[Theory] Best impulse response in Bass Reflex enclosures

[Jose Hidalgo]

Hi everybody,

This topic aims to answer a simple theoretical question.

A "friend" of mine (well, not really a friend in fact - a quite knowledgeable person when it comes to audio theory, but a person that I appreciate much less as a human being - God, I hope he comes across this topic and reads this ), has written this :

IMHO, the optimal BR tuning is the one that gives the best transient response, even if this means a F-3dB a bit higher.
From this point of view, the optimal tuning is generally close to Fb = Fs with n = 4.
In fact, the best transient response is obtained when Fb/Fs is between 0,9 and 1, with a consistent Vb.

Unfortunately, the person who has written this hasn't even bothered about trying to justify it (he probably thought whe should just take His Word for it and praise Him for that ). While I strongly disagree with this kind of attitude, I think that he still may be right (like I said, the guy has some audio knowledge when it comes to theory).

So here are my questions :
1/ Can someone explain in cartesian terms why Fb/Fs between 0,9 and 1 gives the best transient response ?
2/ What happens if we lower Fb (Fb/Fs = 0,8 for example) ?
3/ Is digital active crossover (e.g. Izotope Ozone) really capable of actually correcting the transient response ? How, and what's the price to pay ? (I have some trouble understanding this point - sorry).

Thanks in advance.

[Dag]

A BR is a closed box with a resonance to increase the bass output. But this resonance will also delay the bass signal around the resonance. Therefore you will get the best impulse responce if you make the port resonance as low as possible (but the you are loosing the wanted increase of the bass level). Actually the best impulse responce is if you tune the port to zero herts...a closed box.
So, it is always a tradeoff between bass output and impulse responce (and maybe other things like distortion).

Fb/Fs between 0,9 and 1 might be a good compromise but you should really do a simulation for your application. Or better do some real life tests!!

DSP or a digital crossover can compensate for the delay but the delay can change at different bass levels making the compensation incorrect at high levels.

[Jose Hidalgo]

Hi Dylan, thanks.

BTW, the bass level is not an issue here, since we're talking about qualitative criteria (impulse response), not about SPL.

Of course I agree with you. I have made lots of simulations, and they all seem to prove the benefits of a very low tuning, in order to approach the behaviour of a closed box (impulse response, low group delay) while keeping the benefits of the bass reflex (keeping the excursion below Xmax and lower it around Fb).

But all this would mean that "the lower the Fb, the better the impulse response", plain and simple. And this is not exactly what my "friend" said. Thus my two questions : in your opinion, why only between 0,9 and 1, and what can happen if we go below 0,9 ?

BTW, let's remain on a theoretical basis, because the aim of this topic is only to understand and to try to demonstrate, not to experiment.

Thanks again.

[SY]

Impulse response and the shape of the cutoff region (analogous to Q, which properly speaking is only a parameter used in second order) are related 1:1. So, if you start off by saying , "A Q of 0.9 is the best," then the impulse response has already been defined. Can't change one without the other.

Now what Q is "best?" Clearly, that depends on what you're trying to do in the design. There is no one correct answer, with all due respect to your friend. Likewise, the ratio of the box tuning frequency to the driver resonance defines the Q.

[oshifis]

A bass-reflex box is equivalent to a 4-th order high-pass filter. The flattest group delay tuning gives the best transient response. That type of filter is known as Bessel filter.

[bjorno]

…A bass-reflex box is equivalent to a 4-th order high-pass filter…

No many are third order or intermediate order too.

…The flattest group delay tuning gives the best transient response. That type of filter is known as Bessel filter…

Quite right. Adding one more remark: The Bessel alignment is only valid for discrete values of Qts in a close range i.e. it exists only for one particular value of Qts at a time with all other parameters locked to specific values.

If Ql is chosen to the common value of 7 and fb/fs is set to = 0.97(35) and a box is made with a volume of Vas /1.91 (1.9076) then a driver with Qts= 0.33(12) is necessary, f-3dB will be about =1.49(42) x fs.

Obviously If Qts slightly off i.e. preferably lower like 0.32 a new Ql value must be re-calculated and so on.

For nearly 35 years ago I used drivers like Kef B110 (SP1057) Qts=0.33 and by measuring the Ql in the box with real + virtual volume magnification from the stuffing giving a equivalent volume about 12.37 L it was possible to come very close to a true acoustic Bessel transfer function.

Many designers overstuffed the boxes at that time so a common recommendation was to design for a real volume of at least 9 L but not over 10 L before stuffing was added.

My opinion at that time was, that the B110 in a Bessel box used below 1 kHz was superior to other BR boxes using similar driver sizes, as they mostly were cutting off higher.

The B110 reached below (60 Hz) avoiding the very critical area about 70 Hz where at and above many BR drivers had cut-off not reaching enough fidelity with respect to male voices.

…So here are my questions :
1/ Can someone explain in cartesian terms why Fb/Fs between 0,9 and 1 gives the best transient response ?…

Fb/fs> about 0.97 and higher gives the best transfer function, TF.

.2/ What happens if we lower Fb (Fb/Fs = 0,8 for example) ?…

Peaky response.

….3/ Is digital active crossover (e.g. Izotope Ozone) really capable of actually correcting the transient response ?…

Yes but also as well use analogue design like the LT or any positive feedback system but only if a closed box is used. Has Izotope Ozone a built in digital active crossover too?

…How,…

Bad transient response is usually a peaky response having a certain TF and when multiplied with a correcting TF the new TF can be chosen arbitrarily. The resulting TF is the only that matters.

…and what's the price to pay ?…

You have to pay for the need of added electronics, added complexity when active design is used and not to forget all the power losses involved.

b

[Jose Hidalgo]

Thanks everyone for your answers. I think I see more clearly now.

Some comments on bjorno's answer :

Originally posted by bjorno
Fb/fs> about 0.97 and higher gives the best transfer function, TF.
[...]
Peaky response.
I understand. Well, in my case I guess that shouldn't be a problem, because I am using digital active crossover (my source being of course a dedicated audio PC). So if I understand well, the transfer function & peaky response thing will only be a real problem for people who don't use digital active xover. Am I right ?

Has Izotope Ozone a built in digital active crossover too?
Of course. See for yourself, you might even like it : http://www.izotope.com/products/audio/ozone

You have to pay for the need of added electronics, added complexity when active design is used and not to forget all the power losses involved.
Of course : active design means more complexity (but also more control), and flattening the response (Ozone has a very good "matching" function for that) involves power losses. But hey, that's a reasonable price to pay IMHO given the benefits. Thanks again.

[oshifis]

Originally posted by bjorno
... The Bessel alignment is only valid for discrete values of Qts in a close range i.e. it exists only for one particular value of Qts at a time with all other parameters locked to specific values.

If Ql is chosen to the common value of 7 and fb/fs is set to = 0.97(35) and a box is made with a volume of Vas /1.91 (1.9076) then a driver with Qts= 0.33(12) is necessary, f-3dB will be about =1.49(42) x fs.

Obviously If Qts slightly off i.e. preferably lower like 0.32 a new Ql value must be re-calculated and so on...

I checked the Bessel (BL4) alignment in the R. H. Small article in JAES 1973 October issue (Vented-Box Loudspeaker Systems - Part IV.: Appendices). Not taking Ql into account, the ideal Qts is 0.316, Vb = Vas / 2.33, f3 = 1.5 * fs. Quite close to your figures with a more realistic Ql.

[bjorno]

…Quite close to your figures with a more realistic Ql…

I always tune my closed mid bass boxes enclosures to Bessel Qtc=1/sqrt (3) but for the lowest octaves more towards the critical damping of 0.5.

Crossing over to a sub is then only a matter of FR smoothness and in the reality it’s difficult to get good flatness from a Bessel tuned mid-bass crossed over to a sub, move the speakers and any fine-tuned FR will be lost.

My opinion and more important is that the group delay should stay below about 0.75/f for mid-bass but can be relaxed below about 40 Hz where it’s better trying to achieve an even FR in the room with a minimum power input to the sub driver which often is the main distortion source in the chain.

b

[Ron E]

The question is poorly formed and cannot be answered. Transient response depends on more than just Fb/Fs, it also depends on alpha = Vas/Vb and damping parameters of leakage, absorption and port losses...

There is one JAES article that claims the SC4 filter has the best impulse response - and SC4 has Fb/Fs greater than 1 in all cases. I would be inclined to believe that article before your nameless "technical friends".

BTW, maximally flat delay is just that, it is not "optimized for transient response"; although it is better than many of the other alignments, it is not the best.

Bass enclosure design is a set of tradeoffs. If you were to optimize for transient response, you might find that the actual optimum requires a very large box with a very low tuning - and then what you basically have is a leaky infinite baffle.

To get the practical optimum, you need to give more constraints. An example: Say you wanted to design a dissolving pill that is disc shaped and dissolves quickest. The "optimum" would be a disc one molecule thick, but that wouldn't fit in the cup of water - so you need to add constraints like maximum dimensions, then consider packaging cost, manufacturing costs, etc.... and you end up with a pill that dissolves fast and is inexpensive to make....