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I am studying the meaning of the power spectral density (PSD) and I am wondering whether I can see from the PSD of the input signal and output signal if a system is a LTI system?

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    $\begingroup$ Note that the power spectral density is a function describing a time series, not a system. This is why your question as it is formulated now does not make any sense. $\endgroup$
    – Matt L.
    Nov 25, 2014 at 10:35
  • $\begingroup$ Thank you @MattL, you're right. I slightly changed the question. $\endgroup$
    – paterpeng
    Nov 26, 2014 at 11:27

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Suppose that the input signal to the system has power spectral density (PSD) $S_i(f)$ while the output has PSD $S_o(f)$. Then, if the system is an LTI system, then it must be that $$S_o(f) = |H(f)|^2 S_i(f) \tag{1}$$ where $H(f)$ is the transfer function of the system. So, for a given pair of PSDs, is there a $H(f)$ such that $(1)$ holds? One obvious case when $(1)$ cannot hold for any choice of $H(f)$ is when the supports of $S_i(f)$ and $S_o(f)$ are different (in the sense that there are frequencies $f$ for which $S_o(f) > 0$ while $S_i(f) = 0$). So in this case we can say unequivocally that the system is not an LTI system.

What about when the supports of $S_i(f)$ and $S_o(f)$ are identical or the support of $S_o(f)$ is a subset of the support of $S_i(f)$ (think bandstop or notch filter)? Well, in this case, we do have a $|H(f)|^2$ from which we can deduce $|H(f)|$. Throw in a conjugate-symmetric $\angle H(f)$ and we have constructed the transfer function $H(f) = |H(f)|e^{\angle H(f)}$ of an LTI system whose output PSD is $S_o(f)$ when the input PSD is $S_i(f)$. Note that there can be infinitely many LTI systems that will provide this specific input-output relationship.


But, the above does not allow us to say that the system that actually produced the given output PSD from the given input PSD is an LTI system, let alone one having the specific transfer function that we have constructed. The actual system that we are looking at might be nonlinear or time-varying, etc. We cannot say. After all, the $|H(f)|e^{\angle H(f)}$ that we have come up with might be a physically unrealizable transfer function.

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    $\begingroup$ "... nonlinear or time-varying" is probably what you meant. $\endgroup$
    – Matt L.
    Nov 28, 2014 at 17:03
  • $\begingroup$ @MattL. Thanks for the careful reading. I am editing the answer right away. $\endgroup$ Nov 28, 2014 at 17:13
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Yes, the PSD of a time signal can give you infos about the linearity of the system under study.

For example, assume you are studying a symmetric duffing's oscillator described by the equation:

$m \ddot{x} + c \dot{x} + k x + k_2 x^3 = p(t) $

$p(t) = A \sin{\omega t}$

If you compute the PSD of the acceleration signal you will see a peak at the excitation frequency and then some harmonics generated by the non linearity.

Since the nonlinearity is odd, the harmonics will be at odd multiples of the excitation frequency, so if $\omega$ is the excitation frequency, the harmonics will be at $3\omega$, $5\omega$ and so forth.

For linear system these harmonics are not present, because the response of a linear system excited by a monoharmonic signal will be monoharmonic at the same frequency (after transients have died out).

I suggest you to read the book written by Worden and Tomlinson if you are interested in nonlinear dynamics.

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