# Prove that the filter is stable, causal and minimum phase

I have a system which has the following transfer function $$H(s)=\frac{\beta + s}{s^{2} + 2\alpha s + \beta^{2}}$$ where $$\beta = \sqrt{\omega^{2} + \alpha^{2}}$$ and $$\alpha>0$$.

This system, as already discussed in a dsp question is stable. How can I check the other two properties?

EDIT

If a random (wide-sense stationary) process $$n(t)$$ passes through a time-invariant filter $$h(\tau)$$, then the autocorrelation function is $$R_{n^\prime}(t)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} R_{n}(t-\tau+\tau^\prime) h(\tau) h(\tau^\prime) d \tau d \tau^\prime$$ For white noise input, $$R_{n}(t)=\delta(t)$$ and $$R_{n^\prime}(t)=\int_{-\infty}^{\infty} h(\tau-t) h(\tau) d \tau$$ The power spectrum of the output is the Laplace transform of $$R_{n^\prime}(t)$$ and it is possible to prove that $$S_{n^\prime}(s)= H(-s) \int_{-\infty}^{\infty} h(\tau) e^{-s \tau}d \tau=H(-s) H(s)$$ In my case I have $$H(s)=\frac{\beta + s}{s^{2} + 2\alpha s + \beta^{2}}$$. I would like to prove that the filter is stable, causal and minimum phase.

• What are your thoughts about this? If you know the definition of "minimum phase", what problems did you encounter when trying to find out if the given system satisfies the respective conditions? A good answer to that question should be quite long because given only the expression for the transfer function, you can't say anything about stability or causality. You need more information. Jul 10, 2022 at 11:05
• I've added an answer to your other question, clarifying a few points that are also relevant to this question. Jul 10, 2022 at 12:05
• What makes a filter “unique”? Stable, causal, and minimum phase I get, but what is “unique”?
– Peter K.
Jul 10, 2022 at 14:54
• And -- since this looks like homework -- what work have you done so far to answer it on your own? Jul 10, 2022 at 22:56
• Thanks to all. I edited my post
– Mark
Jul 11, 2022 at 20:01