aanl.lib
A library for antialiased nonlinearities. Its official prefix is aa.
This library provides aliasing-suppressed nonlinearities through first-order and second-order approximations of continuous-time signals, functions, and convolution based on antiderivatives. This technique is particularly effective if combined with low-factor oversampling, for example, operating at 96 kHz or 192 kHz sample-rate.
The library contains trigonometric functions as well as other nonlinear functions such as bounded and unbounded saturators.
Due to their limited domains or ranges, some of these functions may not suitable for audio nonlinear processing or waveshaping, although they have been included for completeness. Some other functions, for example, tan() and tanh(), are only available with first-order antialiasing due to the complexity of the antiderivative of the x * f(x) term, particularly because of the necessity of the dilogarithm function, which requires special implementation.
Future improvements to this library may include an adaptive mechanism to set the ill-conditioned cases threshold to improve performance in varying cases.
Note that the antialiasing functions introduce a delay in the path, respectively half and one-sample delay for first and second-order functions.
Also note that due to division by differences, it is vital to use double-precision or more to reduce errors.
The environment identifier for this library is aa. After importing
the standard libraries in Faust, the functions below can be called as aa.function_name.
References
- https://github.com/grame-cncm/faustlibraries/blob/master/aanl.lib
- Reducing the Aliasing in Nonlinear Waveshaping Using Continuous-time Convolution, Julian Parker, Vadim Zavalishin, Efflam Le Bivic, DAFX, 2016
- http://dafx16.vutbr.cz/dafxpapers/20-DAFx-16_paper_41-PN.pdf
Auxiliary Functions
(aa.)clip
Clipping function.
(aa.)Rsqrt
Real-valued sqrt().
(aa.)Rlog
Real-valued log().
(aa.)Rtan
Real-valued tan().
(aa.)Racos
Real-valued acos().
(aa.)Rasin
Real-valued asin().
(aa.)Racosh
Real-valued acosh()
(aa.)Rcosh
Real-valued cosh().
(aa.)Rsinh
Real-valued sinh().
(aa.)Ratanh
Real-valued atanh().
(aa.)ADAA1
Generalised first-order ADAA function.
Usage
_ : ADAA1(EPS, f, F1) : _
Where:
EPS: a threshold to handle ill-conditioned casesf: a function that we want to process with ADAAF1: f's first antiderivative
(aa.)ADAA2
Generalised second-order ADAA function.
Usage
_ : ADAA2(EPS, f, F1, F2) : _
Where:
EPS: a threshold to handle ill-conditioned casesf: a function that we want to process with ADAAF1: f's first antiderivativeF2: f's second antiderivative
Main functions
Saturators
These antialiased saturators perform best with high-amplitude input signals. If the input is only slightly saturated, hence producing negligible aliasing, the trivial saturator may result in a better overall output, as noise can be introduced by first and second ADAA at low amplitudes.
Once determining the lowest saturation level for which the antialiased functions perform adequately, it might be sensible to cross-fade between the trivial and the antialiased saturators according to the amplitude profile of the input signal.
(aa.)hardclip
First-order ADAA hard-clip.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.hardclip : _
(aa.)hardclip2
Second-order ADAA hard-clip.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.hardclip2 : _
(aa.)cubic1
First-order ADAA cubic saturator.
The domain of this function is ℝ; its theoretical range is [-2.0/3.0; 2.0/3.0].
Usage
_ : aa.cubic1 : _
(aa.)parabolic
First-order ADAA parabolic saturator.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.parabolic : _
(aa.)parabolic2
Second-order ADAA parabolic saturator.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.parabolic : _
(aa.)hyperbolic
First-order ADAA hyperbolic saturator.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.hyperbolic : _
(aa.)hyperbolic2
Second-order ADAA hyperbolic saturator.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.hyperbolic2 : _
(aa.)sinarctan
First-order ADAA sin(atan()) saturator.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.sinatan : _
(aa.)sinarctan2
Second-order ADAA sin(atan()) saturator.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.sinarctan2 : _
(aa.)softclipQuadratic1
First-order ADAA quadratic softclip.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.softclipQuadratic1 : _
(aa.)softclipQuadratic2
Second-order ADAA quadratic softclip.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.softclipQuadratic2 : _
(aa.)tanh1
First-order ADAA tanh() saturator.
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.tanh1 : _
(aa.)arctan
First-order ADAA atan().
The domain of this function is ℝ; its theoretical range is [-π/2.0; π/2.0].
Usage
_ : aa.arctan : _
(aa.)arctan2
Second-order ADAA atan().
The domain of this function is ℝ; its theoretical range is ]-π/2.0; π/2.0[.
Usage
_ : aa.arctan2 : _
(aa.)asinh1
First-order ADAA asinh() saturator (unbounded).
The domain of this function is ℝ; its theoretical range is ℝ.
Usage
_ : aa.asinh1 : _
(aa.)asinh2
Second-order ADAA asinh() saturator (unbounded).
The domain of this function is ℝ; its theoretical range is ℝ.
Usage
_ : aa.asinh2 : _
Trigonometry
These functions are reliable if input signals are within their domains.
(aa.)cosine1
First-order ADAA cos().
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.cosine1 : _
(aa.)cosine2
Second-order ADAA cos().
The domain of this function is ℝ; its theoretical range is [-1.0; 1.0].
Usage
_ : aa.cosine2 : _
(aa.)arccos
First-order ADAA acos().
The domain of this function is [-1.0; 1.0]; its theoretical range is [π; 0.0].
Usage
_ : aa.arccos : _
(aa.)arccos2
Second-order ADAA acos().
The domain of this function is [-1.0; 1.0]; its theoretical range is [π; 0.0].
Note that this function is not accurate for low-amplitude or low-frequency input signals. In that case, the first-order ADAA arccos() can be used.
Usage
_ : aa.arccos2 : _
(aa.)acosh1
First-order ADAA acosh().
The domain of this function is ℝ >= 1.0; its theoretical range is ℝ >= 0.0.
Usage
_ : aa.acosh1 : _
(aa.)acosh2
Second-order ADAA acosh().
The domain of this function is ℝ >= 1.0; its theoretical range is ℝ >= 0.0.
Note that this function is not accurate for low-frequency input signals. In that case, the first-order ADAA acosh() can be used.
Usage
_ : aa.acosh2 : _
(aa.)sine
First-order ADAA sin().
The domain of this function is ℝ; its theoretical range is ℝ.
Usage
_ : aa.sine : _
(aa.)sine2
Second-order ADAA sin().
The domain of this function is ℝ; its theoretical range is ℝ.
Usage
_ : aa.sine2 : _
(aa.)arcsin
First-order ADAA asin().
The domain of this function is [-1.0, 1.0]; its theoretical range is [-π/2.0; π/2.0].
Usage
_ : aa.arcsin : _
(aa.)arcsin2
Second-order ADAA asin().
The domain of this function is [-1.0, 1.0]; its theoretical range is [-π/2.0; π/2.0].
Note that this function is not accurate for low-frequency input signals. In that case, the first-order ADAA asin() can be used.
Usage
_ : aa.arcsin2 : _
(aa.)tangent
First-order ADAA tan().
The domain of this function is [-π/2.0; π/2.0]; its theoretical range is ℝ.
Usage
_ : aa.tangent : _
(aa.)atanh1
First-order ADAA atanh().
The domain of this function is [-1.0; 1.0]; its theoretical range is ℝ.
Usage
_ : aa.atanh1 : _
(aa.)atanh2
Second-order ADAA atanh().
The domain of this function is [-1.0; 1.0]; its theoretical range is ℝ.
Usage
_ : aa.atanh2 : _