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I was reading through slides about Hilbert Huang Transform. In slide 14, which talks about the motivations of a new method instead of Fourier Transform (FT), the author provides those two reasons in addition to other reasons

  1. Physical processes are mostly nonstationary.
  2. Physical processes are mostly nonlinear.

I understand the first reason, where the FT cannot locate the frequencies in time. However, I do not understand why the FT does not work for the nonlinear processes. Based on my understanding, the FT is a method to decompose the signal into different and simple components. Why does condition of non-linearity play a role here?

My first thought for reasoning that assumption, was that we cannot represent a non-linearity in the signal by a linear combinations of sinusoids. However, I think sinusoids themselves will handle the non-linearity.

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  • $\begingroup$ Non linear systems often don't have simple eigen-components. $\endgroup$
    – mathreadler
    Commented May 14, 2017 at 13:04

3 Answers 3

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Because complex exponentials $e^{\jmath \omega t}$, which are results of Fourier transform, are the eigenfunctions for linear, time invariant (LTI) systems. See eigenfunction of LTI. Also see this answer on SP.SE.

Thus, Fourier transform is useful for analyzing linear (not suitable for non-linear one) time invariant (can be intepreted as stationnary) system.

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  • $\begingroup$ For more information, can you add a reference that explains what you said in the first paragraph? $\endgroup$
    – hbak
    Commented May 14, 2017 at 0:42
  • $\begingroup$ I have added a reference. You can find other ones easily by a simple Google search. :) $\endgroup$
    – mascara
    Commented May 14, 2017 at 7:35
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    $\begingroup$ @mascara I added another link to an answer I gave a while back that I believe is relevant. $\endgroup$
    – Peter K. ♦
    Commented May 16, 2017 at 18:21
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Looking at an unknown system relates to finding relations between inputs and outputs. A first question is: are there specific inputs that are "almost" unchanged by the system? Those are sometimes called "root" signals. The effect of the system on some other signal is often simpler to analyze by rewriting or approximating them by a combination of several root signals.

When a system is linear, the root signals are precisely sines. Hence, the Fourier transform is appropriate in that case. Indeed, it was made for that. However, it still can work when this is not true, depending on the nature of the nonlinearity. This is discussed in this question on this question on linear and time invariance

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Part of the confusion I think comes from the fact that FFTs are in practice used in modeling nonlinear systems. FFTs are linear; they're just a linear combination of sinusoidal signals (the "root" signals). But often, we can assume linearity over a short time period (or small space in the case of two dimensions). And doing so gives us an advantage because LTI systems are powerful and easier to implement. For instance, many audio processing applications need to model nonlinear systems (e.g. the transfer function of a real-life speaker is nonlinear). But the input to the system that models nonlinearities (e.g. Volterra filter or neural network) is FFT values.

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