##
Nonlinear Filter Example:

Dynamic Range Compression

A simple practical example of a
*nonlinear* filtering operation
is *dynamic range compression*, such as occurs in Dolby or DBX
noise reduction when recording to magnetic tape (which, believe it or
not, still happens once in a while). The purpose of dynamic range
compression is to map the natural dynamic range of a signal to a
smaller range. For example, audio signals can easily span a range of
100 dB or more, while magnetic tape has a linear range on the order of
only 55 dB. It is therefore important to compress the dynamic range
when making analog recordings to magnetic tape. Compressing the
dynamic range of a signal for recording and then expanding it on
playback may be called
*companding*
(compression/expansion).

Recording engineers often compress the dynamic range of individual tracks to intentionally ``flatten'' their audio dynamic range for greater musical uniformity. Compression is also often applied to a final mix.

Another type of dynamic-range compressor is called a *limiter*,
which is used in recording studios to ``soft limit'' a signal when it
begins to exceed the available dynamic range. A limiter may be
implemented as a very high compression ratio above some amplitude
threshold. This replaces ``hard clipping'' by ``soft limiting,''
which sounds less harsh and may even go unnoticed if there were no
indicator.

The preceding examples can be modeled as a variable *gain* that
automatically ``turns up the volume'' (increases the gain) when the
signal level is low, and turns it down when the level is high. The
signal level is normally measured over a short time interval that
includes at least one period of the lowest frequency allowed, and
typically several periods of any pitched signal present. The gain
normally reacts faster to attacks than to decays in audio compressors.

### Why Dynamic Range Compression is Nonlinear

We can model dynamic range compression as a *level-dependent
gain*. Multiplying a signal by a constant gain (``volume control''),
on the other hand, is a linear operation. Let's check that the
scaling and superposition properties of linear systems are satisfied
by a constant gain: For any signals , and for any constants
, we must have

Dynamic range compression can also be seen as a *time-varying
gain* factor, so one might be tempted to classify it as a linear,
time-varying filter. However, this would be incorrect because the
gain , which multiplies the input, *depends on the input
signal* . This happens because the compressor must estimate the
current signal level in order to normalize it. Dynamic range
compression can be expressed symbolically as a filter of the form

*rms level*(the ``root mean square'' [84, p. 75] computed over a sliding time-window). Since many successive samples of are needed to estimate the current level, we cannot correctly write for the gain function, although we could write something like (borrowing matlab syntax), where is the number of past samples needed to estimate the current amplitude level. In general,

In general, any signal operation that includes a multiplication in which both multiplicands depend on the input signal can be shown to be nonlinear.

**Next Section:**

A Musical Time-Varying Filter Example

**Previous Section:**

Showing Linearity and Time Invariance, or Not