Mathematical Treatment

The mathematics are actually not too difficult. I worked it out by plotting the standard set up outlined (16 dB GR @ 2:1 +1 dBsc) and then worked out the relationships on a calculator.

Firstly we work out the system threshold. For this example I will assume the standard set-up above.

System Threshold

= [ Gain Reduction x Ratio ] / Ratio -1

= - 16 x 2 (:1) / 1

= - 32 dB

Next we calculate the set point for + 1 dB of system gain (dBsc). The set-point is how much the secondary path is offset relative to the main path.

Set-point

= [ ( antilog ( system gain increase / 20 ) - 1) ] log20

= [ ( antilog ( 1 / 20 ) - 1 ) ] log 20

= - 18.27 dB

From the set-point and threshold we derive the level added to the main signal path at the threshold.

Compressor's effective output at threshold

= (threshold / ratio) + set-point

= ( -32 / 2 ) + ( -18.27 )

= -34.27 dB

Adding the input threshold and the compressor's effective output at that threshold gives us the total system output at the threshold.

System output at threshold

= [( antilog (output at threshold / 20)) + ( antilog (threshold / 20))] log20

= [( antilog -34.27 / 20 ) + ( antilog -32 / 20 )] log 20

= -27 dB

The difference between the system threshold and the output at that threshold is the amount of lift.

Lift at threshold

= system output - threshold

= ( -27 ) - ( -32 )

= 5 dB

We work out the resultant gain law as follows.

Ratio

= threshold / ( system output at threshold - system gain )

= -32 / (( -27 ) - ( +1 ))

= 1.14:1

Some general patterns emerge if multiple settings are plotted against each other. Figure 8 shows how altering different parameters affects the resultant transfer characteristic. The general 'rules' are as follows:

Increasing gain reduction lowers the threshold and increases the lift at that point.
Decreasing gain reduction raises the threshold and decreases the lift at that point.
Tighter ratios on the compressor with the same gain reduction and set-point give the same threshold with a greater amount of lift.
Given the same GR and set-point the steeper compression ratios give progressively less boost at the threshold.

I have written a Microsoft Excel Spreadsheet that does all this math AND draws graphs of the resultant transfer characteristics. Email me if you want a copy of this.