Prove that the filter is stable, causal and minimum phase

Question

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.

Answer 1

In general, the algebraic expression of a transfer function alone doesn't uniquely describe a single system. For the example in your question with two complex conjugate poles in the left-half plane, the given transfer function describes two systems: one is causal and stable, the other one is non-causal and unstable.

A complete (external) description of an LTI system is given by the transfer function and the corresponding region of convergence (ROC). For a causal system, the ROC is a right half-plane. Since the ROC to the right of the two poles includes the imaginary axis, the corresponding causal system is stable. Since the ROC to the left of the two poles does not include the imaginary axis, the corresponding non-causal system is unstable.

So, assuming that the given transfer function is meant to describe a causal system, BIBO (bounded-input, bounded-output)-stability follows immediately.

For answering the question if that causal system is a minimum-phase system, you just have to ask if it is possible to invert the system by a causal and stable system. If that is the case, the system is a minimum-phase system, otherwise it isn't.