# What is the precise definition of a nonlinear signal?

I tried to find the the precise definition of a nonlinear signal. However it is not easy. How can we define a nonlinear signal?

• Do you really mean a nonlinear signal? Or maybe a nonlinear system? – Max Apr 18 at 7:46
• I mean a nonlinear signal. – cabal Apr 18 at 10:08
• @cabal: The term really doesn't make much sense. Can you provide a reference where it's being used or did you come up with it yourself? – Hilmar Apr 18 at 10:46
• I found it in "Encyclopedia of Information Science and Technology". Here is the link to the definition: igi-global.com/dictionary/… – cabal Apr 18 at 10:50
• Well, the definition is there. A signal, that comes out of a nonlinear system. Refer to Matt's answer. – Max Apr 18 at 11:52

In my opinion, the term nonlinear signal doesn't make much sense. If you refer to nonlinear signal processing then this is not about the processing of nonlinear signals, but about the nonlinear processing of signals, i.e., about the use of nonlinear systems.

A system $$\mathcal{T}$$ is nonlinear if it doesn't obey the superposition principle. I.e., if $$y_1(t)=\mathcal{T}\{x_1(t)\}$$ is the response to input $$x_1(t)$$, and $$y_2(t)=\mathcal{T}\{x_2(t)\}$$ is the response to input $$x_2(t)$$ then

$$\mathcal{T}\big\{\alpha x_1(t)+\beta x_2(t)\big\}=\alpha\mathcal{T}\big\{x_1(t)\big\}+\beta\mathcal{T}\big\{x_2(t)\big\}$$

is generally not satisfied.

Well-known examples of such nonlinear systems are median filters and Volterra filters, but there are many trivial operations, such as taking the magnitude of a signal, which must be classified as nonlinear processing.

A nonlinear signal is generally defined as the signal generated by the system that does not obey superposition and scaling properties.

You can certainly define it that way, but it seems like a useless definition since it doesn't reference any property of the signal itself.

Consider a case where you have a non-linear system followed by an linear "pass through" system. Per the definition, the output of the non-linear system would be a "non-linear signal" and the output of the pass-through would be a "linear" signal. Since the linear system is pass-through the input and output are the same, so now you have two signals that are identical but one of them is "non-linear" and the other one isn't. That really doesn't make any sense, so the definitions seem pointless and not useful for anything.

There are materials where the strain/stress relationship becomes nonlinear at high signal amplitude that are studied in nonlinear acoustics.

The same can be true in optics where high amplitude fields changes a linear propagation to a nonlinear one.

I believe action potentials in nerves are nonlinear.

The term “small signal” has been associated with linear propagating signals.

I can’t say I’ve seen the terms linear or nonlinear signals used.