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Author Topic: Minimum phase and acoustics  (Read 14791 times)

JohnM

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Re: Minimum phase and acoustics
« Reply #15 on: October 08, 2010, 01:07:32 pm »

Here are the pole and zero locations when the filter gain has been set to -15dB. Applying a cut shifts the poles inwards (towards lower decay times) and the zeroes outwards (towards longer decay times).
index.php/fa/15564/0/
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JohnM

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Re: Minimum phase and acoustics
« Reply #16 on: October 08, 2010, 01:10:11 pm »

Q also has an effect on the pole/zero locations, reducing the Q moves both the poles and zeroes inwards, in this case reducing Q to 4.15 puts the zero back where it was in the first plot and shifts the pole in even further.
index.php/fa/15565/0/

If this filter were being used to target a 15dB response resonance at 50Hz whose decay time was 433ms (and hence had a pole at the 50Hz, 433ms position), the filter setting of 15dB cut and Q=4.15 would place the zero on the resonance pole, cancelling it, leaving us with the filter’s pole – but the filter’s pole has a 60dB decay time of only 77ms, so the net effect has been to replace a 433ms decay time and 15dB peak with a flat response (in that region) and 77ms decay.
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JohnM

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Re: Minimum phase and acoustics
« Reply #17 on: October 08, 2010, 01:26:00 pm »

To try and make all that a bit less abstract, here is an example of a real response that has a strong resonance at 50Hz, with a 1232ms 60dB decay time. First the pole-zero plot showing the response poles and zeroes (note many zeroes outside the unit circle, so not minimum phase) and, in red, the pole and zero of the 50Hz filter, which has a cut of 18dB and a Q of 9.94.

index.php/fa/15566/0/
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JohnM

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Re: Minimum phase and acoustics
« Reply #18 on: October 08, 2010, 01:28:18 pm »

Next the frequency responses with and without the filter overlaid.
index.php/fa/15567/0/
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JohnM

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Re: Minimum phase and acoustics
« Reply #19 on: October 08, 2010, 01:30:03 pm »

Waterfall of the response without filter.
index.php/fa/15568/0/
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JohnM

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Re: Minimum phase and acoustics
« Reply #20 on: October 08, 2010, 01:33:52 pm »

Finally, waterfall response with filter.
index.php/fa/15569/0/

P.S. Between having to use a separate post for every image and having to wait a while between posts to avoid triggering the "post flood" detector, posting on here is a bit of a chore. Maybe I should be a bit less visual Smile
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Constantin

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Re: Minimum phase and acoustics
« Reply #21 on: October 08, 2010, 01:39:37 pm »

Thanks for all the work you do for us John Smile

I will need some time to translate and understad this topic, but this is a important topic when it comes to listening room design.

thanks again
cheers
constantin

jimmyjazz

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Re: Minimum phase and acoustics
« Reply #22 on: October 08, 2010, 09:18:42 pm »

John, this is fascinating stuff.  I'm clearing the cobwebs and dredging poles and zeroes back to the forefront.  In the examples you show, is anything an actual measurement, or is it all theory?
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JohnM

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Re: Minimum phase and acoustics
« Reply #23 on: October 09, 2010, 02:33:11 am »

jimmyjazz wrote on Sat, 09 October 2010 02:18

In the examples you show, is anything an actual measurement, or is it all theory?
The room response shown is an actual measurement, but I chose it as it provides a nice illustration of the principle that is easily seen in the before and after data with a single filter because the room's resonance at 50Hz is relatively isolated (the next resonances either side are at approx 41 and 58Hz and much lower amplitudes). All the usual caveats still apply, however, the response is for a single location and will be different at other locations - the even response through the 50Hz region would show a substantial dip in a measurement from a location where the resonance was not as pronounced.
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Geoff Emerick de Fake

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Re: Minimum phase and acoustics
« Reply #24 on: October 09, 2010, 06:32:56 pm »

JohnM wrote on Fri, 08 October 2010 12:06

Geoff Emerick de Fake wrote on Fri, 08 October 2010 15:01

I don't get it...
I'm not too clear from your post what you don't get exactly, but I'll expand on my comments, which may help either make clear what I'm saying or provide some specifics for objection.
Basically, what I don't get is you consider the influence of poles decay and apparently neglect that of zeros.
Quote:

 I suspect you may be considering this in terms of providing a perfect inverse for a room's transfer function by means of suitably chosen biquads. Whilst the TF of a room response from a specific source position to a specific receiver position can be broken down into its constituent poles and zeroes, which can then be grouped as biquads, that of course does not make it invertible - there are non-minimum phase zeroes in most such responses, even at very low frequencies, hence stable inverses do not exist whether constructed from biquads or anything else.
As I mentioned earlier, this discussion would not even have started if we didn't agree on the fact that there is a domain where the approximation is valid, and all arguments/discussions/fights are based on that assumption. Smile
Quote:

 My post was about EQ filters however, not biquads in general. EQ filters have constraints on their biquad coefficients to allow them to function as intended, they do not allow independent placement of the filter poles and zeroes.
I agree, that's why I've expressed the transfer function in s notation earlier.
Quote:

 I'll illustrate this with some Z-plane plots (I operate in a sampled data world Smile) and since the poles and zeroes come in conjugate pairs I'll only show the upper half plane - in fact, I've produced the plots to span a frequency range of 210Hz and will look at a 50Hz filter, so the area of interest is in the upper right quadrant. The first plot shows the pole and zero locations for a 50Hz EQ filter with a Q of 10 (using the RBJ definition of Q from filter bandwidth at half gain) and with gain of zero. The pole and zero are at the same location. The T60 annotation (433ms) expresses the radius at which the pole and zero are located in terms of the time the pole resonance would take to decay by 60dB, as a convenience for use when examining decays of acoustic resonances.
index.php/fa/15563/0/
That's where you're starting to lose me. Having left school 40 years ago doesn't help either. Sad
I don't understand what is an EQ filter with a gain of zero...
An EQ filter has some boost or cut, so do you mean 0dB boost (or is it cut?)??? But a flat EQ has neither poles nor zeros...although I understand that, for the sake of demonstration, you may consider a flat EQ as identical denominator and numerator, which would obviously have their zeros and poles at the same location. Which is a perfect demonstration of the interaction between zeros and poles when they are very close.
Your example showing the dramatic reduction of decay at 50Hz goes to show that there is something that reduces the overall decay of the room + EQ combo. If you took into account only the poles, you would add both decays.
In fact it is a superb demonstration that EQ can compensate some of the time-related effects of resonance, just like a 10dB cut can perfectly compensate a 10dB boost at same frequency and same BW. The way the decay at 50Hz aligns almost perfectly with the 60-90Hz range decay says for me Mission accomplished.
I deliberately don't use Q notion because it just doesn't work for EQ's. IMO it works only for RLC bandpass filters. The academic definition of Q suppose asymptotic zero response and narrow BW. Eq filters never obey the 1st condition and 2nd condition is not applicable for less than 3dB boost/cut. [/aside]
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JohnM

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Re: Minimum phase and acoustics
« Reply #25 on: October 09, 2010, 07:49:15 pm »

Geoff Emerick de Fake wrote on Sat, 09 October 2010 23:32

Basically, what I don't get is you consider the influence of poles decay and apparently neglect that of zeros.
That is not the case, both are taken into account but I am not attempting to do anything about the zeroes of the room transfer function. That is primarily because they are the part of the TF which varies with source and/or mic position, the pole locations actually remain the same - they are a property of the room rather than positions within it, the variability we encounter in the level of any particular modal resonance as we move about the room is due to the shifting locations of the TF zeroes. For papers that go into detail on that google "common acoustical poles".

To address the influence of a particular modal resonance pole pair an EQ filter can be adjusted so that the filter's zeroes overlay the resonance's poles, which cancels them, effectively removing both the filter's zeroes and the resonance's poles from  the TF. We are then left with the filter's poles, which, because our filter has cut, have faster decay than the poles of the resonance we started with.

Quote:

I don't understand what is an EQ filter with a gain of zero...An EQ filter has some boost or cut, so do you mean 0dB boost (or is it cut?)??? But a flat EQ has neither poles nor zeros...although I understand that, for the sake of demonstration, you may consider a flat EQ as identical denominator and numerator, which would obviously have their zeros and poles at the same location. Which is a perfect demonstration of the interaction between zeros and poles when they are very close.
Yes, I mean 0dB, which as you say means the numerator and denominator are the same and the poles and zeroes are on top of each other. Signals pass through the filter unaffected, though the poles and zeroes are still there (though in practice of course the EQ designer would simply ignore filters with 0dB gain, no sense wasting time implementing a filter that has no effect). As the filter is adjusted its poles and zeroes move, by adjusting it appropriately we can place the zeroes on the poles of one of the room's resonances, eliminating the effect of that resonance and replacing it with the effect of the filter's poles.

Quote:

I deliberately don't use Q notion because it just doesn't work for EQ's. IMO it works only for RLC bandpass filters. The academic definition of Q suppose asymptotic zero response and narrow BW. Eq filters never obey the 1st condition and 2nd condition is not applicable for less than 3dB boost/cut.
"Q" can have a variety of meanings depending on how the filter designer chooses to relate the value to the biquad parameters. Here is a paper that reviews the various Q/bandwidth definitions used for biquad EQ filters. In REW's "Generic" EQ filters I use the definition RBJ, the author of that paper, proposes, which is to define the filter Q using the bandwidth measured at the half gain points, which avoids problems of undefined Q at gains/cuts less than 3dB. Whilst the various definitions of Q do cause some ambiguity it is still an easier notion for most people than a list of biquad coefficients Smile
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Geoff Emerick de Fake

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Re: Minimum phase and acoustics
« Reply #26 on: October 10, 2010, 09:45:10 am »

Tell me if I understand you well.
"common acoustical poles" seems to be a way of representing the invariant room accidents, in particular room modes, in a way that suit itself to equalization, which is the basis of this discussion. In this representation, room accidents are represented by biquads, with poles and zeroes.
Now it seems that you don't do anything with this particular biquad zeroes because they are almost an act of God  Confused .
But you take into account the poles of the EQ because they are perfectly and easily accountable.
Honestly trying to understand; somewhat goes against 40 years of electronics engineering and a droplet of acoustics.

The RBJ method is interesting; it's much better than the -3dB method, which many think is an absolute truth, cast-in-stone, when in fact it shows its limits as soon as you combine a boost and a cut at the same frequency.
But personally, I use a different method, based on users expectations. I use the "1/4-the-dB" method, which in the case of +/- 12dB boost/cut EQ, is consistent with the auditive concept that 3dB is the audible corner. And I benever dare calling it Q, I refer to it as bandwidth.
Q is supposed to be a portable notion, which in fact turns out to be utterly unusable.
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bruno putzeys

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Re: Minimum phase and acoustics
« Reply #27 on: October 10, 2010, 11:56:00 am »

What algorithm do you use to get from a response measurement to a p/z representation? The reasoning that the poles remain in place while the zeros shift about is entirely correct as I see it. I seem to remember our company being contacted by a Finnish joint with similar ideas.
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JohnM

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Re: Minimum phase and acoustics
« Reply #28 on: October 10, 2010, 05:24:44 pm »

Geoff Emerick de Fake wrote on Sun, 10 October 2010 14:45

Tell me if I understand you well.
"common acoustical poles" seems to be a way of representing the invariant room accidents, in particular room modes, in a way that suit itself to equalization, which is the basis of this discussion. In this representation, room accidents are represented by biquads, with poles and zeroes.
Now it seems that you don't do anything with this particular biquad zeroes because they are almost an act of God  Confused .
But you take into account the poles of the EQ because they are perfectly and easily accountable.
Honestly trying to understand; somewhat goes against 40 years of electronics engineering and a droplet of acoustics.
I think something got lost in translation there, accidents? Anyway, the  transfer function from a source position to a receiver position in a room has a numerator and a denominator. The roots of the numerator are the zeroes of the TF, the roots of the denominator are the poles, which correspond to individual resonant frequencies in the TF. The closer the poles are to the unit circle in the Z plane or the imaginary axis in the S plane, the slower the associated resonance decays and the greater the peak in the frequency response due to that particular resonance - the overall response is the combined effect of all the poles and zeroes of course, but each has its contribution.

If we could move the room TF poles further away from the unit circle we would have a response that had smaller peaks and faster decay. We can't actually alter the poles of the room TF, but what we can do is put a biquad EQ filter in the path to the source for each resonance we want to target. The filter adds its own pair of poles and its own pair of zeroes to the TF. We can adjust the EQ filter cut and bandwidth so that its zero locations fall exactly on top of a particular pair of poles in the room TF. The net effect of a pole and a zero in the same location is nil, so doing this effectively eliminates the room TF pole pair we have targeted. We are left with the pole pair of the filter itself, but the nature of EQ cut filters is that their poles are further from the unit circle than their zeroes, so overall the effect is as if we moved the room TF pole pair to the locations of the filter pole pair, where they have much smaller influence on the frequency response than they had in their original locations.
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JohnM

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Re: Minimum phase and acoustics
« Reply #29 on: October 10, 2010, 05:55:34 pm »

bruno putzeys wrote on Sun, 10 October 2010 16:56

What algorithm do you use to get from a response measurement to a p/z representation? The reasoning that the poles remain in place while the zeros shift about is entirely correct as I see it. I seem to remember our company being contacted by a Finnish joint with similar ideas.
There's a bit more info on REW's modal analysis, which produces the pole-zero plot,
here. It is a parametric analysis of the impulse response to determine the set of damped complex exponentials that produce the time series, in a similar manner to ARMA modelling or Linear Predictive Coding. I've been through a variety of methods to arrive at something reasonably robust, including Decimated Signal Diagonalisation and Filter Diagonalisation, neither of which responded well to the noisy signals we are faced with in acoustic measurement. I eventually adapted a technique used in processing NMR data, but all such methods are very sensitive to the signal-to-noise ratio of the measured IR. One group of Finns developed a technique they called "Frequency Zooming ARMA" to do the job, which looks from their papers like it should work well also.
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