# Given Gain function, how to design a causal, stable and minimum phase IIR filter?

I am given $|H(\omega)|$, I wonder if minimum phase stable causal filter is unique and how to calculate it.

• if you're wanting to design a finite-order IIR (determine the order $N$ and the $2N+1$ coefficients, you have to use something like the Prony method or Greg Berchin's FDLS. – robert bristow-johnson Mar 19 '15 at 21:04
• @robertbristow-johnson: Prony's method designs an IIR filter given a prescribed impulse response. How would you use it here when the magnitude of the frequency response is given? – Matt L. Mar 20 '15 at 8:08
• @MattL., i guess i would first create a minimum-phase complex frequency response, and compute an impulse response from that. or use Greg's FDLS, since it's frequency domain. – robert bristow-johnson Mar 20 '15 at 19:40

If $H(\omega)=e^{\alpha(\omega)+j\phi(\omega)}$ is a minimum phase frequency response, then the attenuation $\alpha(\omega)$ and the phase $\phi(\omega)$ are related by the following Hilbert transform relationship:
$$\phi(\omega)=-\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{\alpha(\Omega)}{\omega-\Omega}d\Omega$$
So $\phi(\omega)$ is uniquely determined by $\alpha(\omega)$. This is the corresponding wikipedia entry.