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.

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    $\begingroup$ 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. $\endgroup$ 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. $\endgroup$ 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). $\endgroup$ Jan 30, 2021 at 17:36


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