Minimum phase non-rational transfer function: Hilbert transform between log magnitude and phase

Question

In Signal Processing literature, it is well known that a minimum phase sequence with rational transfer function ('zeros' and 'poles' in unit circle) has Hilbert transform relation between log magnitude and phase. Does the relation hold for non-rational transfer function, where we don't have the concept of zeros and poles? The general definition of minimum phase sequence would be a sequence whose z-transform, along with the z-transform of its inverse, are analytic for $|z|\geq 1$

Answer 1

If you have a trigonometric series $$ S=\sum_{n=-\infty}^{\infty}a_{n}e^{inx}, $$ then the conjugate series is $$ T=-i\sum_{n=-\infty}^{\infty}(\mbox{sgn}(n))a_{n}e^{inx}, $$ where $\mbox{sgn}(n)$ is $1$ for $n > 0$, is $-1$ for $n < 0$, and is $0$ if $n=0$. Then $$ S+iT= a_{0}+2\sum_{n=1}^{\infty}a_{n}e^{inx}. $$ You'll recognize the conjugate series.

If $S(z)=\sum_{n=-\infty}^{\infty}a_{n}r^{|n|}e^{inx}$ converges in the unit disk $(z= re^{ix})$, then $S$ is harmonic in the unit disk, and its conjugate $T(z)=-i\sum_{n=-\infty}^{\infty}(\mbox{sgn}(n))a_{n}r^{|n|}e^{inx}$ is also harmonic, with holomorphic series $$ S(z)+iT(z) = a_{0}+\sum_{n=1}^{\infty}a_{n}z^{n}. $$ That is, $T$ is the harmonic conjugate of $S$ in the unit disk. $T$ is normalized so $T(0)=0$.

So, if you start with a function $S$ which is harmonic inside (outside) the unit disk, then $S+iT$ is holomorphic inside (outside) the unit disk. Any $f\in L^{2}(|z|=1)$ has a series expansion $\sum_{n=-\infty}a_{n}e^{inx}$ with square summable coefficients. The harmonic series $S(re^{inx})=\sum_{n=-\infty}^{\infty}a_{n}r^{|n|}e^{inx}$ has $L^{2}$ boundary function $f$ on the unit circle. $T$ is the harmonic conjugate of $S$, and $S+iT$ is holomorphic in the unit disk.

You can think in terms of outside the disk just as easily. So there's always an underlying holomorphic structure present for $f+i\mathcal{H}f$.