... Filters¹

Adapted from arXiv.org publication arXiv:1606.06154 [cs.CE].

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

....²

In engineering, it is more common to use the unilateral Laplace transform, in which the integral traverses $[0,\infty)$ , and the differentiation theorem becomes ${\cal L}_s\{x^{(1)}\} = s\,X(s)-x(0)$ .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... parts.³

https://ccrma.stanford.edu/~jos/filters/Differentiation.html

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...,⁴

A function of time $f(t)$ is said to be causal if $f(t)=0$ for all $t<0$ .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... formula.⁵

https://en.wikipedia.org/wiki/Cauchy_formula_for_repeated_integration

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... function⁶

https://en.wikipedia.org/wiki/Gamma_function

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... differintegral,⁷

https://en.wikipedia.org/wiki/Differintegral

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

....⁸

Our definition of $\alpha$ is the negative of its usual definition in fractional calculus. We keep it so that $\alpha$ can refer to the slope of the Bode magnitude plot, instead of the negative slope. Note also that we allow $\alpha>0$ , but only consider $\alpha\in\mathbb{R}$ .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... calculus⁹

http://nrich.maths.org/1369

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... Toolbox,¹⁰

https://www.mathworks.com/

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... distribution.¹¹

http://www.gnu.org/software/octave/

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

...VossAndClarke78.¹²

http://123.physics.ucdavis.edu/week_3_files/voss-clarke.pdf

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... Web¹³

https://ccrma.stanford.edu/~jos/sasp/Example_Synthesis_1_F_Noise.html

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... distribution.¹⁴

faust.grame.fr

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... rates.¹⁵

http://www.firstpr.com.au/dsp/pink-noise/

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... log-frequency.¹⁶

https://en.wikipedia.org/wiki/Bode_plot

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

... behavior.¹⁷

https://en.wikipedia.org/wiki/Phase_margin

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.