sgn ( sin ( x ) ) d x {displaystyle int operatorname {sgn}(sin(x)),dx,}

A simple equation with a period of 4, with y ( 0 ) = 1 {displaystyle y(0)=1} . 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: y ( x ) = x mod 4 2 1 {displaystyle y(x)=x,{bmod {,}}4-2-1}

or, a more complex and complete version of the above equation with a period of p {displaystyle p} , amplitude a {displaystyle a} , and starting with y ( 0 ) = a / 2 {displaystyle y(0)=a/2} : y ( x ) = 2 a p ( ( x mod p ) p 2 p 4 ) {displaystyle y(x)={frac {2a}{p}}{Biggl (}{biggl }left(x{bmod {p}}right)-{frac {p}{2}}{biggr }-{frac {p}{4}}{Biggr )}}

The function (1) is a specialization of (2), with a=2 and p=4: y ( x ) = 2 2 4 ( ( x mod 4 ) 4 2 4 4 ) {displaystyle y(x)={frac {2times 2}{4}}{Biggl (}{biggl }left(x{bmod {4}}right)-{frac {4}{2}}{biggr }-{frac {4}{4}}{Biggr )}Leftrightarrow } y ( x ) = ( ( x mod 4 ) 2 1 ) {displaystyle y(x)={Biggl (}{biggl }left(x{bmod {4}}right)-2{biggr }-1{Biggr )}}

An odd version of the function (1) can be made, just shifting by one the input value, which will change the phase of the original function: y ( x ) = ( x 1 ) mod 4 2 1 {displaystyle y(x)=(x-1),{bmod {,}}4-2-1}

Generalizing the formula (3) to make the function odd for any period and amplitude gives: y ( x ) = 4 a p ( ( ( x p 4 ) mod p ) p 2 p 4 ) {displaystyle y(x)={frac {4a}{p}}{Biggl (}{biggl }left((x-{frac {p}{4}}){bmod {p}}right)-{frac {p}{2}}{biggr }-{frac {p}{4}}{Biggr )}}

In terms of sine and arcsine with period p and amplitude a: y ( x ) = 2 a arcsin ( sin ( 2 p x ) ) {displaystyle y(x)={frac {2a}{pi }}arcsin left(sin left({frac {2pi }{p}}xright)right)}

Note: sin y = cos x

ARC LENGTH

The arc length per period "s" for a triangle wave, given the amplitude "a" and period length "p": s = ( 4 a ) 2 + p 2 {displaystyle s={sqrt {(4a)^{2}+p^{2}}}}

SEE ALSO

* List of periodic functions * Triangle
Triangle
function * Waves * Sawtooth wave
Sawtooth wave
* Sound
Sound
* Zigzag
Zigzag

REFERENCES

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

* v * t * e

Waveforms

* Sine wave

* Non-sinusoidal

* Rectangular wave * Sawtooth wave
Sawtooth wave
* Square wave
Square wave
*