Triangle wave

Non-sinusoidal waveform
Triangle wave
A bandlimited triangle wave pictured in the time domain and frequency domain.
A bandlimited triangle wave[1] pictured in the time domain (top) and frequency domain (bottom). The fundamental is at 220 Hz (A3).
General information
General definition x ( t ) = 4 | t − ⌊ t + 3 / 4 ⌋ + 1 / 4 | − 1 {\displaystyle x(t)=4\left\vert t-\left\lfloor t+3/4\right\rfloor +1/4\right\vert -1}
Fields of applicationElectronics, synthesizers
Domain, codomain and image
Domain R {\displaystyle \mathbb {R} }
Codomain [ − 1 , 1 ] {\displaystyle \left[-1,1\right]}
Basic features
ParityOdd
Period1
Specific features
Root { n 2 } , n ∈ Z {\displaystyle \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} }
DerivativeSquare wave
Fourier series x ( t ) = − 8 π 2 ∑ k = 1 ∞ ( − 1 ) k ( 2 k − 1 ) 2 sin ⁡ ( 2 π ( 2 k − 1 ) t ) {\displaystyle x(t)=-{\frac {8}{{\pi }^{2}}}\sum _{k=1}^{\infty }{\frac {{\left(-1\right)}^{k}}{\left(2k-1\right)^{2}}}\sin \left(2\pi \left(2k-1\right)t\right)}
Triangle wave sound sample
5 seconds of triangle wave at 220 Hz
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Additive Triangle wave sound sample
After each second, a harmonic is added to a sine wave creating a triangle 220 Hz wave
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A triangular wave or 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).

Definitions

Sine, square, triangle, and sawtooth waveforms

Definition

A triangle wave of period p that spans the range [0, 1] is defined as

x ( t ) = 2 | t p − ⌊ t p + 1 2 ⌋ | , {\displaystyle x(t)=2\left|{\frac {t}{p}}-\left\lfloor {\frac {t}{p}}+{\frac {1}{2}}\right\rfloor \right|,}
where ⌊   ⌋ {\displaystyle \lfloor \ \rfloor } is the floor function. This can be seen to be the absolute value of a shifted sawtooth wave.

For a triangle wave spanning the range [−1, 1] the expression becomes

x ( t ) = 2 | 2 ( t p − ⌊ t p + 1 2 ⌋ ) | − 1. {\displaystyle x(t)=2\left|2\left({\frac {t}{p}}-\left\lfloor {\frac {t}{p}}+{\frac {1}{2}}\right\rfloor \right)\right|-1.}

Triangle wave with amplitude = 5, period = 4

A more general equation for a triangle wave with amplitude a {\displaystyle a} and period p {\displaystyle p} using the modulo operation and absolute value is

y ( x ) = 4 a p | ( ( x − p 4 ) mod p ) − p 2 | − a . {\displaystyle y(x)={\frac {4a}{p}}\left|\left(\left(x-{\frac {p}{4}}\right){\bmod {p}}\right)-{\frac {p}{2}}\right|-a.}

For example, for a triangle wave with amplitude 5 and period 4:

y ( x ) = 5 | ( ( x − 1 ) mod 4 ) − 2 | − 5. {\displaystyle y(x)=5\left|{\bigl (}(x-1){\bmod {4}}{\bigr )}-2\right|-5.}

A phase shift can be obtained by altering the value of the − p / 4 {\displaystyle -p/4} term, and the vertical offset can be adjusted by altering the value of the − a {\displaystyle -a} term.

As this only uses the modulo operation and absolute value, it can be used to simply implement a triangle wave on hardware electronics.

Note that in many programming languages, the % operator is a remainder operator (with result the same sign as the dividend), not a modulo operator; the modulo operation can be obtained by using ((x % p) + p) % p in place of x % p. In e.g. JavaScript, this results in an equation of the form 4*a/p * Math.abs((((x - p/4) % p) + p) % p - p/2) - a.

Relation to the square wave

The triangle wave can also be expressed as the integral of the square wave:

x ( t ) = ∫ 0 t sgn ⁡ ( sin ⁡ u p ) d u . {\displaystyle x(t)=\int _{0}^{t}\operatorname {sgn} \left(\sin {\frac {u}{p}}\right)\,du.}

Expression in trigonometric functions

A triangle wave with period p and amplitude a can be expressed in terms of sine and arcsine (whose value ranges from −π/2 to π/2):

y ( x ) = 2 a π arcsin ⁡ ( sin ⁡ ( 2 π p x ) ) . {\displaystyle y(x)={\frac {2a}{\pi }}\arcsin \left(\sin \left({\frac {2\pi }{p}}x\right)\right).}
The identity cos ⁡ x = sin ⁡ ( p 4 − x ) {\textstyle \cos {x}=\sin \left({\frac {p}{4}}-x\right)} can be used to convert from a triangle "sine" wave to a triangular "cosine" wave. This phase-shifted triangle wave can also be expressed with cosine and arccosine:
y ( x ) = a − 2 a π arccos ⁡ ( cos ⁡ ( 2 π p x ) ) . {\displaystyle y(x)=a-{\frac {2a}{\pi }}\arccos \left(\cos \left({\frac {2\pi }{p}}x\right)\right).}

Expressed as alternating linear functions

Another definition of the triangle wave, with range from −1 to 1 and period p, is

x ( t ) = 4 p ( t − p 2 ⌊ 2 t p + 1 2 ⌋ ) ( − 1 ) ⌊ 2 t p + 1 2 ⌋ . {\displaystyle x(t)={\frac {4}{p}}\left(t-{\frac {p}{2}}\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor \right)(-1)^{\left\lfloor {\frac {2t}{p}}+{\frac {1}{2}}\right\rfloor }.}

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 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 triangle ( t ) = 8 π 2 ∑ i = 0 N − 1 ( − 1 ) i n − 2 sin ⁡ ( 2 π f 0 n t ) , {\displaystyle x_{\text{triangle}}(t)={\frac {8}{\pi ^{2}}}\sum _{i=0}^{N-1}(-1)^{i}n^{-2}\sin(2\pi f_{0}nt),}
where N is the number of harmonics to include in the approximation, t is the independent variable (e.g. time for sound waves), f 0 {\displaystyle f_{0}} is the fundamental frequency, 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 converges quickly to the triangle wave as N tends to infinity, as shown in the animation.

Arc length

The arc length per period for a triangle wave, denoted by s, is given in terms of the amplitude a and period length p by

s = ( 4 a ) 2 + p 2 . {\displaystyle s={\sqrt {(4a)^{2}+p^{2}}}.}

See also

References

  1. ^ Kraft, Sebastian; Zölzer, Udo (5 September 2017). "LP-BLIT: Bandlimited Impulse Train Synthesis of Lowpass-filtered Waveforms". Proceedings of the 20th International Conference on Digital Audio Effects (DAFx-17). 20th International Conference on Digital Audio Effects (DAFx-17). Edinburgh. pp. 255–259.
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