What is the definition of an incrementally linear system?

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"?

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 $$(5)$$.

• Nicely done! Augments this question and answers well.
– Peter K.
Jan 4, 2023 at 11:41
• I take it any general $x[n] = x_0[n] + \sum_i \alpha_i x_i[n]$ qualifies? Jan 4, 2023 at 12:40
• @OverLordGoldDragon: Sure, but note that that's no generalization. Using just $\alpha_1x_1[n]$ and $\alpha_2x_2[n]$ is sufficient, just as it's the case for the definition of linearity. Jan 4, 2023 at 13:28
• I'm more used to the term "affine system", which appears to be what you're defining here. Is there a difference between a system that's affine and one that's incrementally linear? Jan 4, 2023 at 15:51
• @TimWescott: Every affine system is incrementally linear (my first example), but I'm not sure if we should call a system as shown in my second example "affine". The additive term could be any (non-constant) sequence $a[n]$ that doesn't depend on the input signal. Would you use "affine" in that case? Jan 4, 2023 at 16:03