NOTES.txt

Compressor musings

  If an envelope follower consists of a one-pole IIR filter that obeys

    out(t) = out(t-1) * alpha + in(t) * (1 - alpha)
  
  Then a step input of size x at t=0 will produce:

    out(t) = x - x * e^(t*ln(alpha))
           = x * (1 - e^(t*ln(alpha)))

The process of measuring attack times and converting them to values usable
in this software.

  Measure or prepare the step test gain:
  
  s = step size, raw ratio (e.g 13dB = 10^1.3 = 20)
  
  Measure or assume the expansion ratio:
  
  r = response_level / pre_response_level
  (or, using log scale, r = response_dB - pre_response_dB)
  
  Calculate:
  
  y = s - s * e^-1 = s (1 - s^-1)
  
    This is the -1 neper response value; the point at which the
    follower estimation of the step input has climbed so that the
    error between the estimation and the input (step input - the estimation),
    is equal to step_input * -1 neper.
 
  Deriving t - the attack time unit:
  
  Now find the time at which the follower estimation has climbed to y.
  This is t, the attack time in the neper basis.
  
  Measuring the real follower's estimation value when looking at signals from
  the full expander is complicated because we can't directly see it. We can
  only observe the envelope follower's output by its effect on the gain that
  has been applied to the signal. We have to work backwards by observing the
  gain and removing the expansion ratio.
    
  To do this, we take the observed signal level (in dB), subtract the level
  of the observed signal at the first instant the change in input levels is
  encountered (again, in dB), and then derate for the expansion ratio.
  
  o(0) = observed signal level (in dB) at start of step pulse
  o(n) = observed signal level (in dB) at time n.
  
  We can then derive v(n), which is the inferred value of the envelope
  follower at time n.
  
            [ o(n) - o(0) ]
  v(n) = 10^[ ----------- ]
            [   (r - 1)   ]
            
  The final step is to examine v(n) and find the point in time, n, where
  v(n) = y.



Example: How I probed the DBX expander in my tape unit.
  
  Test signal: 1 kHz tone alternating between 0 dB RMS and -10 dB RMS, recorded
               to tape without encoding, and with enough gain that it
               alternated between 0 dB and -10 dB on the tape meter when played
               back (without decoding).

  This gives us our first value, the step input gain:
    
  s = 10.0 (not decibels; raw gain value)

  We can also compute the -1 neper error in response to this step:
   
  y = s * (1 - e^-1) = 10 * (1 - e^-1) = 6.3

  This is the raw value at which a one-pole envelope follower's error will have
  been reduced to -1 neper of the step input size.
  
  Now the first measurements. I played this recorded signal back, making
  sure to engage the DBX decoder, and digitized it. Examining the recording,
  I observed that the quieter tone was digitized at -39 dBfs RMS and the louder
  tone eventually settled in at -18.5 dbFS RMS.
  
  This yields our first measurement: the expansion ratio.
  
  r = (-18.5 dB - -39 dB) / 10 dB
    = 2.05 (dimensionless)
  
  The next step is to figure out the signal level we should expect to see once
  the internal envelope follower's response has reached y times its starting
  value.
  
  Since y is a ratio in this scenario, we can rephrase it as decibels of
  gain, z:
  
  z = dB(y) = 8.00 dB (~= 6.3)
  
  Now we make a measurement of the input signal to determine o(0). I found that
  the first peak of the higher sine wave section was digitized at -26.25 dB.
  Since all my other measurements are in RMS, however, I must reduce this
  value by 3 dB to arrive at the RMS value: -29.25 dB.
  
  o(0) = -29.25 dB
  
  We can now compute what o(n) will have to be when n is the neper attack
  time:
  
  o(attack_time) =
       = o(0) + z * (r - 1)
       = -29.25 dB + 8.00 dB * (r - 1)
       = -29.25 dB + 8.00 dB * 1.05
       = -29.25 dB + 8.40 dB
       = -20.85 dB
  
  I must now search the decoder response signal until I find that the signal
  has risen to -20.85 dB (RMS). Since it is much easier to observe sine wave
  peaks than RMS values, I adjust this to:
  
  o(attack_time) peak = -20.85 dB RMS + 3 dB = -17.85 dB peak
  
  The signal reaches this value 27.479 ms after the step input. Thus,
  
  attack_time (neper-based) = 27.479 milliseconds
  
  I stress that this is the NEPER-BASED attack time. I think the industry
  standard attack time base is decibel, and thus, to convert to that time
  base, you'd have to multiply the neper-based time by some constant, probably
  1 / log10(e). That would make its decibel attack time:
  
  attack_time (decibel-based) = 11.93 milliseconds