# Response of an unstable LTI system to random signals

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.

• 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. Jan 30, 2021 at 17:02
• @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. Jan 30, 2021 at 17:23
• for the first problem, you'd do a different modelling (see: zero-pole diagram, stability of control systems: pretty rich field of mature methods). Jan 30, 2021 at 17:36