rePhase, loudspeaker phase linearization, EQ and FIR filtering tool

[kgrlee]

Originally Posted by Barleywater

... Still, in terms or reciprocity something doesn't jibe quite right in my mind.

I don't understand your use of the word reciprocity. What does it have to do with measurement?

Mantra: Linear time invariant system is periodic and has an inverse.

Can you explain this mantra? It is a VERY strange LTI system that (has a) periodic (impulse response?)

Kirkeby may be used to generate inverse for swept sine measurements that lets swept sine effectively measure DC to Nyquist. Kirkeby may be applied to MLS to generate inverse too.

I'm not sure why you need Kirkeby to measure a system. It is an EQ method. In fact as you say

..thus possibility of IR recovery of same result using swept sine, MLS, and potentially any broad band signal.

No need for Kirkeby to do this.
__________________

Originally Posted by Barleywater

My room, as most has issues, and to keep perspective, I use same correction for left and right,

I sorta suspected this from my own experience. Can you post the 2 frequency responses for left & right at your listening position?

Did you try getting both left & right 'perfect' using different EQ for the 2 channels? What did it sound like?

[Barleywater]

My intention is not to correct listening position, but speaker. Idea is to get original waveform into room. Entry point is not the listening position. A soloist performs in front of listener, and of course room affects sound. Soloists sound bad in anechoic space too.

Reciprocity: Same result would be obtained if microphone and speaker position are swapped. True for omni speaker and microphone, thus my interest in Pluto type speaker.

LTI has same output for same input every time. As length of stimulus signal is shorted to single sample, output is impulse response of system.

You don't need Kirkeby to measure system, it allows you to build convolution pairs that produce nearly perfect correlation result for desired number of samples. When this sample period is greater than a linear time invariant system's impulse response, the impulse response of the system is returned without aliasing.

Correction EQ with individual filters for each peak and dip requires best fit methods with iteration. Kirkeby inverse is simultaneous solution for phase and amplitude of all frequency bins in FFT of time domain IR, and happens to be great correction EQ.

If I didn't think sound at listening position is fantastic, I wouldn't be sharing this. My hope is that scientifically minded individual would attempt to duplicate my results so comparison and discussion of results are possible.

[kgrlee]

Originally Posted by Barleywater

My intention is not to correct listening position, but speaker. Idea is to get original waveform into room. Entry point is not the listening position.

Can you clarify what you are doing?

Are you EQing the 'anechoic' response of the speaker using Kirkeby inversion?

Is this 'anechoic perfect' speaker than placed in your room with no further efforts to compensate for the room?

[bohdan1232000]

Hi,

Some comments on Kirkeby and Inverse HBT.

Kirkeby inverse equalization method involves frequency-dependent regularization parameter E(freq). The role of E(freq) is to maintain the inverse filter as calculated, but only within the frequency band of interest. Outside this frequency band, the E(freq) must have such value, that it extinguishes the inverting filter’s operation.

The effect of convolving loudspeaker response with such inverted filter produces flat response within the frequency band of interest, and returns to the original frequency response outside this band.

In order to accomplish this, the E(freq) function obviously must include transition bands on both sides of the frequency band of interest. There is possibly an infinite number of ways that E(freq) may transit from one level to another. Farina proposed logarithmic transition over 1/3 octave in AES convention paper “Implementation of a double StereoDipole system on a DSP board – Experimental validation and subjective evaluation inside a car cockpit”. Kirkeby in his AES Preprint 4916 gave some hints, but nothing specific.

The point here is, that in any case, within the transition bands, the E(freq) is a manually imposed function, that has nothing to do with the original loudspeaker frequency response.


In order to understand Inverted HBT difference, please have a look into http://www.bodziosoftware.com.au/Square_Wave.pdf and focus on Figure 5.

The dark-blue curve is the inverted filter SPL from 91Hz to 5220Hz. The orange curve is the corresponding inverted phase response. But something interesting happens on the high-side of 5220Hz and low-side of 91Hz.

In both instances, the phase response tapers-off to 0deg (as you would expect). The shape of phase response within both transition bands is mathematically related (calculated) to where the original loudspeaker frequency/phase response was at transition points, and also includes what happens on both sides of the transition points. So, at 91Hz, the phase response tapers off between 100Hz to 40Hz (more than 1octave). Above 5220Hz, the phase response tapers-off all the way to 50kHz (2.2 octaves). All this is based on the original loudspeaker frequency response – and is not arbitrary selected transition function.

This is the power and mathematical elegance of Inverted HBT. It gives you mathematically correct amplitude/phase response of the inversion filter across the whole bandwidth.

Best Regards,
Bohdan

[kgrlee]

Originally Posted by bohdan1232000

Some comments on Kirkeby and Inverse HBT.

... In order to understand Inverted HBT difference, please have a look into http://www.bodziosoftware.com.au/Square_Wave.pdf and focus on Figure 5.
... This is the power and mathematical elegance of Inverted HBT. It gives you mathematically correct amplitude/phase response of the inversion filter across the whole bandwidth.

Bohdan, if I understand you correctly, you

1. select a bandwidth to EQ and straighten the amplitude response outside these limits.
2. apply the Hilbert Transform to this 'truncated' (in freq. domain) amplitude response to get phase. But this is also called getting the Minimum Phase !.
3. You invert (in complex freq) this Minimum Phase response to get your EQ
4. This leaves you with an amplitude response which is flat in the passband but exhibits the usual Minimum Phase behaviour. ie a non-flat (but minimum) phase response
   
   
5. You take this non-flat phase response and use it to make an all-pass network with the same phase response but flat amplitude
6. You take this resultant impulse response and time reverse it to get a further EQ which when applied to the Minimum Phase EQ in 2 turns it into a Linear Phase EQ

[john k...]

Originally Posted by kgrlee

Bohdan, if I understand you correctly, you

1. select a bandwidth to EQ and straighten the amplitude response outside these limits.
2. apply the Hilbert Transform to this 'truncated' (in freq. domain) amplitude response to get phase. But this is also called getting the Minimum Phase !.
3. You invert (in complex freq) this Minimum Phase response to get your EQ
4. This leaves you with an amplitude response which is flat in the passband but exhibits the usual Minimum Phase behaviour. ie a non-flat (but minimum) phase response
   
   
5. You take this non-flat phase response and use it to make an all-pass network with the same phase response but flat amplitude
6. You take this resultant impulse response and time reverse it to get a further EQ which when applied to the Minimum Phase EQ in 2 turns it into a Linear Phase EQ

Not Bohdan, but to answer your questin that is basicly the idea, but since it is all done with FIR filters the minimum phase EQ and the phase linearization are both done with a single convolution in the frequency domain.

[Barleywater]

Originally Posted by kgrlee

Can you clarify what you are doing?

Are you EQing the 'anechoic' response of the speaker using Kirkeby inversion?

Is this 'anechoic perfect' speaker than placed in your room with no further efforts to compensate for the room?

Direct response is what human hearing uses for direction and timbrel identification.

What I DO know is that you don't want to achieve 'anechoic' results.

No, I don't listen to live music in anechoic venue, and I don't listen in anechoic chamber at home.



At what point, and under what conditions is a speaker suppose to reproduce source waveform?

[pos]

You cannot measure in anechoic situation at all frequencies in a room anyway, so you have to deal with multiple measurement methods for different frequency ranges, and deal with the different shortcomings...

The main grip I have with the automated inversion methods leading to a completly flat response (correcting even high Q disparities) as shown above, is that they entierly rely on the accuracy of the (single) measurement.
If the measurement is not spacially averaged over at least small angle around the listening axis it will have a lot of articats specific to its exact position, such as diffraction for example (coming from object as well as from the box, or the driver frame itself).
"Correcting" these artifacts is not a good idea IMHO...
I would be curious to see the look of the perfectly flat corrected curve with another mic position than the one used to construct the correction.

For automated corrections I find DRC-FIR, or PORC (using a finite number of biquads) much more "stable" approaches.
And I even prefer the manual "full-fledged" approach (which is the topic of that thread )

[bohdan1232000]

Hi kgrlee,

Yes, as John commented, this is essentially the gist of Inverted HBT.
Please note, that this approach offers you the option of running your equalized system (loudspeaker+equalizer) in minimum-phase or linear-phase modes, and outside the equalized bandwidth, the whole system is still clearly defined as linear-phase or minimum-phase, depending on your choice of phase characteristics.

Now, back to Kirkeby. I do not own a device, that is based on this algorithm, therefore I am struggling to understand the implications of their method.

• When you create a system (loudspeaker+Kirkeby equalizer) to equalize the loudspeaker between F1 and F2 – is the system between F1 and F2 “linear phase”, or “minimum phase”.

• As I understand, such system has transition regions below F1 and above F2. So my next question is: within the transition regions, is the system “minimum-phase”, or “linear-phase” or “undetermined-phase”.

• Do the transition regions fall into the audio bandwidth for any driver?. If yes are they audible?. How do you deal with this issue?.

• Can you run Kirkeby equalizer in minimum-phase mode, or it imposes straight away linear-phase mode?.

Can anybody answer these questions?.

Best Regards,
Bohdan

[john k...]

Bohdan,

Just to make things perfectly clear. The UE allows the system to run in linear phase mode, or with just minimum phase EQ applied. When just the EQ is applied the system phase remains essentially that of the specified crossover. It will reduce to a minimum phase system only if the crossover does not introduce its own nonlinear phase, such as a 1st order acoustic crossover. I just wanted to make that clear.

POS, while I understand what you are saying about equalizing flat based on a single measurement, what happens off axis is largely dependent on the design of the speaker. If constant directivity is maintained off axis then the single point reference for eq is actually pretty good. If you don't want to EQ every small ripple you can start with a smoothed response, like 1/3 octave smoothing applied to the reference measurement.

In any event it is still possible to take any number of measurements and average them with the UE. The danger with that approach is that such an averaged response can actually result in a reference response that is worse than the measurement at any of the points contributing to the average. This is because you are not dealing with a simple average of amplitude but of a vector sum. This becomes very sensitive at higher frequency where the averaged response may exhibit comb-filtering effects.