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Minimum Phase

DonH56

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But audio is just as often not steady state...

Yes, had something else in mind and that was wrong. Or rather it is true but too limited. I corrected it to this:
This can be applied to time-varying signals (like audio -- it is the system response that is time-invariant)

Sorry about that, I did say to check my off-the-cuff babbling, and you did!

Thanks,
Don
 
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andreasmaaan

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The quote is misleading:
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.

Yes, the time domain performance is worsened. But that is because we are now dealing with a summed response that has been high-pass filtered at 24dB/octave as opposed to 12dB/octave. Of course the transient response of a 24dB/octave filter is worse than the transient response of a 12dB/octave filter.

And I don't know much about that source, but the statement "the LF extension is obtained... not on transients" doesn't make any sense. The LF extension is there. Play a steady state signal and you have LF extension. Play a transient and you have LF extension (with double the transient smearing of a closed box system). Perhaps it's just a clumsily worded sentence.

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.

Think of a multiple driver system with crossovers. Despite the fact that the output comes from multiple outputs (drivers), the summed response can be phase corrected.

The same is true of a vented system. There are multiple outputs (port and woofer) that each have their own phase. These outputs sum outside the box, and the summed response itself has a (summed) phase. This phase can be corrected.

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.

It seems that you're imagining the port and the woofer as though they both output at all frequencies. But the opposite is the case. The driver is not outputting anything at the port resonance. So if DSP corrects the phase response at resonance, that does not affect the direct radiation from the driver.
 

Cosmik

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There are multiple outputs (port and woofer) that each have their own phase.
They each have their own phase, but not individually controllable because the port output is derived from the cone output. If we change the phase of the cone we automatically change the phase of the port.
It seems that you're imagining the port and the woofer as though they both output at all frequencies. But the opposite is the case. The driver is not outputting anything at the port resonance. So if DSP corrects the phase response at resonance, that does not affect the direct radiation from the driver.
All steady state-centric. When the driver 'kicks off' for the first time, the port isn't yet resonating. Once it gets going, it all works perfectly - until you try to stop it.
 

andreasmaaan

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They each have their own phase, but not individually controllable because the port output is derived from the cone output. If we change the phase of the cone we automatically change the phase of the port.

All steady state-centric. When the driver 'kicks off' for the first time, the port isn't yet resonating. Once it gets going, it all works perfectly - until you try to stop it.

The driver isn't "kicking off" at the port resonance. It's still, and not producing any output. At higher frequencies, where the driver is producing output, the port isn't. The whole system can be corrected.
 

mansr

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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).
A pure time delay is linear phase, not maximum.
 

mansr

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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.
1/2 the sample rate
Not quite. The so-called ringing is at the filter cut-off frequency. For reconstruction filters, this is typically at or slightly below half the sample rate, but it doesn't have to be.
 

mansr

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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.
Linear phase filters always have a symmetric impulse response, and conversely.

But for me the post-impulse decay can be equally annoying; it is often larger and longer,
Given a linear phase filter, a minimum phase one with the same magnitude response has twice as much post-ringing.
 

DonH56

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A pure time delay is linear phase, not maximum.

Thank you, must have not remembered that one rightly. Long, long time since those graduate courses and I am fundamentally an analog guy. Should have pulled out O&S or one of my other references but I was tracking down a test bug and didn't take the time yesterday (Sunday, working again...)

All these corrections should serve to keep me from posting on things I am not actively using or designing, too easy for senility to creep in and cause misinformation.
 

Cosmik

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Supposing I don't want 24dB/octave roll-off from my bass reflex speaker, but in fact I want 12dB/octave. Why
The driver isn't "kicking off" at the port resonance. It's still, and not producing any output. At higher frequencies, where the driver is producing output, the port isn't. The whole system can be corrected.
In the face of the unbridgeable gap between frequency domain 'steady state' and the concept of the time domain transient, I give up! :)
 

andreasmaaan

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Let me try again @Cosmik, and please banish from your mind that I’m interested for present purposes in the steady state amplitude response.

At the frequency of the port resonance, the driver output is zero (or in the real world, negligible). If we correct the phase at this frequency, it is only the output from the port that we are correcting.

Do you agree thus far?
 
OP
RayDunzl

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Right, is 22kHz or more then something we need to worry about?

As for the frequency, not for me.

There may be a "spreading" of the energy. In the example, the (illegal) Impulse is full scale, but after a little manipulation to create ringing example (upsample, downsample), it is 0.444dB down at the peak.


1546884059079.png


In this contrived example, the ringing maximum level is at -25dBfs (or so).
 
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mansr

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There may be a "spreading" of the energy. In the example, the (illegal) Impulse is full scale, but after a little manipulation to create ringing example (upsample, downsample), it is 0.444dB down at the peak.
You've removed all the high frequencies, so the total energy has to be less. This translates to a lower peak amplitude.
 

Keith_W

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I have been struggling to understand Minimum Phase so I did a search on google, and this old thread on ASR popped up. I have read John Mulcahy's article (also linked above), the Wikipedia article, and watched some videos on Youtube. The problem with most resources on explaining Minimum Phase is that it is impenetrable to the layman - as @mansr says, it is half an engineering degree. I would appreciate it if ASR members can check if I have understood it correctly and let me know if I am wrong. Yes, ASR is full of pedants and I am asking for some full on pedantry here ;)

"A minimum phase system is where the phase response can be calculated directly from the frequency response using a mathematical function known as the Fourier transform. It has two important properties - it can be inverted (and thereby cancel the original signal perfectly), and it has minimum delay. Anything that increases the delay, for example room reflections, baffle diffraction, group delay from crossovers, etc. will render the system non minimum phase."

"Loudspeakers are minimum phase devices, however they are rendered non minimum phase as soon as drivers are placed in a cabinet (because of internal reflections in the cabinet), have a crossover (because the crossover changes the phase response, the only exception being a digital FIR filter), and in a room (because reflections from the room render the system non-minimum phase by introducing delayed signals which may be at lower or higher amplitude to the main signal). Regions of non-minimum phase behaviour increase as wavelength gets shorter, i.e. above the Schroder frequency where the room stops behaving in a modal fashion. Bass dips are usually regions of non-minimum phase behaviour (which is why we recommend not to fill nulls by boosting the volume), however bass peaks are usually minimum phase, which is why they can be corrected".

"Excess phase is the difference between minimum phase which is calculated from the frequency response, and the measured phase. The calculation can be windowed to take into account how many cycles are permitted when making the calculation." <-- In Acourate there is an option to calculate excess phase by inputting the number of cycles. Should I think of it as the phase equivalent of a Frequency Dependent Window (FDW)?

I do not have a complete understanding of "linear phase" - the definition is simple enough, which is that all the samples are shifted by the same amount of time over a given frequency band. However, I do not understand how this is different to group delay. Aren't they the same thing?
 

DonH56

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There's probably not a real easy way to explain it without a lot of words to describe the equations. Group delay Gd is the negative derivative of phase P(f) where phase is a function of frequency f: Gd = -dP(f)/f. In words, group delay is the negative of the change in phase over frequency, or how much phase changes (as a function of frequency) with respect to frequency. If phase is linear, then the phase change is proportional to frequency, and the change in phase over frequency is a constant. If the phase change is constant over frequency, then each frequency is delayed by the same amount of time, a constant group delay. Note phase has units of degrees, and group delay has units of time.

How much that (constant group delay) matters for audio has been debated but it is critical in other applications like radar. Why? If all frequencies are delayed equally, then the signal coming out of a system is the same as what entered the system except for a constant delay. If a signal has more than one frequency, like a pulse or square wave that theoretically is an infinite series of frequencies, then a linear-phase system means the pulse coming out is identical to that going in, just delayed in time. If group delay is not constant, phase is not linear over frequency, and some frequencies are delayed more than others, thus the pulse shape will be changed (corrupted). If you apply a perfect square wave to an ideal speaker with linear phase, you get a square wave out. If the speaker does not have linear phase, the square wave that comes out will no longer look like an ideal square wave, but have "ripples" in the wave shape sine all the frequency components are not perfectly aligned (see https://www.audiosciencereview.com/.../composition-of-a-square-wave-important.1921/).

HTH - Don
 

gnarly

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I have been struggling to understand Minimum Phase so I did a search on google, and this old thread on ASR popped up. I have read John Mulcahy's article (also linked above), the Wikipedia article, and watched some videos on Youtube. The problem with most resources on explaining Minimum Phase is that it is impenetrable to the layman - as @mansr says, it is half an engineering degree. I would appreciate it if ASR members can check if I have understood it correctly and let me know if I am wrong. Yes, ASR is full of pedants and I am asking for some full on pedantry here ;)

"A minimum phase system is where the phase response can be calculated directly from the frequency response using a mathematical function known as the Fourier transform. It has two important properties - it can be inverted (and thereby cancel the original signal perfectly), and it has minimum delay. Anything that increases the delay, for example room reflections, baffle diffraction, group delay from crossovers, etc. will render the system non minimum phase."

"Loudspeakers are minimum phase devices, however they are rendered non minimum phase as soon as drivers are placed in a cabinet (because of internal reflections in the cabinet), have a crossover (because the crossover changes the phase response, the only exception being a digital FIR filter), and in a room (because reflections from the room render the system non-minimum phase by introducing delayed signals which may be at lower or higher amplitude to the main signal). Regions of non-minimum phase behaviour increase as wavelength gets shorter, i.e. above the Schroder frequency where the room stops behaving in a modal fashion. Bass dips are usually regions of non-minimum phase behaviour (which is why we recommend not to fill nulls by boosting the volume), however bass peaks are usually minimum phase, which is why they can be corrected".

"Excess phase is the difference between minimum phase which is calculated from the frequency response, and the measured phase. The calculation can be windowed to take into account how many cycles are permitted when making the calculation." <-- In Acourate there is an option to calculate excess phase by inputting the number of cycles. Should I think of it as the phase equivalent of a Frequency Dependent Window (FDW)?

I do not have a complete understanding of "linear phase" - the definition is simple enough, which is that all the samples are shifted by the same amount of time over a given frequency band. However, I do not understand how this is different to group delay. Aren't they the same thing?

I'll offer what I've come to believe are the pragmatic points about minimum phase, linear phase, and group delay.
They no doubt will be in err in strict engineering and academic terms, but they are what I've learned I can act on, and get consistent tuning results....so textbooks be damned hahaha

For me, minimum phase means frequency response and phase change together. Flatten one, and you flatten the other.
It only works fully well with a single source....like a single driver.
Anytime two or more sources combine, they each may individually be minimum phase, but their summation is a vector out at any particular point in space.
And together they are no longer min phase.

A minimum phase source can have constant delay and still be minimum phase. In most fact all do, given acoustic time of flight (ToF) , which is a constant delay.

Which brings me to my definition of group delay.....mine does not include constant delay of any type (ToF, fixed computational latency, FIR filter delays, etc.)

Nor does it include the delay observed with all-pass filters, one end of the spectrum compared to the other. Sure, there is a relatively constant delay, one end of the spectrum vs the other, but it takes a frequency dependent delay between the two ends to get there. Which to me means it's NOT a constant delay across all frequencies.

Imo, Group delay is all about tone burst envelope distortion, and other than that idea,.... should be utterly purged from audio for all the confusion it causes :p .....

For me, linear phase means a flat phase trace at zero degrees across the spectrum.
Flat phase can also be had with no constant delay, although it's limited to a single driver full-ranger, or a two-way using a first order xover.

Flat phase via FIR, comes with the price of a constant delay, accomplished via impulse centering within the FIR filter.

For me, to contrast minimum phase xovers with linear phase ones, for me simply means one has phase rotation through the xover region, the other doesn't.
One has no constant delay, the other does.

Hey, Here's an old post of Bruno P's you might like https://repforums.prosoundweb.com/index.php?topic=16529.msg216033#msg216033
 

dc655321

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"A minimum phase system is where the phase response can be calculated directly from the frequency response using a mathematical function known as the Fourier transform. It has two important properties - it can be inverted (and thereby cancel the original signal perfectly), and it has minimum delay. Anything that increases the delay, for example room reflections, baffle diffraction, group delay from crossovers, etc. will render the system non minimum phase."

Frequency Response has two components, an Amplitude Response and a Phase Response.
The former is colloquially and imprecisely referred to as "frequency response"....

Imagine a "black box" system - you don't know what's inside or how it works internally.
But, you can apply input signals and measure output signals to get an idea of the characteristics of the "black box".
The black box could represent a speaker driver, an electric circuit, a vehicle's suspension system, a potato... whatever.

A minimum phase system is one that transfers energy, from input to output, faster than any other system of identical Amplitude Response.
Read that again, it's the most important aspect.

The mathematical characteristics all follow from the physics.
Unless you ask a mathematician... :rolleyes:

"Loudspeakers are minimum phase devices, however they are rendered non minimum phase as soon as drivers are placed in a cabinet (because of internal reflections in the cabinet), have a crossover (because the crossover changes the phase response, the only exception being a digital FIR filter), and in a room (because reflections from the room render the system non-minimum phase by introducing delayed signals which may be at lower or higher amplitude to the main signal). Regions of non-minimum phase behaviour increase as wavelength gets shorter, i.e. above the Schroder frequency where the room stops behaving in a modal fashion. Bass dips are usually regions of non-minimum phase behaviour (which is why we recommend not to fill nulls by boosting the volume), however bass peaks are usually minimum phase, which is why they can be corrected".

A system composed of minimum phase elements arranged serially will always be minimum phase.
Minimum phase components arranged in parallel may not constitute a minimum phase system.

Note which category multi-driver speaker systems typically fall into.
They could be (and typically are) non-minimum phase systems even if all environmental reflections were prevented, simply by the nature of their topology (eg: woofer, mid, tweeter in parallel).

"Excess phase is the difference between minimum phase which is calculated from the frequency response, and the measured phase. The calculation can be windowed to take into account how many cycles are permitted when making the calculation." <-- In Acourate there is an option to calculate excess phase by inputting the number of cycles. Should I think of it as the phase equivalent of a Frequency Dependent Window (FDW)?

I would not think of excess phase as you suggest. I don't see the equivalence at all.
Or, are you referring to the possible equivalence with the Acourate option?

I do not have a complete understanding of "linear phase" - the definition is simple enough, which is that all the samples are shifted by the same amount of time over a given frequency band. However, I do not understand how this is different to group delay. Aren't they the same thing?

Yes, all samples shifted by the same amount of delay (or phase - this is a property of the Fourier Transform).
But no, linear phase and group delay are not the same thing.
Linear phase is a term intended to express the notion that graphically, group delay would appear as a flat line when plotted against frequency (i.e. a constant).

But, a plot of group delay vs frequency can have almost any shape imaginable.
Linear phase is a (very) special case.


Hopefully I haven't muddied the waters too much.
 

Keith_W

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Thank you all for your replies. I think it confirms that I am more or less on the right track and I do not have any major errors in my understanding.

I would not think of excess phase as you suggest. I don't see the equivalence at all.
Or, are you referring to the possible equivalence with the Acourate option?

Yes, I am thinking of the Acourate "Excess Phase" correction, which allows you to specify the amount of correction in terms of cycles.

Unlike the "FDW" for amplitude response correction (I am using the right term now after you taught me!) which is easy to see, I do not see the effect of "Excess Phase" correction in the measurement, unless I am looking at the wrong graph. The phase graph, which is wrapped, looks as chaotic as ever. But if I choose a low correction (say 2 cycles) vs. a high correction (15 cycles), then the step response looks poor. Excessively high correction makes the predicted step response look great, but the listening experience has audible artefacts, most notably a very short tone that occurs before transients. I have learnt to go for a middle ground (about 5 cycles) that makes the step response look good, but does not introduce audible artefacts.
 

René - Acculution.com

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"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?
Causality has to do with the output coming after the input, which is true for any physical system. Stability has to do with the output being unbounded (going berserk) for a bounded input. Invertibility has to do with undoing what a previous system did, getting back to the response of a wire. So you generally have two situations where you cannot find a system that inverts a previous system, and those (previous systems) are both non-minimum phase. One is a time delay, because to invert it, you would need a negative time delay, such that you output a signal before you have an input, to output the proper output at time zero even though the previous system has not yet woken up. So your inverse system would not be causal, and so not physical. The other situation is that your original system has so-called zeros in the right-hand plane, which may not make sense if you haven't had much signal processing, but the issue here is that to undo/invert such an NMP system, you will need to cancel those zeros with so-called poles, and if poles reside in the right-hand plane, your inverting system will not be stable.

Also, @dc655321 has a good post above that I totally agree with regarding how to view this from a loudspeaker perspective.
 

René - Acculution.com

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I have been struggling to understand Minimum Phase so I did a search on google, and this old thread on ASR popped up. I have read John Mulcahy's article (also linked above), the Wikipedia article, and watched some videos on Youtube. The problem with most resources on explaining Minimum Phase is that it is impenetrable to the layman - as @mansr says, it is half an engineering degree. I would appreciate it if ASR members can check if I have understood it correctly and let me know if I am wrong. Yes, ASR is full of pedants and I am asking for some full on pedantry here ;)

"A minimum phase system is where the phase response can be calculated directly from the frequency response using a mathematical function known as the Fourier transform. It has two important properties - it can be inverted (and thereby cancel the original signal perfectly), and it has minimum delay. Anything that increases the delay, for example room reflections, baffle diffraction, group delay from crossovers, etc. will render the system non minimum phase."

"Loudspeakers are minimum phase devices, however they are rendered non minimum phase as soon as drivers are placed in a cabinet (because of internal reflections in the cabinet), have a crossover (because the crossover changes the phase response, the only exception being a digital FIR filter), and in a room (because reflections from the room render the system non-minimum phase by introducing delayed signals which may be at lower or higher amplitude to the main signal). Regions of non-minimum phase behaviour increase as wavelength gets shorter, i.e. above the Schroder frequency where the room stops behaving in a modal fashion. Bass dips are usually regions of non-minimum phase behaviour (which is why we recommend not to fill nulls by boosting the volume), however bass peaks are usually minimum phase, which is why they can be corrected".

"Excess phase is the difference between minimum phase which is calculated from the frequency response, and the measured phase. The calculation can be windowed to take into account how many cycles are permitted when making the calculation." <-- In Acourate there is an option to calculate excess phase by inputting the number of cycles. Should I think of it as the phase equivalent of a Frequency Dependent Window (FDW)?

I do not have a complete understanding of "linear phase" - the definition is simple enough, which is that all the samples are shifted by the same amount of time over a given frequency band. However, I do not understand how this is different to group delay. Aren't they the same thing?
Should read 'Hilbert transform', not 'Fourier transform'.
 
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