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The Info List - Triangle Wave


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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. 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[edit]

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 summing odd harmonics of the fundamental while multiplying every other odd harmonic by −1 (or, equivalently, changing its phase by π) and multiplying the amplitude of the harmonics by one over the square of their mode number, n, (which is equivalent to one over the square of their relative frequency to the fundamental). The above can be summarised mathematically as follows:

x

t r i a n g l e

( t )

=

i = 0

N

( − 1

)

i

n

− 2

(

sin ⁡ [ n t ]

)

displaystyle begin aligned x_ mathrm triangle (t)& =sum _ i=0 ^ N (-1)^ i n^ -2 left(sin[nt]right)end aligned

where N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), and i is the harmonic label which is related to its mode number by

n = 2 i + 1

displaystyle n=2i+1

. This infinite Fourier series
Fourier series
converges to the triangle wave as N tends to infinity, as shown in the animation. Definitions[edit]

Sine, square, triangle, and sawtooth waveforms

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

− 1

displaystyle x(t)=2left2left( t over a -leftlfloor t over a + 1 over 2 rightrfloor right)-1right-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

π

arcsin ⁡

(

sin ⁡

(

2 π

p

x

)

)

displaystyle y(x)= frac 2a pi arcsin left(sin left( frac 2pi p xright)right)

Arc Length[edit] 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[edit]

List of periodic functions Sawtooth wave Sound Triangle
Triangle
function Wave Zigzag

References[edit]

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

v t e

Waveforms

Sine wave Non-sinusoidal

Rectangular wave Sawtooth wave Square

.