Validity of an argument that two transfer functions are minimum-phase based on their ratio being minimum-phase Update I think the essence of my question below is this: If the ratio of two transfer functions may be represented exactly as a minimum-phase filter (MPF) plus a pure delay (in the title, I simply refer to this representation as “minimum-phase” for brevity, even though it’s not accurate), does that imply that each transfer function in the ratio must also be represented exactly as an MPF plus a pure delay?
I was reading a paper where an argument is made in the Appendix regarding the validity of representing a transfer function as a minimum-phase filter (MPF) plus a pure delay.
In particular, the goal is to show that two transfer functions $H_L$ and $H_R$ can each be represented as a minimum-phase filter (MPF) plus a pure delay.
That is, if we have $H_L = |H_L| \text{e}^{\text{i}\phi_L}$ and $H_R = |H_R| \text{e}^{\text{i}\phi_R}$, where $\phi_L$ and $\phi_R$ are their respective phase responses, then given that these are rational transfer functions that can be decomposed into a cascade of minimum-phase and all-pass transfer functions, we must show that $\phi_L = \phi_L^{\text{mp}} + \omega\,t_L$ and $\phi_R = \phi_R^{\text{mp}} + \omega\,t_R$, for some constants $t_L$ and $t_R$ (i.e., the all-pass phase response is purely linear), and where the superscript &quot;mp&quot; stands for minimum-phase.
As I understand it, the argument by the authors of the paper goes as follows:
Give $H_L$ and $H_R$ as above, we have $$\ln \left(\frac{H_R}{H_L}\right) = \underbrace{\left(n_R-n_L\right)}_{\Delta n} + \text{i}\underbrace{\left(\phi_R-\phi_L\right)}_{\Delta \phi},$$
where $n_R = \ln |H_R|$ and $n_L = \ln |H_L|$.
Since, in general, we have $\phi_L = \phi_L^{\text{mp}} + \phi_L^{\text{ap}}$ and $\phi_R = \phi_R^{\text{mp}} + \phi_R^{\text{ap}}$, where the superscripts &quot;mp&quot; and &quot;ap&quot; stand for &quot;minimum-phase&quot; and &quot;all-pass&quot;, respectively, this means $\Delta \phi = \Delta \phi^{\text{mp}} + \Delta \phi^{\text{ap}} = \mathcal{H}\left(\Delta n\right) + \Delta \phi^{\text{ap}}$, where $\mathcal{H}$ denotes the Hilbert transform operator.
The authors show that this may be expressed as $\Delta \phi = \mathcal{H}\left(\Delta n\right) + \omega t_0$ for a constant, $t_0$.
Let us assume that this equation (which corresponds to Eq. (A8) in their paper) is valid.
The authors then go on to suggest (incorrectly in my opinion) that since $\Delta \phi^{\text{ap}} = \omega t_0$, $H_L$ and $H_R$ must each always be representable exactly as an MPF plus pure delay (when this is, in fact, not known).
I think this argument is flawed because it assumes that $\Delta \phi^{\text{ap}} = \phi_R^{\text{ap}}-\phi_L^{\text{ap}} = \omega t_0$ implies that $\phi_R^{\text{ap}}$ and $\phi_L^{\text{ap}}$ are also each linear functions of $\omega$ (since only then do $H_L$ and $H_R$ each have MPF plus delay representations).
However, this is not the only possibility correct? I think a function like $\omega t_0$, which is linear in $\omega$ can, in general, be obtained as the difference of two non-linear functions of $\omega$, each corresponding to either $\phi_R^{\text{ap}}$ or $\phi_L^{\text{ap}}$, which would mean that an MPF plus delay representation for $H_L$ and $H_R$ as described earlier is not always necessary, right? For example, if $\phi_R^{\text{ap}} = \alpha \omega^2$ and $\phi_L^{\text{ap}} = \alpha \omega^2 + t_0 \omega$, for some scalar $\alpha$, then $\Delta \phi^{\text{ap}} = \omega t_0$. Am I going wrong somewhere?
Thanks in advance for any inputs!
two transfer functions HL and HR can each be represented as a minimum-phase filter (MPF) plus a pure delay.

That is generally not true and it's easy enough to disprove it by counter example. Let's simply look at a first order all pass
$$H(z) = \frac{p-1/z}{1-p/z} = H_{MP}(z) \cdot H_{AP}(z)$$ which has a pole at $p$ and a zero at $1/p$. The &quot;minimum phase&quot; part, $H_{MP}(z)$ is simply unity, i.e. $H_{MP}(z)=1$ and the all pass part is just the original all pass. The all pass phase response is NOT that a of a pure  delay, we clearly have $\Phi_{MP}(\omega) \neq \omega \cdot t_0$
I think you misunderstand the point in the paper: I think they are just saying that a specific class of transfer functions can be approximated well with this model. In this case this class is &quot;HRTF ratios for human heads&quot; and they show that this works particularly well for spherical heads. Since binaural perception is mainly based on interaural differences, approximating the frequency dependent interaural time delay by a single constant value can be quite useful.