Dynamic Range Compression/Expansion

• Dynamic range compression or expansion involves the modification, typically via a time-varying gain control, of a signal's dynamic range.

• The reduction of a signal's dynamic range is referred to as compression, while a range increase is referred to as expansion

• Applications of dynamic range reduction/compression include:
  • increase perceptual loudness;
  • peak limiting;
  • transmission to a system with lower dynamic range;
  • reproduction of material with wide dynamic range in noisy environments;
  • to achieve timbre variations.

• Applications of dynamic range expansion include:
  • noise suppression (noise gate or downward expansion);
  • restore dynamic level of a previously compressed signal;
  • add dynamic range for perceptual effect.

• If a signal $x(t)$ has a short-term signal level given by $\lambda( x(t), \tau)$ over the period $\tau$, the dynamic range of the signal is given by $\max(\lambda) / \min(\lambda)$, and is usually expressed in decibels.

• Various metrics exist for estimating or evaluating the short-term level of a signal, including peak, average, and average root-power levels.

• The short-term peak value of a signal (i.e., the greatest absolute value within the last $\tau$ seconds) is given by:
  $$\lambda_\infty(x(t), \tau) = \max_{t-\tau<\xi<t} \vert x(\xi)\vert$$

  Figure 5: A signal and its short-term peak and average root-power level estimates.

  

• The short-term average root-power level can be defined as:
  $$\lambda_2(x(t), \tau) = \sqrt{x^2(t) * \left[ \frac{e^{-t/\tau}}{\tau} \cdot u(t)\right]},$$
  where $*$ denotes convolution and $u(t)$ is the unit step function. The value of $\lambda_2$ is thus formed by squaring the signal $x(t)$, performing a “leaky” integration, and taking the square root of the result.

• A “leaky” integrator can be implemented in discrete time as:
  $$y[n] = b_0 x[n] + a_1 y[n-1] \hspace{0.2in} \Longrightarrow \hspace{0.2in} H(z) = \frac{b_0}{1 - a_1z^{-1}},$$
  where $b_0 = 1 - a_1$ for unity gain and $\vert a_1\vert < 1$ for filter stability.

• Automatic gain control involves the application of a time-varying gain $g(t)$ to a signal, based on an estimate of the signal level.

• The compression/expansion gain control is typically specified as a memoryless function that takes a signal level estimate as input and produces a desired gain level as output (levels in dB). An example compression curve is shown in Fig. 6 (the dashed line indicates an input-to-output ratio of 1).

  Figure 6: A static compression curve (top: in dB; bottom: on a linear scale.

  

• The dB output level is equal to the dB input level up to a threshold level $l_T$ (near -50 dB in Fig. 6). Beyond $l_T$, a constant compression ratio is defined by $R$ such that there is an increase of $1/R$ dB in output for every dB increase of the input.

• For example, if the compression ratio is 3:1, an input signal that is 9 dB over the threshold will be attenuated (by subtracting 6 dB from the input signal level) to a level 3 dB over the threshold. That would correspond to a linear attenuation factor of 0.5 being applied to the input signal.

• The time duration $\tau$ over which the input signal level estimates are calculated has important influence on the response of the compression/expansion system.

• It is often desirable that a compressor limit peak amplitudes. This ability requires a relatively small $\tau$ value. However, small values of $\tau$ result in more level variance, which is generally undesirable. For this reason, different integration constants are often used for the attack and release portions of a signal.

• Further information on this topic can be found on the Universal Audio Blog.