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Waveshaper

Uses[edit]

Waveshapers are used mainly by electronic musicians 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 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 tubeordiode limiter.^[2]

How it works[edit]

A waveshaper is an 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 functionortransfer function, to the input signal (the term shaping function is preferred to avoid confusion with the transfer function from systems theory).^[3] The function can be any function at all.

Mathematically, the operation is defined by the waveshaper equation

${\displaystyle y=f(a($t)x(t))}

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.^[4] This parameter a is often used as a constant gain factor called the distortion index.^[5] 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[edit]

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[edit]

Apolynomial is a function of the form

${\displaystyle f($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_{n}x^{n}}

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 Nth harmonic of the sinusoid. To prove this, consider a sinusoid used as input to the general polynomial.

${\displaystyle \sum _{n=0}^{N}a_{n}(\alpha \cos(\omega t+\phi ))^{n}}$

Next, use the inverse Euler's formula to obtain complex sinusoids.

${\displaystyle \sum _{n=0}^{N}a_{n}{\Bigg (}\alpha {\frac {e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )}}{2}}{\Bigg )}^{n}=a_{0}+\sum _{n=1}^{N}{\frac {a_{n}\alpha ^{n}}{2^{n-1}}}{\frac {(e^{j(\omega t+\phi )}+e^{-j(\omega t+\phi )})^{n}}{2}}}$

Finally, use the binomial formula to transform back to trigonometric form and find coefficients for each harmonic.

${\displaystyle a_{0}+\sum _{n=1}^{N}{\Bigg [}{{\frac {a_{n}\alpha ^{n}}{2^{n-1}}}\sum _{k=0}^{n}{{n \choose k}{\frac {e^{j(n-k)(\omega t+\phi )}e^{-jk(\omega t+\phi )}}{2}}}{\Bigg ]}}=a_{0}+\sum _{n=1}^{N}{\Bigg [}{{\frac {a_{n}\alpha ^{n}}{2^{n-1}}}\sum _{k=0}^{n}{{n \choose k}{\frac {e^{j(n-2k)(\omega t+\phi )}}{2}}}{\Bigg ]}}}$

${\displaystyle =a_{0}+\sum _{n=1}^{N}{\Bigg [}{{\frac {a_{n}\alpha ^{n}}{2^{n-1}}}\sum _{k=0}^{\lfloor n/2\rfloor }{{n \choose k}\cos {((n-2k)(\omega t+\phi ))}}{\Bigg ]}}}$

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 ${\displaystyle N\omega }$.
• All odd monomial terms ${\displaystyle x^{n}}$ 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 ${\displaystyle a_{n}}$ coefficient and the amplitude of the input by ${\displaystyle {\frac {a_{n}\alpha ^{n}}{2^{n-1}}}}$

Problems associated with waveshapers[edit]

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]}.

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]}. 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]}.

Sources[edit]