Compressors: Formula for Maximum Volume

Suppose we have an audio signal which peaks at L decibels. We apply a compressor with a threshold T (with T being smaller than L, otherwise the compressor will not spring into action) and ratio r. How does this effect the maximum volume of the audio signal? Let’s derive a formula for that. Remember that the compressor leaves the parts of the signal that are below the threshold unchanged and dampens the excess volume (threshold to signal level) by the ratio we set. So the dynamic range from the threshold to the peak, which is L – T, is compressed to (L – T) / r. Hence, the peak volume after compression is:

L’ = T + (L – T) / r

For example, suppose our mix peaks at L = – 2 dB. We compress it using a threshold of T = – 10 dB and a ratio r = 2:1. The maximum volume after compression is:

L’ = – 10 dB + ( – 2 dB – (- 10 dB) ) / 2 = – 10 dB + 8 dB / 2 = – 6 dB