Multiple Small Subs - Geddes Approach

[john k...]

Originally posted by gedlee



There are so many misleading points in the discussion that its hard to follow. John uses the term "linear" in a manner not consistant with "linear systems theory" and this gets confusing. He seems to mean "flat" or "straight" when he says "linear", but I guess I don't know what he means when he says that the modes are not "linear". The wave equation IS linear and anything derived from it is linear.


But one thing that has not been shown is why it is assumed that because a room does not meet the conditions for MP that it should have the same characteristics as an electronic circuit that is not MP. I do not see how one can make this assumption. In other words, since a real room has a multitude of transfer functions can one assume that minimizing the "mean magnitude response" of these functions does not minimize their "mean decay" just because one or more of those transfer functions is not MP, or even if all of them are not MP. I just don't see how this leap can be made. Its certainly not been shown.

Earl,

I use the term linear in the only way it applies: A system is a linear system if the equation governing the behavior of the system is linear.

F = M x" + B x' + k x

is the equation for a linear spring, mass damper system.

F = M x" = B x' + k x^2

is a nonlinear system

However the transfer function (or system function) for the linear system need not be linear and in general is not.

For example, a 2nd order LP filter has

T(s) = 1/(1 +Qs + s^2)

This is not a linear function (of s).

But

O(s) = T(s) x I(s)

constitutes a linear relationship between input and output because T(s) is not a function of I(s). It is just a scale factor which is constant for any fixed value of s, but varies (nonlinearly) with s. That was the point I was trying to get across. I was trying to clarify the difference between a system being linear and whether or not the system's transfer function is linear or not. I guess I confused things instead of making the distinction clearer.

I would also agree that even if the room response is MP at some x,y,z point in the room, so what? In that case the response can be corrected in both time and amplitude, but what about the rest of the room? As you and I have both indicated the in room response is composed of an infinite number of transfer functions, even for fixed source location, one for ever x,y,z coordinate defining a point within the room boundaries.

[mat02ah]

Originally posted by gedlee

But one thing that has not been shown is why it is assumed that because a room does not meet the conditions for MP that it should have the same characteristics as an electronic circuit that is not MP.

For a given amplitude response there's only one way for a system to be MP but there are many ways (infinite?) to be non-MP. So, being "non-MP" doesn't make systems comparable.

Originally posted by gedlee

In other words, since a real room has a multitude of transfer functions can one assume that minimizing the "mean magnitude response" of these functions does not minimize their "mean decay" just because one or more of those transfer functions is not MP, or even if all of them are not MP. I just don't see how this leap can be made. Its certainly not been shown.

I was trying to make the leap in the other direction: If the transfer function(s) were MP, improving the flatness (I guess that's what you meant instead of 'minimizing') of the mean magnitude response of these functions will minimize decay.

In terms of logics: (flat AND minimum-phase) -> perfect transient.
The negation of this

NOT(flat AND minimum-phase) -> NOT (perfect transient)

is of course not necessarily true.

"Ex falso sequitur quodlibet " - From falsehood follows anything.

I still think that the (near) MP property of the speaker-room system at LFs is one of the reasons why your optimizing method works and is justified. I'm stubborn, I know ...

[gedlee]

John

I still think that your use of the term "linear" in the context that you were using it was not consistant with "linear systems theory" where the filter is still linear even though the transfer function is not a linear function of omega. The assumption of superposition still holds and hence the transfer function is "linear" in the systems theory sense. Only confusion will result if you change this convention.

[gedlee]

Originally posted by mat02ah

I was trying to make the leap in the other direction: If the transfer function(s) were MP, improving the flatness (I guess that's what you meant instead of 'minimizing') of the mean magnitude response of these functions will minimize decay.

I don't know that this is true. I am not talking about a single transfer function, but an average of many. I don't think that it follows that minimizing the average magnitude deviation from the mean (to be more precise) will minimize the average decay, or any decay. You would need to prove this, not just state it. But don't bother, because its really academic. If it were important I might look at it myself, but I don't see it as important. What I want to do is minimize the average deviation from the mean and I really don't care what else this does. (Although I suspect that it does work as you suggest, I just don't see this as something that you can assume.)

I would be careful mixing the terms "system" and "transfer function" as they are not synonimous. A system, such as a room, can have many transfer functions, but is still only one system. You can talk about any one transfer function being MP, but I am not sure that the system can be so easily described if there are many transfer functions and not all of them need be MP. Any one of them might be, but the system would not then become MP for all the others.

""Ex falso sequitur quodlibet " - From falsehood follows anything."

Thats a great phrase. I'll remember it.

[gedlee]

Originally posted by Sheldon
This thread and other reading convinced me to start playing with sub placement and measurements, which has further convinced me to add additional subs. Right now, my "subs" are incorporated into the mains. These subs are crossed over to the mid bass at 80 Hz using a transient perfect filter, calculated with John's program. My plan is to leave those as is, and add additional subs to smooth out the overall room response.

However, the midbass unit has significant output below 80 Hz. Dr. Geddes recommends allowing the mains to run to their full extension and blend in the subs. The question is; should I treat the sub portion of my mains as just an extended main speaker and blend in the additional subs with the mains as they are now? Or, should I treat the mains subs, as I do the added subs and blend them as I would the added subs? I'm leaning toward the former, but eliminating the EQ that I have added to the lower couple of octaves.

Sheldon

I would make the mains as smooth as possible for a free field situation, then leave them fixed and place them for best usage as stereo sources. Then add in the subs and manipulate only those. If there is room EQ used then this can be global in the sense that all LF sources are affected, but you can also do this only the subs only. Although there could be situations that may be more effective globally applied. In other words, the mains COULD excite a particular mode that the subs had a hard time correcting, in which case EQing only the subs would not be very effective. But this is completely situation specific.

[mat02ah]

Originally posted by gedlee
I don't know that this is true. I am not talking about a single transfer function, but an average of many.

It's true that I've been thinking about a single transfer function (fixed speakers, one listening position).

Originally posted by gedlee

I would be careful mixing the terms "system" and "transfer function" as they are not synonimous. A system, such as a room, can have many transfer functions, but is still only one system.

This of course depends on the definiton of the system. If you make the listener (and the speakers) part of the system, the system has one transfer function. If you have multiple listeners, it's certainly useful to split it up into one system (speakers + room) with multiple outputs (listeners). Point taken.

Originally posted by gedlee
"Ex falso sequitur quodlibet " - From falsehood follows anything."

It's a basic principle of logics. From:

it rains -> the street is wet

does not necessarily follow (but may follow)

it does NOT rain -> the street is NOT wet

because someone might be hosing the street ...

To summarise: Methods for multiple sub placement strictly optimize for minimum variations of the amplitude response at multiple listening positions. Transients, decay are not taken into account. Welti and Devantier (2006) are very clear about this - they never mention it (they do however mention (non-)minimum phase transfer functions). Your statements here about this are also clear. And while the assumption of minimum-phase at LFs may sound reasonable and may explain, why optimising amplitude response using multiple subs can or could also improve transient response (decay), it's just that: An unproven assumption.

It's embarrassing: Earl, is there a paper of you describing your method? Up to now my informations are "second hand" ... I didn't find anything in the AES E-library and google wasn't helpful neither. If there is one, do you happen to have an electronic pre-print?

I am glad that we managed not to turn this into a flame war. Sorry if I was a bit direct (rude...) at times. A discussion like this would be so much easier sitting at a table and having paper and pencil at hand ...

Thomas

[john k...]

Originally posted by gedlee
John

I still think that your use of the term "linear" in the context that you were using it was not consistant with "linear systems theory" where the filter is still linear even though the transfer function is not a linear function of omega. The assumption of superposition still holds and hence the transfer function is "linear" in the systems theory sense. Only confusion will result if you change this convention.

I guess I have to disagree. Superposition holds only because because the the system equation (the PDE of the system) is linear. The transfer function is just a convenient means of expressing the the solution of the system equation to any input that can be expressed as exp(st) where s is s general complex number s = a + jb. Superposition allows us to say that the solution to and input of the form I(s) = exp(s1 x t) + exp(s2 x t) + ....+ exp(sn x t) is the sum of the solutions for each additive component the input.

But let's agree to drop it because it isn't going to help here.

[john k...]

Originally posted by mat02ah

And while the assumption of minimum-phase at LFs may sound reasonable and may explain, why optimizing amplitude response using multiple subs can or could also improve transient response (decay), it's just that: An unproven assumption.

Not to wax on endlessly on this but it's not something that can be subjected to proof. However, in any specific case, like your listening environment, it is a simple matter to set up a sub, or multiple subs, measure the response and decompose it into minimum phase and an all pass components and see if the allpass is a pure time delay.

Here is such a result for a single woofer.



I have presented the data in two equivalent formats. The upper plot shows the the measured phase data in green and amplitude in fuchsia. The thin blue line overlaying the phase data is the MP computed from the measured amplitude data with the addition of a delay of about 14 msec. There is good agreement up to about 35 Hz but above that the deviation from MP plus delay indicated that the measurement is not just MP with a delay, but some non-MP response.

In the lower figure I removed a 14 msec delay from the measured data rather than add it to the MP result. The thin blue phase line is therefore just the MP computed from the measured amplitude. Again, there is agreement to only about 35 Hz. Here we see a phase wrap associated with the notch at 40 Hz where as the MP phase shows a wiggle. above 40 Hz it looks like the MP response follows the measured data for a while but remember to compare the data the 360 degree phase rap must be accounted for. This is a clear indication of non-MP response.

For what it's worth, here is a simulation of the response of the woofer system to a 1. msec pulse:



Green is the pulse, brownish red the response when the woofer is MP and blue the response of the woofer based on the measured phase. No amplitude eq has been applied in either case.

And while I'm at my desk, here is a simulation of the impulse response when the amplitude is eq'ed to the smooth response shown in red in the upper plot of the figure below. The red impulse trace is what would happen is the woofer response was MP and the blue response is what happens when the actual woofer response is eq'ed to the smooth response. As can be seen, due to the non-MP nature of the real woofer response, when the amplitude is eq'ed there is still a lot of trailing "noise in the impulse.



Nothin' say lovin' like something from the oven.

[gedlee]

Originally posted by mat02ah

To summarise: Methods for multiple sub placement strictly optimize for minimum variations of the amplitude response at multiple listening positions. Transients, decay are not taken into account. Welti and Devantier (2006) are very clear about this - they never mention it (they do however mention (non-)minimum phase transfer functions). Your statements here about this are also clear. And while the assumption of minimum-phase at LFs may sound reasonable and may explain, why optimising amplitude response using multiple subs can or could also improve transient response (decay), it's just that: An unproven assumption.

It's embarrassing: Earl, is there a paper of you describing your method? Up to now my informations are "second hand" ... I didn't find anything in the AES E-library and google wasn't helpful neither. If there is one, do you happen to have an electronic pre-print?

I am glad that we managed not to turn this into a flame war. Sorry if I was a bit direct (rude...) at times. A discussion like this would be so much easier sitting at a table and having paper and pencil at hand ...

Thomas

Thomas

No, I have never gotten arround to writting it up - thats embarassing! Life has been just too hectic for me the last few years. Markus wrote up the proceedure, but I have never published my results comparing my approach with Welti.

Discussions like this can often get out of hand and would be better in person, but alas thats not going to happen, so struggling with the difficulties of the format are what we have to learn to do.

John

Interesting results. What software do you use for the simulations?

[john k...]

Originally posted by gedlee




John

Interesting results. What software do you use for the simulations?

SoundEasy. Actually, SE will allow me to compute the MP eq to make the amplitude response match the target and then allow me to play that filter. In other words, I should be able to play and measure the impulse response shown in blue in the lower curve.