Is wavelet a Nonlinear transform, or Not?
specifically, continuous wavelet transform with morlet function.
I am studying behavior of a dynamic system, and it has nonlinear behaviour. can I employ wavelet transform?


A transform being linear has very little to do with its ability to analyze linear or nonlinear systems.

The wavelet transform $W[s(t)]$ of a signal $s(t)$ is linear because $$W[a s_1(t) + b s_2(t)]=a W[s_1(t)]+b W[s_2(t)]$$ for real or complex $a$ and $b$.

The signal you're analyzing is just a signal, it has no concept of linearity. However, if you try to come to conclusions about system properties of a nonlinear system, then you cannot break the analysis down to just a set of base signals to understand the system. In the worst case you would have to look at every possible intput/output pair. Often this can be simplified using known system properties like symmetries (i.e. time invariance).

  • $\begingroup$ Actually, I am writing a report, I used cwt to study my nonlinear system, can I acclaim that wavelet transform is suitable for analyzing non-linear system? @Jazzmaniac $\endgroup$ – SAH Nov 28 '13 at 11:08
  • $\begingroup$ It depends. How do you plan to analyze it? What kind of features or parameters are you looking for? There is nothing that speaks against wavelets for nonlinear system analysis from a theoretical point of view. However if it is useful depends on how you use it. $\endgroup$ – Jazzmaniac Nov 28 '13 at 13:52
  • $\begingroup$ I am working on a mechanical system, aim to estimate damping and frequencies in the system oscillations. the system consist of number of nonlinear elements. I need to put some reference that says nonlinear system can be analyze fine with wavelet and it's not matter that wavelet is linear and system in nonlinear. if there is no reference I need to prove that fact. @Jazzmaniac $\endgroup$ – SAH Nov 28 '13 at 14:34
  • $\begingroup$ I don't think you need a reference for that. There's no a-priori reason why you cannot analyze a nonlinear system with a linear transformation. Like I said, the two have very little to do with each other. $\endgroup$ – Jazzmaniac Nov 28 '13 at 22:15
  • $\begingroup$ as you know wavelet can analyze nonstationary signal very well. can I say nonlinear element of system can cause nonstationary signal, hence wavelet method can be a good tool for analysing these signals? @Jazzmaniac $\endgroup$ – SAH Nov 29 '13 at 7:31

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