# Minimum Phase

#### RayDunzl

##### Major Contributor
Central Scrutinizer
"In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable."

Could someone who understands the above (and whatever practical problems it leads to) give a practical explanation to someone who knows squat (yes, that's me!) about control theory and signal processing?

#### mansr

##### Senior Member
You're basically asking for someone to provide you with half an engineering degree in a forum post. I can't offer that, so instead I'll point you to this excellent online resource: https://www.dsprelated.com/freebooks/filters/Minimum_Phase_Filters.html

If you have more specific concerns or questions, articulating those as best you can may allow someone to give you a meaningful answer without covering all the underlying theory.

#### tomelex

##### Addicted to Fun and Learning
Patreon Donor
Ray, it looks like you are going to have to scrutinize yourself on this one buddy!

#### DonH56

##### Major Contributor
Technical Expert
Patreon Donor
Basic control theory, any junior/senior college class should suffice... Here's a quick simplified hand-waving crack at it:

LTI = constant linear relationship between input and output independent of time -- the old y = mx + b bit where m (slope) and b (intercept) are constants and no matter when you measure the result is the same (same result if you do it now or some other time, just a shift in time). This can be applied to time-varying signals (like audio -- it is the system response that is time-invariant), and means if y =ax and y = bx then y = ax + bx (i.e. linear mathematical relationships like superposition and scaling can be applied).

Causal = output follows input based upon current and past values, no dependence upon future values, no output not related to the input.

Stable = just like it sounds, does not generate unbounded outputs (like oscillations) for any input.

Inverse may be a little trickier... If y = H1(x) where H is some function of x, then the inverse H2(y) = x; it is the function needed to generate x from y instead of y from x.

Mathematically the definition is unintuitive without the afore-mentioned class(es): a minimum-phase function sampled-time (z) domain has all its roots (poles) inside the unit circle defined by z = 1 (ensuring stability); a maximum-phase filter has all its zeros outside the unit circle. This for functions like y = [(x-z1)(x-z2)...]/[(x-p1)(x-p2)...] where zn are zeros and pn are poles.

Maximum-phase filters are also useful; the usual example is an all-pass filter, i.e. a filter that delays all frequencies equally (a pure time delay).

Off the cuff with a couple of lookups so somebody should check my memory...

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#### dc655321

##### Senior Member
Could someone who understands the above (and whatever practical problems it leads to) give a practical explanation to someone who knows squat (yes, that's me!) about control theory and signal processing?
@RayDunzl, did your curiosity about minimum phase systems arise because you have seen that phrase appear when speaker drivers are discussed?

To get at why drivers (or other devices) are minimum phase, you would have to analyze the transfer functions for those devices. You'll find that their transfer functions obey the pole/zero relationship and other characteristics summarized by @DonH56.

#### Cosmik

##### Major Contributor
Looking at the operation of the bass reflex speaker cabinet:
...the bass-reflex loudspeaker system exhibits the characteristics of a minimum-phase highpass filter.
But we also have:
The low frequency driver in a resonant speaker enclosure system such as a ported cabinet or passive radiator cabinet cannot start and stop instantly like it can in a sealed-box cabinet. In order to achieve their bass output, ported speaker enclosures stagger two resonances. One from the driver and boxed air and another from the boxed air and port. This a more complex case than an equivalent sealed box. It causes increased time delay (increased group delay imposed by the twin resonances), both in the commencement of bass output and in its cessation. Therefore, a flat steady-state bass response does not occur at the same time as the rest of the sonic output. Instead, it starts later (lags) and accumulates over time as a longish resonant "tail". Because of this complex, frequency-dependent loading, ported enclosures generally result in poorer transient response at low frequencies than in well-designed sealed-box systems.
So is the system minimum phase?

#### RayDunzl

##### Major Contributor
Central Scrutinizer
did your curiosity about minimum phase systems arise because you have seen that phrase appear when speaker drivers are discussed?
More in the context of "room correction" (I think).

Ray, it looks like you are going to have to scrutinize yourself on this one buddy!
Looks like there is no easy answer for we, the unclean, so, while I welcome more replies, I suppose I'll have to study the very very basics, and see where that leads.

#### DonH56

##### Major Contributor
Technical Expert
Patreon Donor
One thing missing from the math is the "why"... Again bearing in mind this not really my area of expertise, minimum-phase filters do not necessarily provide the best impulse (time-domain) response, often much worse than a linear-phase filter (which best preserves waveshapes in the time domain since every frequency is equally delayed -- a pulse into the filter produces the same general pulse at the output after filtering), but minimum-phase filters have no "preshoot" or ringing before the pulse... Look at some DAC reviews of components with different filter responses and view the plots of the impulse response. Minimum-phase or hybrid filters typically have nothing before the start of the impulse (no preshoot/pre-ringing) then a damped ringing waveform after the filter. Linear-phase filters usually have a better-looking impulse, much cleaner with much less "ringing" around it, but it is often symmetric with a little ringing after and before the impulse. That signal before the impulse is what folk are concerned about; it can blur the leading edge of transient attacks, e.g. a click or drum hit may not start as "sharply". Some claim to readily hear this; I have problems hearing it myself unless very carefully going back and forth listening for it, or unless the filter has pretty large pre-ringing. A mastering engineer I respect put together a video demo but I have not been able to find it; he claims the effect is very obvious even via YouTube; I did not find it so, but could pick it out after a trial or three. It's kind of like some little thing stuck in your teeth; you may not notice it until you do, then it becomes impossible to ignore. But for me the post-impulse decay can be equally annoying; it is often larger and longer, but since most instruments (etc.) decay relatively slowly compared to the filter it is not as objectionable.

I have almost zero experience relating filter theory and practice to speaker design and so will (try to) stay out of that part of the discussion.

IMO and all that jazz - Don

#### amirm

Staff Member
CFO (Chief Fun Officer)
More in the context of "room correction" (I think).
Nulls are not minimum phase (can't be reversed). Peaks tend to be (if not they are not composites mixed with nulls).

#### restorer-john

##### Major Contributor
Linear-phase filters usually have a better-looking impulse, much cleaner with much less "ringing" around it, but it is often symmetric with a little ringing after and before the impulse. That signal before the impulse is what folk are concerned about; it can blur the leading edge of transient attacks, e.g. a click or drum hit may not start as "sharply". Some claim to readily hear this; I have problems hearing it myself unless very carefully going back and forth listening for it, or unless the filter has pretty large pre-ringing.
I'll be honest, I can't hear it on musical signals either.

It was only the other month after all those scope shots I made on first generation machines (Sony CDP-101 16/44 vs later FIR O/S machines). I tried with music (not test signals) and simply came to no useful conclusions whatsoever. I was in that 'convincing myself I can hear it' frame of mind, so I pulled the plug and forgot about it.

here: https://www.audiosciencereview.com/...log-reconstruction-filter-is-it-a-thing.2634/

Yes, it's easy with impulses or square waves to see it (pre-ringing vs none+overshoot) directly. What music/instruments are going to show up (audibly) the brick wall chebyshev multi element filter vs the 8xOS noise shaped FIR filters?

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#### Cosmik

##### Major Contributor
"In control theory and signal processing, a linear, time-invariant system is said to be minimum-phase if the system and its inverse are causal and stable."
Consider a speaker, a listener and a wall. Direct sound will reach the listener's ear, followed by a delayed echo of the direct sound.

Intuitively, I can kill that echo at the listener's ear as follows: after I send out an impulse from the speaker, I wait for a certain time, then send out an inverted version of that impulse. If I get it right, it will cancel out the echo at the listener's ear. But... the wall will create an echo of the inverted impulse too, and it is already winging its way to the listener's ear. After the same time delay I need to send out an inversion of that impulse to cancel it at the listener's ear. And then I need to send out an inversion of that, etc.

However, if the echo is louder than the direct sound, the inverted impulse will need to be getting louder each time ad infinitum.

Could an expert comment on whether either, neither, or both of these scenarios (gain <=1; gain >1) can be considered minimum phase?

• Time delay systems are of non-minimum phase behavior.
... which makes sense to me.

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#### gvl

##### Major Contributor
I'll be honest, I can't hear it on musical signals either.

It was only the other month after all those scope shots I made on first generation machines (Sony CDP-101 16/44 vs later FIR O/S machines). I tried with music (not test signals) and simply came to no useful conclusions whatsoever. I was in that 'convincing myself I can hear it' frame of mind, so I pulled the plug and forgot about it.

here: https://www.audiosciencereview.com/...log-reconstruction-filter-is-it-a-thing.2634/

Yes, it's easy with impulses or square waves to see it (pre-ringing vs none+overshoot) directly. What music/instruments are going to show up (audibly) the brick wall chebyshev multi element filter vs the 8xOS noise shaped FIR filters?
What frequency is ringing anyway? Is it ultrasonic? I thought ringing isn't really a big concern with music unless it's badly mastered with clipping.

#### RayDunzl

##### Major Contributor
Central Scrutinizer

#### Cosmik

##### Major Contributor
LTI = constant linear relationship between input and output independent of time -- the old y = mx + b bit where m (slope) and b (intercept) are constants and no matter when you measure the result is the same (same result if you do it now or some other time, just a shift in time). This can be applied to steady-state AC signals (like audio)
But audio is just as often not steady state...

#### gvl

##### Major Contributor
1/2 the sample rate
Right, is 22kHz or more then something we need to worry about?

#### andreasmaaan

##### Major Contributor
Patreon Donor
Right, is 22kHz or more then something we need to worry about?
Absolutely not

#### andreasmaaan

##### Major Contributor
Patreon Donor
Looking at the operation of the bass reflex speaker cabinet:

But we also have:

So is the system minimum phase?
The Wikipedia quote is wrong on sealed boxes. The driver/box system creates a 12dB/octave high pass filter with associated group delay, so it’s incorrect to state that the driver can start and stop instantly in a sealed box.

The other aspects of this you already know my views on

#### Cosmik

##### Major Contributor
The Wikipedia quote is wrong on sealed boxes. The driver/box system creates a 12dB/octave high pass filter with associated group delay, so it’s incorrect to state that the driver can start and stop instantly in a sealed box.
The quote says "...like it can in a sealed cabinet". In other words, DSP can be used to produce ideal behaviour (I presume). Maybe some overall latency is required to do it, however (not suitable for live monitoring perhaps).

The question is, can DSP produce such ideal behaviour with a bass reflex arrangement, seeing as it staggers two systems - and produces separate physical outputs for both of them. Steady state, no problem: it works. But transients..?

If you bung the hole up, no problem, but also no benefit from the resonator. If you want resonance, you've got to accept the transient smear. Correct?

#### andreasmaaan

##### Major Contributor
Patreon Donor
The quote says "...like it can in a sealed cabinet". In other words, DSP can be used to produce ideal behaviour (I presume). Maybe some overall latency is required to do it, however (not suitable for live monitoring perhaps).

The question is, can DSP produce such ideal behaviour with a bass reflex arrangement, seeing as it staggers two systems - and produces separate physical outputs for both of them. Steady state, no problem: it works. But transients..?

If you bung the hole up, no problem, but also no benefit from the resonator. If you want resonance, you've got to accept the transient smear. Correct?
Methinks that's a generous reading. In any case, DSP can be used to produce ideal transient behaviour in both minimum phase and non-minimum phase systems. So whether the ported system is minimum phase or not, its transient behaviour can be corrected.

Anyway, although I can't pretend to fully understand the physics myself, I do accept on the basis of what I do know that ported systems are indeed minimum phase.

#### Cosmik

##### Major Contributor
Another quote, this time from The Loudspeaker and Headphone Handbook:
(have had to get this text by OCR)
The reflex, ABR. bandpass and transmission line principles certainly do extend the
frequency response of loudspeakers, or permit a reduction is size when compared to
a conventional scaled enclosure and this has led to their wide adoption. However,
concentrating on the frequency domain alone does not tell the whole story. Although
the frequency domain performance is enhanced, the time domain performance is
worsened. The LF extension is obtained only on continuous tone and not on tran-
sients.
In this respect the bandpass enclosure is the worst offender whereas the true
transmission line causes the least harm.

None of these traditional techniques can offer sufficient accuracy in the time
domain to allow the mechanism of Fig. 2.60 to operate. Instead LF transients suffer
badly from linear distortion because the leading edges of the transients are removed
and reproduced after the signal has finished to give the phenomenon of hangover.

The Iow-frequency content 0f the sound lags behind the high frequencies in an unnat-
ural way. This can be measured as an exaggerated rearward shift in the acoustic
source position in the lowest frequencies reproduced. In other words the input waveform
is simply not reproduced accurately using these techniques, as is easily revealed
by comparison with the original sound.

Today the shortcomings of these techniques need no longer he suffered as active
technology clearly outperforms them using simply constructed scaled enclosures of
compact size.
Where the greatest. precision is not a requirement, passive techniques
will continue primarily because of tradition, and also where economy is paramount.
In the sealed case, the transducer is obviously a cone attached to a coil. Energise the coil and the cone will start moving instantaneously. Reverse the current and it will start to reverse. For sure, there's a phase lag, springiness of air in the box etc. but it is easy to see that pre-processing the signal can lead to ideal behaviour because there's only one output.

The bass reflex box is simply not the same. The output of the port is derived from the backwave of the cone. For its output to be in phase with the front wave it has lag behind by half a cycle - it cannot be coincident with, or ahead of, the front wave even with the magic of DSP; if we advance the port output we also advance the cone output. A transient must be smeared unavoidably. Once it gets going with a repeating waveform, no problem.