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Given only a magnitude response $A(\omega)$ of a minimum phase filter, one can calculate the phase response using the Hilbert Transform:

$$θ(ω) = -\mathcal{H}\{\ln(A(\omega)\}$$

This paper suggests that there is another Hilbert Transform relationship between magnitude and phase for maximum phase filters, but I can't find anything beyond that in the cited references and elsewhere.

Does anybody know what the exact Hilbert Transform relationship is between magnitude and phase for maximum phase filters?

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Note that a Hilbert transform relationship for maximum phase systems can only exist if a maximum phase system is defined to be an anti-causal stable system with an anti-causal stable inverse (cf. these MIT course notes, p.4). Considering only systems with rational transfer functions, this means that not only the system's zeros but also its poles must be in the right half-plane (for continuous-time systems) or outside the unit circle (for discrete-time systems), respectively. The reason for this is that for the Hilbert transform relationships to hold, $\ln(H(s))$ (or $\ln(H(z))$) must be analytic, either for $\text{Re}\{s\}\ge 0$ (or $|z|\ge 1$) (minimum-phase), or for $\text{Re}\{s\}\le 0$ (or $|z|\le 1$) (maximum-phase). Since both poles and zeros of $H(s)$ cause singularities in $\ln(H(s))$ we need to make sure that for maximum phase systems there are neither poles nor zeros in the left half-plane $\text{Re}\{s\}\le 0$ (or inside the unit circle for discrete-time systems).

Note that the above definition of a maximum phase system is different from another common definition which says that a maximum phase system is a causal and stable system with all its zeros in the right half-plane (or outside the unit circle), but - since it's causal and stable - all its poles in the left half-plane (inside the unit circle).

According to the first definition above, a maximum phase system is obtained from a minimum phase system by time-reversal, which leaves the magnitude of the frequency response unchanged but inverts the sign of the phase. Consequently, for a maximum phase system according to above definition, we simply get the following Hilbert transform relationship between phase and magnitude:

$$\theta(\omega)=\mathcal{H}\{\ln(A(\omega))\}\tag{1}$$

Note that the sign inversion in the Hilbert transform relationship between magnitude and phase can also be seen from the fact that a minimum phase system has a causal cepstrum (from which the Hilbert transform relationship follows), and a maximum phase system has an anti-causal cepstrum, resulting in the same Hilbert transform relationship between magnitude and phase but with an inverted sign.

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