Dynamic range compressor
Audio Toolbox / Dynamic Range Control
The Compressor block performs dynamic range compression independently across each input channel. Dynamic range compression attenuates the volume of loud sounds that cross a given threshold. The block uses specified attack and release times to achieve a smooth applied gain curve.
Data Types 

Direct Feedthrough 

Multidimensional Signals 

VariableSize Signals 

ZeroCrossing Detection 

The Compressor block processes a signal frame by frame and element by element.
The Npoint signal, x[n], is converted to decibels:
$${x}_{\text{dB}}[n]=20\times {\mathrm{log}}_{10}\leftx[n]\right$$
x_{dB}[n] passes through the gain computer. The gain computer uses the static compression characteristic of the Compressor block to attenuate gain that is above the threshold.
If you specified a soft knee, the gain computer has the following static characteristic:
$${x}_{\text{sc}}({x}_{\text{dB}})=\{\begin{array}{cc}{x}_{\text{dB}}& {x}_{\text{dB}}<\left(T\frac{W}{2}\right)\\ {x}_{\text{dB}}+\frac{\left(\frac{1}{R}1\right){\left({x}_{\text{dB}}T+\frac{W}{2}\right)}^{2}}{2W}& \left(T\frac{W}{2}\right)\le {x}_{\text{dB}}\le \left(T+\frac{W}{2}\right)\\ T+\frac{\left({x}_{\text{dB}}T\right)}{R}& {x}_{\text{dB}}>\left(T+\frac{W}{2}\right)\end{array}\text{\hspace{1em}},$$
where T is the threshold, R is the compression ratio, and W is the knee width.
If you specified a hard knee, the gain computer has the following static characteristic:
$${x}_{\text{sc}}({x}_{\text{dB}})=\{\begin{array}{cc}{x}_{\text{dB}}& {x}_{\text{dB}}<T\\ T+\frac{\left({x}_{\text{dB}}T\right)}{R}& {x}_{\text{dB}}\ge T\end{array}$$
The computed gain, g_{c}[n], is calculated as
$${g}_{\text{c}}[n]={x}_{\text{sc}}[n]{x}_{\text{dB}}[n].$$
g_{c}[n] is smoothed using specified attack and release time parameters:
$${g}_{\text{s}}\text{[}n]=\{\begin{array}{cc}{\alpha}_{\text{A}}{g}_{\text{s}}[n1]+(1{\alpha}_{\text{A}}){g}_{\text{c}}[n],& {g}_{\text{c}}[n]\le {g}_{\text{s}}[n1]\\ {\alpha}_{\text{R}}{g}_{\text{s}}\text{[}n1]+(1{\alpha}_{\text{R}}){g}_{\text{c}}[n],& {g}_{\text{c}}[n]>{g}_{\text{s}}[n1]\end{array}$$
The attack time coefficient, α _{A}, is calculated as
$${\alpha}_{\text{A}}=\mathrm{exp}\left(\frac{\mathrm{log}(9)}{Fs\times {T}_{\text{A}}}\right)\text{\hspace{0.17em}}.$$
The release time coefficient, α _{R}, is calculated as
$${\alpha}_{\text{R}}=\mathrm{exp}\left(\frac{\mathrm{log}(9)}{Fs\times {T}_{\text{R}}}\right)\text{\hspace{0.17em}}.$$
T _{A} is the attack time period, specified by the Attack time (s) parameter. T_{R} is the release time period, specified by the Release time (s) parameter. Fs is the input sampling rate, specified by the Inherit sample rate from input or the Input sample rate (Hz) parameter.
If Makeup gain (dB) is set to Auto
, the
makeup gain is calculated as the negative of the computed gain for a 0 dB input:
$$M={{x}_{\text{sc}}}_{{x}_{\text{dB}}=0}.$$
Given a steadystate input of 0 dB, this configuration achieves a steadystate output of 0 dB. The makeup gain is determined by the Threshold (dB), Ratio, and Knee width (dB) parameters. It does not depend on the input signal.
The makeup gain, M, is added to the smoothed gain, g_{s}[n]:
$${g}_{\text{m}}[n]={g}_{\text{s}}[n]+M$$
The calculated gain in dB, g_{dB}[n], is translated to a linear domain:
$${g}_{\text{lin}}[n]={10}^{\left(\frac{{g}_{\text{m}}[n]}{20}\right)}$$
The output of the dynamic range compressor is given as
$$y[n]=x[n]\times {g}_{\text{lin}}[n].$$
[1] Giannoulis, Dimitrios, Michael Massberg, and Joshua D. Reiss. "Digital Dynamic Range Compressor Design –– A Tutorial And Analysis." Journal of Audio Engineering Society. Vol. 60, Issue 6, 2012, pp. 399–408.