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There are few notes online stating that an all-pass filter is a maximum phase filter (e.g., Link). The core of the claim is that an all-pass filter is a maximum phase filter since its zeros are outside the unit circle, which is the definition of a maximum phase filter.

However, according to the MIT lecture note, a maximum phase filter is stable and anti-causal while its inverse is also stable and anti-causal. As we all know, for an anti-causal filter and its inverse to be stable, its poles must also be outside the unit circle, so its ROC includes the unit circle. The same claim can also be found in other notes (definition D. 3. 6.)

But going back to the definition of an all-pass filter, its transfer function can be expressed as:
$$H(z)=\frac{z^{-1}-a^*}{1-az^{-1}}$$

Assuming $|a|<1$, its pole is inside the unit circle, but its zero is outside the unit circle. However, since its pole is inside the unit circle, the filter cannot be stable if this is an anti-causal filter. If this is not an anti-causal filter, then it is difficult to see it as a maximum phase filter.

Can anyone point out what I am missing here?

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    $\begingroup$ Nice question!! $\endgroup$
    – Peter K. ♦
    Dec 15, 2022 at 23:39
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    $\begingroup$ Given the most meaningful definition of maximum-phase system, I would say the answer is "yes". All non-trivial all-pass filters are maximum phase. $\endgroup$
    – robert bristow-johnson
    Dec 16, 2022 at 21:49

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You're not missing anything. There are just two different commonly used definitions of what a maximum-phase filter is.

Oppenheim and Schafer write [1]

Maximum-phase sequences are stable sequences whose poles and zeros are all outside the unit circle.

According to this definition, if $H(z)$ is a minimum-phase system, then $H(1/z)$ is maximum-phase. This system is stable but anti-causal (left-sided).

In the time domain, this means that if $h[n]$ is the impulse response of a minimum-phase system, then $h[-n]$ is the impulse response of a maximum-phase system.

(For complex-valued systems, we use $H^*(1/z^*)$ and $h^*[-n]$ for the maximum-phase system.)

According to above definition, no allpass filter can be a maximum-phase system.

The other definition, which can for instance be found in [2], says that a maximum-phase system is a causal and stable filter with all its zeros outside the unit circle. Of course, causality and stability imply that all poles need to be inside the unit circle.

Following this definition, a stable allpass filter is a maximum-phase system.

Note that in the case of FIR filters, both definitions are closely related. If $H(z)$ is a minimum-phase FIR filter, then its maximum-phase version is $H(1/z)$ according to the first definition, and $z^{-N}H(1/z)$ according to the second definition, where $N$ denotes the filter order. So both filters are just shifted versions of each other. Of course, this doesn't work for IIR systems, because shifting an anti-causal IIR filter will never result in a causal system.

[1] Oppenheim, Schafer, Discrete-Time Signal Processing, 3rd ed., p. 990.

[2] Proakis, Digital Signal Processing, 3rd ed., section 4.6.2., p. 359.

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  • $\begingroup$ Thanks for the clarification!!! $\endgroup$
    – Emm386
    Dec 15, 2022 at 20:57
  • $\begingroup$ I thought that a maximum-phase filter was when all the zeros are outside the unit circle, while all the poles are inside. $\endgroup$
    – robert bristow-johnson
    Dec 16, 2022 at 8:16
  • $\begingroup$ @robertbristow-johnson: Yes, that's one of the two common definitions. But Oppenheim and Schafer use the other definition, with also the poles outside the unit circle. $\endgroup$
    – Matt L.
    Dec 16, 2022 at 9:13
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    $\begingroup$ yeah, there are a couple of other semantics (like the definition of the "Nyquist rate") in O&S that disappoint me. I didn't know about this one. I have always maintained that a filter, in general, can be broken down and separated into the cascade of: 1. pure delay, 2. minimum-phase filter, and 3. all-pass filters. A filter is minimum phase if and only if 1 and 3 are empty (just wires). A maximum-phase filter is a minimum-phase filter in which every zero inside the unit circle is canceled with a pole of an all-pass filter in cascade. $\endgroup$
    – robert bristow-johnson
    Dec 16, 2022 at 21:40
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    $\begingroup$ And the problem is, with the other definition you mention (where poles and zeros are outside the unit circle) besides the obvious stability problem (so we're not building these maximum-phase filters) is that the poles outside the unit circle will tend to reduce the phase shift of the zeros outside the unit circle. The phase of a stable maximum-phase filter (that is the meaningful definition of "maximum-phase filter" with poles inside and zeros outside of the unit circle) will be greater than the phase of the that O&S definition you point to. So it doesn't really make any sense. $\endgroup$
    – robert bristow-johnson
    Dec 16, 2022 at 21:45
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The difference in the definitions are due to what can be physically implemented as a filter. Not to imply that non-causal filters are not usable for signal processing, but when we restrict ourselves to physically realizable (causal) filters, the definition of a maximum phase as simply being a filter with all zeros outside the unit circle (and no poles outside the unit circle) is useful.

With this alternate definition, the reverse of a minimum phase FIR filter is a maximum phase filter. The reverse filter is $z^{-N}H(1/z^*)$, where the added delay given by $z^{-N}$ ensures causality and is formed by simply reversing the order of the coefficients in the filter (as well as taking the complex conjugate if the filter coefficients are complex). Thus will then meet the definition of having only zeros outside the unit circle while being stable and causal (all poles inside the unit circle.)

Why the maximum phase definition of simply being all zeros outside the unit circle (for a stable causal filter) is useful is that for any given magnitude response, this maximum phase system will be the one that has the longest possible delay over any other implementable filter of the same filter order and same magnitude response; it will have the furthest excursion in phase over frequency- so indeed is “maximum phase”! (Just as the minimum phase filter will be the one filter with the same magnitude response that has the least delay, and the least excursion in phase… minimum phase!).

With that definition, which is used and useful, the all pass filter itself is indeed a maximum phase filter.

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    $\begingroup$ @Emm386 (thank you, and not to take anything away from Matt’s good explanation). I don’t off hand see anything useful in that for all-pass filters specifically; I was more referring to the use of the alternate definition where it applies to making a realizable min phase FIR filter into a max phase by reversing the coefficients (Not inverse, I corrected that) and how that definition does apply to all-pass IIR filters (which cannot be simply reversed) $\endgroup$
    – Dan Boschen
    Dec 15, 2022 at 23:02
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    $\begingroup$ What is perhaps of further interest is that any filter with arbitrary magnitude and phase can be decomposed into the min phase filter for that given magnitude response cascaded with an all pass with the residual phase difference. We note that that all-pass is a maximum phase filter but it is not THE maximum phase filter for that given magnitude response of the original filter (which would be the reverse of the min phase). So calling it a max phase filter other than otherwise saying all of its zeros are outside the unit circle doesn’t have much use that I can think of. $\endgroup$
    – Dan Boschen
    Dec 15, 2022 at 23:14
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    $\begingroup$ Thanks for joining this interesting discussion. It's remarkable that there seems to be no agreement on that definition. I checked several books of different reputable authors, and I found both definitions. Reading your post, I changed my mind about the "usefulness" of one definition or the other; it's really up to what you're trying to do, and none of the two is better than the other, so I'll edit my answer later on. $\endgroup$
    – Matt L.
    Dec 16, 2022 at 8:06
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    $\begingroup$ @MattL. good comment and agreed. In the end I don’t see a significant difference between the two since I often ignore a fixed delay difference as distinguishing two filters that are otherwise equivalent. I can see the small advantage of eliminating the need for compensating for delay when we don’t need to be limited by causality (post processing) such as that done when cascading a filter with it’s non-causal reverse (zero phase filter)— but it’s not a necessity, just convenient. This makes me think the reverse filter itself may also have a causal and non-causal definition. $\endgroup$
    – Dan Boschen
    Dec 16, 2022 at 12:51
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    $\begingroup$ @DanBoschen: Exactly, there could be two definitions of minimum-phase, but here everybody seems to agree that shifting a response to the left to make it "more" minimum-phase is cheating, so we restrict ourselves to causal filters. Following the same argument, maximum-phase filters should be non-causal. But again, none of the two definitions is inherently better I think, mainly a matter of taste. We just need to be very clear about what we mean when we say "maximum-phase". $\endgroup$
    – Matt L.
    Dec 16, 2022 at 14:16

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