2
$\begingroup$

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"?

$\endgroup$

1 Answer 1

3
$\begingroup$

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 $(5)$.

$\endgroup$
6
  • 1
    $\begingroup$ Nicely done! Augments this question and answers well. $\endgroup$
    – Peter K.
    Jan 4, 2023 at 11:41
  • $\begingroup$ I take it any general $x[n] = x_0[n] + \sum_i \alpha_i x_i[n]$ qualifies? $\endgroup$ Jan 4, 2023 at 12:40
  • 1
    $\begingroup$ @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. $\endgroup$
    – Matt L.
    Jan 4, 2023 at 13:28
  • $\begingroup$ 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? $\endgroup$
    – TimWescott
    Jan 4, 2023 at 15:51
  • $\begingroup$ @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? $\endgroup$
    – Matt L.
    Jan 4, 2023 at 16:03

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.