1
$\begingroup$

A convenient approach for studying the response of a stable LTI system with impulse response $h(t)$ to a WSS stochastic input $X(t)$ is to look at the power spectral density (PSD) of the output $Y(t)$ as $S_Y(f) = \vert H(f) \vert^2 S_X(f)$, where $H(f)$ is the Fourier transform of $h(t)$ and $S_X(f)$ and $S_Y(f)$ are the PSDs of the input and output, respectively. For simplicity, let's assume that $X(t)$ is zero-mean.

Now if I have an unstable LTI system for which $h(t)$ may not even have a Fourier transform, this method naturally won't work anymore. Is there a similar straightforward approach for looking at the statistical properties of the output of an unstable system?

The only approach I can think of is to directly look at the convolution integral $R_Y(\tau) = h(\tau) \star R_X(\tau) \star h(-\tau)$ but that's not as intuitive as the frequency-domain analysis we have for stable systems.

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ what do you hope to come out of your analysis? If you put input into an unstable system, then by any chance, the output won't have the nice properties (WSS/boundedness) that the input had, and hence, things like a PSD will also be undefined. $\endgroup$
    – Marcus Müller
    Jan 30, 2021 at 17:02
  • $\begingroup$ @MarcusMüller For example, let's say I have an unstable system, and I want to see what are the chances that the output is bounded for a specific random input signal with a known PSD. Or in general, given a specific input, whether or not the output does have the nice properties you talk about. $\endgroup$
    – Sahand Tabatabaei
    Jan 30, 2021 at 17:23
  • $\begingroup$ for the first problem, you'd do a different modelling (see: zero-pole diagram, stability of control systems: pretty rich field of mature methods). $\endgroup$
    – Marcus Müller
    Jan 30, 2021 at 17:36

0

Reset to default

Your Answer

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