What is the definition of an incrementally linear system?

Question

Even though the concept of an incrementally linear system is mentioned in several textbooks, I haven't found an explicit definition. What exactly do we mean by "incrementally linear"?

Answer 1

In this answer I'll use discrete-time signals and system to explain the concept of "incrementally linear". However, the answer is also valid and essentially identical for continuous-time systems.

Let $y_0[n]$ be a system's response to the input $x_0[n]$. Furthermore, let $y_0[n]+y_i[n],\; i=1,2$ be the responses to inputs $x_0[n]+x_i[n],\; i=1,2$. Then the system is incrementally linear if its response to the input

$$x[n]=x_0[n]+\alpha x_1[n]+\beta x_2[n]\tag{1}$$

is given by

$$y[n]=y_0[n]+\alpha y_1[n]+\beta y_2[n]\tag{2}$$

Of course, every linear system is also incrementally linear. Note that unless the system is linear, $y_1[n]$ and $y_2[n]$ are not the responses to inputs $x_1[n]$ and $x_2[n]$, respectively. However, if the system is incrementally linear, there exists a linear system describing the mapping from an increment in the input signal to the corresponding increment in the output signal. This is clear from rewriting $(1)$ and $(2)$ as

$$\Delta x[n]=x[n]-x_0[n]=\alpha x_1[n]+\beta x_2[n]\tag{3}$$

$$\Delta y[n]=y[n]-y_0[n]=\alpha y_1[n]+\beta y_2[n]\tag{4}$$

Examples:

The most straightforward example of an incrementally linear system is the system described by

$$y[n]=ax[n]+b$$

This is an affine system, and it could be used to describe the behavior of a (very simple) linear system with an additional non-zero initial condition. The incremental output response caused by an increment of the input signal is described by

$$\Delta y[n]=a\Delta x[n]\tag{5}$$

which is clearly a linear relation.

The following system is a simple example of a time-varying incrementally linear system:

$$y[n]=ax[n]+bn\tag{6}$$

The response to $x[n]=x_0[n]+\Delta x[n]$ is

$$y[n]=ax_0[n]+a\Delta x[n]+bn=y_0[n]+\Delta y[n]$$

and the linear system describing the relation between $\Delta x[n]$ and $\Delta y[n]$ is again given by $(5)$.