Why Dynamic Range Compression is Nonlinear

We can model dynamic range compression as an amplitude-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 $x_1,x_2$, and for any constants $\alpha,\beta$, we must have

$$g \cdot [\alpha \cdot x_1(n) + \beta \cdot x_2(n)] = \alpha \cdot [g \cdot x_1(n)] + \beta \cdot [g \cdot x_2(n)].$$

Since this is obviously true from the algebraic properties of real or complex numbers, both scaling and superposition have been verified. (For clarity, an explicit ``$\cdot$'' is used to indicate multiplication.)

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 $g$, which multiplies the input, depends on the input signal $x(n)$. 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

$$y(n) = g_n(x) \cdot x(n)$$

where $g_n(x)$ denotes a gain that depends on the ``current amplitude'' of $x(\cdot)$ at time $n$. Since many successive samples of $x$ are needed to estimate current average amplitude level, we cannot correctly write $g[x(n)]$ for the gain function, although we could write something like $g[x(n-M:n)]$ (borrowing matlab syntax), where $M$ is the number of past samples needed to estimate the current amplitude level. In general,

$$g(x_1 + x_2)\cdot [x_1(n) + x_2(n)] \neq g(x_1) \cdot x_1(n) + g(x_2) \cdot x_2(n) .$$

That is, the compression of the sum of two signals is not generally the same as the addition of the two signals compressed individually. Therefore, the superposition condition of linearity fails. It is also clear that the scaling condition fails.

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