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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 "mp" 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 "mp" and "ap" stand for "minimum-phase" and "all-pass", 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!

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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 "minimum phase" 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 "HRTF ratios for human heads" 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.

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  • $\begingroup$ I do understand that not all transfer functions can be represented as an MPF plus a pure delay. But they very clearly state (in the para after Eq. A8) that “In our case, Eq. A8 is exact because the spherical head transfer functions, as described in Eq. (A1) are minimum phase.” They then “illustrate this idea” by effectively showing what I describe in the original post, which I do not believe shows that the spherical head transfer functions themselves are minimum-phase. I know this may not be the point of their paper overall, but I’m taking issue with their claim and how they “prove” it. $\endgroup$
    – Rahul
    Jan 31 at 14:05
  • $\begingroup$ Spherical head transfer functions have not been shown to be minimum phase (you can see they don’t provide a citation for this claim). Instead they appear to make the claim and them proceed to illustrate it in the way I describe. My question is whether their logical reasoning is flawed given how they try to show this. $\endgroup$
    – Rahul
    Jan 31 at 14:10

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