I am given $|H(\omega)|$, I wonder if minimum phase stable causal filter is unique and how to calculate it.
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$\begingroup$ 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. $\endgroup$– robert bristow-johnsonMar 19, 2015 at 21:04
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$\begingroup$ @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? $\endgroup$– Matt L.Mar 20, 2015 at 8:08
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$\begingroup$ @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. $\endgroup$– robert bristow-johnsonMar 20, 2015 at 19:40
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1 Answer
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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.
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$\begingroup$ @user40129: I just looked around but couldn't find anything convincing. In 'The Fourier Integral and Its Applications' by Papoulis there's proof of a similar relationship using contour integration. And in Oppenheim and Schafer's book (Discrete-Time Signal Processing) there's something like a proof of the equivalent relationship for discrete-time systems, but part of it is left as a 'problem'. $\endgroup$– Matt L.Mar 19, 2015 at 18:31
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$\begingroup$ Proof is at Page 206 of 'The Fourier Integral and Its Applications' by Papoulis. Thanks! $\endgroup$ Mar 19, 2015 at 19:50