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{{for|the Swedish electronic musician|Waveshaper (musician)}}
<!-- Deleted image removed: [[Image:Waveshaper.JPG|frame|right|A screenshot of the Fruity Waveshaper plugin]] -->
In [[electronic music]], '''waveshaping''' is a type of [[distortion synthesis]] in which complex [[Spectrum|spectra]] are produced from simple tones by altering the shape of the [[waveform]]s.<ref>Charles Dodge and Thomas A. Jersey (1997). ''Computer Music: Synthesis, Composition, and Performance'', "Glossary", p.438. {{ISBN|0-02-864682-7}}.</ref>
== Uses ==
Waveshapers are used mainly by [[electronic musician]]s to achieve an extra-abrasive sound. This effect is most used to enhance the sound of a music [[synthesizer]] by altering the waveform or vowel. Rock musicians may also use a waveshaper for heavy [[distortion]] of a guitar or bass. Some synthesizers or virtual software instruments have built-in waveshapers. The effect can make instruments sound noisy or [[Distortion (music)|overdriven]].
In digital modeling of analog audio equipment such as [[tube amplifiers]], waveshaping is used to introduce a static, or memoryless, nonlinearity to approximate the transfer characteristic of a [[vacuum tube]] or [[diode]] limiter.<ref>Yeh, David T. and Pakarinen, Jyri (2009). "A Review of Digital Techniques for Modeling Vacuum-Tube Guitar Amplifiers", ''Computer Music Journal'', 33:2, pp. 89-90</ref>
== How it works ==
A waveshaper is an [[audio signal processing|audio effect]] that changes an audio signal by mapping an input signal to the output signal by applying a fixed or variable mathematical function, called the ''shaping function'' or ''transfer function'', to the input signal (the term shaping function is preferred to avoid confusion with the [[transfer function]] from systems theory).<ref>http://www.music.mcgill.ca/~gary/courses/2012/307/week12/node2.html {{Dead link|date=February 2022}}</ref> The function can be any function at all.
Mathematically, the operation is defined by the ''waveshaper equation''
:<math>y = f(a(t)x(t))</math>
where ''f'' is the shaping function, ''x(t)'' is the input function, and ''a(t)'' is the ''index function'', which in general may vary as a function of time.<ref>Le Brun, Marc (1979). "Digital Waveshaping Synthesis", ''Journal of the Audio Engineering Society'', 27:4, p. 250</ref> This parameter ''a'' is often used as a constant gain factor called the ''distortion index''.<ref>http://www.music.mcgill.ca/~gary/courses/2012/307/week12/node4.html {{Dead link|date=February 2022}}</ref> In practice, the input to the waveshaper, x, is considered on [-1,1] for digitally sampled signals, and f will be designed such that y is also on [-1,1] to prevent unwanted clipping in software.
== Commonly used shaping functions ==
Sin, arctan, polynomial functions, or piecewise functions (such as the hard clipping function) are commonly used as waveshaping transfer functions. It is also possible to use table-driven functions, consisting of discrete points with some degree of interpolation or linear segments.
===Polynomials===
A [[polynomial]] is a function of the form
<math> f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0 = \sum_{n=0}^{N}a_nx^n </math>
Polynomial functions are convenient as shaping functions because, when given a single sinusoid as input, a polynomial of degree ''N'' will only introduce up to the ''N''th harmonic of the sinusoid. To prove this, consider a sinusoid used as input to the general polynomial.
:<math>\sum_{n=0}^{N}a_n(\alpha \cos(\omega t + \phi))^n </math>
Next, use the inverse [[Euler's formula]] to obtain complex sinusoids.
:<math>\sum_{n=0}^{N}a_n \Bigg(\alpha \frac{e^{j(\omega t + \phi)}+e^{-j(\omega t + \phi)}}{2}\Bigg)^n
= a_0 + \sum_{n=1}^{N}\frac{a_n \alpha^n}{2^{n-1}}\frac{(e^{j(\omega t + \phi)}+e^{-j(\omega t + \phi)})^n}{2}</math>
Finally, use the [[binomial formula]] to transform back to trigonometric form and find coefficients for each harmonic.
:<math>a_0 + \sum_{n=1}^{N}\Bigg[{\frac{a_n \alpha^n}{2^{n-1}} \sum_{k=0}^{n} {{n \choose k} \frac{e^{j(n-k)(\omega t + \phi)}e^{-jk(\omega t + \phi)}}{2}}\Bigg]}
=a_0 + \sum_{n=1}^{N}\Bigg[{\frac{a_n \alpha^n}{2^{n-1}} \sum_{k=0}^{n} {{n \choose k} \frac{e^{j(n-2k)(\omega t + \phi)}}{2}}\Bigg]}
</math>
<math>
=a_0 + \sum_{n=1}^{N}\Bigg[{\frac{a_n \alpha^n}{2^{n-1}} \sum_{k=0}^{\lfloor n/2 \rfloor} {{n \choose k} \cos {((n-2k)(\omega t + \phi))}}\Bigg]}
</math>
From the above equation, several observations can be made about the effect of a polynomial shaping function on a single sinusoid:
*All of the sinusoids generated are harmonically related to the original input.
*The frequency never exceeds <math>N\omega</math>.
*All odd monomial terms <math>x^n</math> generate odd harmonics from ''n'' down to the fundamental, and all even monomial terms generate even harmonics from ''n'' down to DC (0 frequency).
*The shape of the spectrum produced by each monomial term is fixed and determined by the binomial coefficients .
*The weight of that spectrum in the overall output is determined solely by its <math>a_n</math> coefficient and the amplitude of the input by <math>\frac{a_n \alpha^n}{2^{n-1}}</math>
== Problems associated with waveshapers ==
The sound produced by digital waveshapers tends to be harsh and unattractive, because of problems with aliasing. Waveshaping is a non-linear operation, so it's hard to generalize about the effect of a waveshaping function on an input signal. The mathematics of non-linear operations on audio signals is difficult, and not well understood. The effect will be amplitude-dependent, among other things. But generally, waveshapers—particularly those with sharp corners (e.g., some derivatives are discontinuous) -- tend to introduce large numbers of high frequency harmonics. If these introduced harmonics exceed the [[Nyquist limit]], then they will be heard as harsh inharmonic content with a distinctly metallic sound in the output signal. [[Oversampling]] can somewhat but not completely alleviate this problem, depending on how fast the introduced harmonics fall off{{Citation needed|date=June 2024}}.
With relatively simple, and relatively smooth waveshaping functions (sin(a*x), atan(a*x), polynomial functions, for example), this procedure may reduce aliased content in the harmonic signal to the point that it is musically acceptable{{Citation needed|date=June 2024}}. But waveshaping functions other than polynomial waveshaping functions will introduce an infinite number of harmonics into the signal, some which may audibly alias even at the supersampled frequency{{Citation needed|date=June 2024}}.
==Sources==
{{reflist}}
[[Category:Electronic music]]