Cardioid Sub Simulations

[NTK]

There seems to be a bit of interest in cardioid subs. Dr Toole thinks monopole sub should be the norm for home use. There are others who obviously disagree. I am also interested in figuring out what differences cardioid subs make.

Since I like playing with modeling/simulations, I thought running some simulations should help. To start simple, I ran 2-D FEM simulations using Mathematica (reference: tech notes). I started with the simplest of 2-D cases to see if the models make sense. In 2-D simulations, where the world is assumed to a flat plane (hello flat earthers), properties in the ignored dimension (height) are constant. Therefore, the sound sources simulated behave like line sources that extend from the floor to the ceiling, with the floor and ceiling being hard rigid boundaries and are perfect reflectors. 2-D analyses won't match real life but these simpler simulations are much faster to setup and run than 3-D ones. They are usually adequate for giving quick rough answers to which directions trends would take.

To create the cardioid radiation pattern, I modeled a dual driver sub which has a front and a rear driver. The rear driver signal is delayed, attenuated, and polarity reversed relative to the front one. The subs simulated are 0.36 m wide and 0.41 m deep (borrowing the dimensions from the Sigberg 10D). To keep things simple for me, both delay and attenuation were kept constant and not changed with frequency. The delay is 1.79 ms and attenuation is 0.75. Below are the free field sound pressure plot and polar plot from 40 Hz to 120 Hz. For the polar plot the sound pressures are normalized such that the pressure in the up (90°) direction is always 1.

Here are the results for the monopole case.

Below are similar plots with the simulated sub operating in "bipole" mode (i.e. dual opposed configuration) and in "monopole" mode (only the front driver is active). Note that in this simulation, the bipole sub is oriented with the drivers facing up (90°) and down (270°), not left (0°) and right (180°). Perhaps counter-intuitively, we get higher sound pressures in the line/plane bisecting the drivers (and perpendicular to the driver axes). Since on this bisection line we have equal distance to either driver, the sound pressures from each drivers always sum constructively and we get maximal pressure. If we are on the driver axes, the distance to the front (closer) driver will be less than the distance to the rear one, the pressure sum will not be totally constructive. The amount lost due to the imperfect coherence between the front and rear driver generated sounds will depend on frequency and the effective distance between the drivers.

[Edit] Corrected the driver orientations for the bipole case.

 

[NTK]

The free field simulations don't give us a lot more than what we already know. They were mostly for checking my simulations. Since the results seemed reasonable, let's go one step further and simulate the subs in half-space (the subs sit in front of an infinitely large and partially reflective wall radiating to free space).

In these half-space simulations an orientation angle (∠) of 0° means the sub is pointing directly away from the wall. Below are the results for the cardioid woofer at 0°, 45°, and 90° orientation. For the polar plot the sound pressures are normalized such that the pressure in the direction the sub is pointing at is always 1.

It is getting late and I'll show the half-space results without comments.

 

[NTK]

[Edit] Updated this post with actual contents.

Here are some results for in-room 2-D simulations. The animated GIFs are "downsized" and frequency steps are 2 Hz due to file size limitation.

The 2-D room simulated is 5.5 m wide and 7.0 m deep. The center of the sub is located at 3/4 of the width and 0.35 m from the front wall. Since a cardioid sub is a "directional source", I simulated 4 orientations: 0° means pointing straight into the room, and 22.5° is rotated 22.5° counter-clockwise, and etc. Note that the simulated sub has a larger depth than width, thus the drivers are on the narrower faces.

I am not going to say too much about my interpretation of the results from these 3 sub configuration. There are differences to be seen, but I don't know how significant they would be in real life. If I squint enough, it looks to me the cardioid configs are more uniform.

The bipole and monopole simulations are does with the sub either pointing straight into the room or rotated 90° counter-clockwise.

These are the room mode plots for the room.

 

[NTK]

Upload my 2-D in-room results. These 2-D simulations are computed in the frequency domain, and they give the response of the entire simulation domain (i.e. the "room") one frequency at a time. Not too convenient as we are more interested in the response of a few positions for a range of frequencies with a high frequency resolution.

What prompted me to start this project was this post. I haven't gotten to the pyroomacoustics part yet. For simulations, I always prefer to be able to do it in more than one way, and if the results agree, then I can have more confident. Now that I have a baseline, I am going to start working on pyroomacoustcs in the days to come, hopefully not too many weeks. Pyroomacoustic simulates in the time domain using the image source method, so it should be more efficient in getting high resolution frequency response at a few listening positions, and the simulations will be in 3-D (but still rectangular rooms) too.

 

[Keith_W]

Thanks for doing these NTK. I have argued for a while that cardioid subs in rooms are pointless, because all they do is excite a different pattern of room modes. However, the counter-argument is that cardioid subs excite fewer lateral modes, and there is less sound leakage from the front and side walls. I am not sure about this. Looking at your simulations, I do not think that the counter-argument is supported. What do you think?

 

[NTK]

Thanks for doing these NTK. I have argued for a while that cardioid subs in rooms are pointless, because all they do is excite a different pattern of room modes. However, the counter-argument is that cardioid subs excite fewer lateral modes, and there is less sound leakage from the front and side walls. I am not sure about this. Looking at your simulations, I do not think that the counter-argument is supported. What do you think?

You are very welcome

My short answer is I am not yet convinced of the cardioid sub benefits, especially when considering the cost. As I mentioned in the open post, Dr. Toole believes omni subs should be the norm. But since Dr. Toole is not a believer of stereo bass, and theoretically cardioid subs may be less problematic with regard to exciting room modes, they may offer some benefits there as multi-sub setups do not work well for stereo bass.

The purpose of the half-space simulations in post #2 was to look at the SBIR only and with no room modal response. By having only the front wall and radiating into open space, there is no room mode effect. Looking back, I should not have stopped at 120 Hz and continue to go higher. The usual co-located cardioid setups with main speakers sitting on top of the cardioid subs work to cross-over frequencies >200 Hz. I will rerun them and see.

Regarding cardioid subs placement and orientation, for these co-located units, the common setups (I think) would be sitting right against the front wall, and pointing toward the prime listening position to match the orientation of the main speaker. That, I guess, would not be the "optimal" orientation for coupling to the room modes. The idea of cardioid is that it couples to both the pressure and velocity modes. When right against a wall, I think that means the cardioid subs should be oriented parallel to the wall instead of perpendicular (because at those locations the velocities in the direction perpendicular to the wall will all be close to zero). However, that means the subs will not match the directivity of the mains, which are pointing toward the listener or straight ahead. I'll run a few more sims by moving the cardioid sub along the front wall and see what effects they have.

I came across this PhD thesis, and I only skimmed over it very quickly. It says different directivity is audibly different, but didn't say definitively which kind sounds the best. In the conclusion it says low frequency (<200 Hz) directivity control is not practical for domestic rooms, and therefore we should only consider it for the higher frequencies. However, the author might have been thinking that these super expensive AsciLab/Kii/Genelec cardioid bass extension units are anything but "practical" for the average consumer.

https://openresearch.surrey.ac.uk/e...f-Loudspeaker-Directivity-upon/99511687202346

 

[Arindal]

There are differences to be seen, but I don't know how significant they would be in real life.

Awesome work, thanks a lot! I have been doing real-life experiments some 15 years ago when the first true cardioid subwoofers appeared on the commercial market, done in a slightly smaller room.

From memory, I can pretty much confirm your simulation results, although I do not have the measurements anymore. What was apparent to me back then, that the zones of peaks and nulls were much smaller with cardioid subs used, and the subjective decay was much faster.

I have argued for a while that cardioid subs in rooms are pointless, because all they do is excite a different pattern of room modes.

If the only goal is to achieve a flat frequency response at a maximum number of positions, I agree. A cardioid sub without properly adjusted positioning and DSP alone would not achieve this.

In practice I found a lot of advantages. The fact that there were more, but smaller zones of peaks and nulls, is exactly the point of cardioid subs. The modes, in my understanding, do not get to be excited to the same degree as with a monopole, so they decay faster and the peaks and nulls are limited and more chaotically distributed.

That was particularly of practical importance, as the result could easily be countered with more narrow-banded DSP bell/notch filters, interestingly without causing an ugly reversed effect at positions were the mode in question was having a minimum instead of a maximum at original mic position. And without causing this typical ´boomy´ remnants of an omnidirectional sub positioned in a maximum pressure zone of a room mode.

With which cardioid subs have you been conducting field experiments?

the author might have been thinking that these super expensive AsciLab/Kii/Genelec cardioid bass extension units are anything but "practical" for the average consumer.

My guess would be these product did not exist at the time the material for this PhD thesis was compiled. The first commercial product that caught my attention, was MEG Basis14k, and this one was practically available from around 2011, but not really well known back in the days.

When right against a wall, I think that means the cardioid subs should be oriented parallel to the wall instead of perpendicular (because at those locations the velocities in the direction perpendicular to the wall will all be close to zero). However, that means the subs will not match the directivity of the mains, which are pointing toward the listener or straight ahead.

From practical experience, I can confirm your assumption on parallel orientation of cardioid subs. Matching the subs´ orientation with those of the main speakers, is not really an important issue, as I doubt we can really ´hear directivity´ at 100Hz or 120Hz, and even under -30deg the response of the cardioid sub is pretty much the same as on axis. Our ears seem to be much more sensitive to the indirect outcome of the room modes being excited more or less than to the direcitivty itself. I assume this is the reason why steps in directivity between omnidirectional and cardioid patterns are audible in some more compact speakers, like Kii.

 

[Juhazi]

At home a subwoofer is usually sitting on the foor and next to wall, perhaps in the corner. Practically it's freefield radiation pattern is irrelevant in it's operating band (very long wavelength) Positioning and adding more subs have much more effect.

Downside of cardiod is loosing 6dB/oct of spl sensitivity which, if compensated with EQ means more distortion, limited spl capacity and compression of dynamics. In case of typical subwoofers operating 2 octaves 20-80Hz this is a catastrophic scenario... https://www.audiosciencereview.com/...adiation-via-lateral-slots-like-d-d-8c.37863/

Cardioid subs do really give benefit in open air concert venues, the musicians won't suffer from extreme spl

Cardioid pattern above 100Hz on the other hand can be considered beneficial in many ways - it really helps with front wall SBIR (first reflection interference), which is a very common problem in small room speaker placement. This is the actual reason why speakers like D&D 8c were developed. The D&D use rearside subwoofers which in near-wall placement get boost from the wall!

 

[Curvature]

Very nice work.

Using this sim you could reproduce the Welti study, other than, perhaps, his simulation of a room with 100 subs. Could be interesting to investigate multisub placement with cardioids and bipoles.

 

[NTK]

Very nice work.

Using this sim you could reproduce the Welti study, other than, perhaps, his simulation of a room with 100 subs. Could be interesting to investigate multisub placement with cardioids and bipoles.

I am thinking all these will be easier with pyroomacoustics. Below is a screen clip from one of the example Juypter notebooks to simulate a room impulse response. The source and mic are easily specified, and the room impulse response (RIR) can be readily generated. Multiple sources can be simulated one at a time and the sum of the response will be the sum of the RIR's. Cardioid radiators can be simulated by applying a delay and adjusting the response magnitude of the back driver relative to the front.

pyroomacoustics/notebooks/rir_demo.ipynb at master · LCAV/pyroomacoustics

Pyroomacoustics is a package for audio signal processing for indoor applications. It was developed as a fast prototyping platform for beamforming algorithms in indoor scenarios. - LCAV/pyroomacoustics

github.com

 

[NTK]

The purpose of the half-space simulations in post #2 was to look at the SBIR only and with no room modal response. By having only the front wall and radiating into open space, there is no room mode effect. Looking back, I should not have stopped at 120 Hz and continue to go higher. The usual co-located cardioid setups with main speakers sitting on top of the cardioid subs work to cross-over frequencies >200 Hz. I will rerun them and see.

Ha! I thought just rerunning my cardioid model to higher frequencies would be simple. Not so. I found out that using fixed delay and fixed attenuation for the rear driver of the cardioid did not work well at all at higher frequencies. As frequency increases, the sound radiation by each driver progressively becomes directional, and the initial assumption that the each driver radiates omnidirectionally starts to break down. So I spent some time "optimizing" the delay and attenuation with frequency, but nothing like the sophistication of P.L.'s method (DIY Active Cardioid Implementation Using Global Optimization Algorithm).

Here are the free field simulation results after optimization from 40 Hz to 300 Hz. My algorithm basically turned off the rear driver at frequencies > ~225 Hz as the directional response of the front driver is sufficiently "directional".

Here are the half space simulation results of the three configurations to specifically look at SBIR. It is unlikely one would want to integrate a non-cardioid sub frequencies significantly above ~120 Hz. However, this can be representative of placing a full range speaker against the front wall. The monopole should be representative of a front firing speaker, and the bipole should be representative of a speaker with symmetrically side mounted woofers. The cardioid does seem to have better control of SBIR.

Note: The polar plots (on the right) are normalized. The on-axis (90°) value is alway 1.

 

[Arindal]

Here are the free field simulation results after optimization from 40 Hz to 300 Hz.

Very interesting! Looking at the polar graphs I assume that already from approx 75Hz upwards, you created a pattern rather resembling what in recording terms is called a super-cardioid, instead of a cardioid. Does the directivity index of that one increase, as expected from a microphone with similar pattern?

 

[NTK]

Very interesting! Looking at the polar graphs I assume that already from approx 75Hz upwards, you created a pattern rather resembling what in recording terms is called a super-cardioid, instead of a cardioid. Does the directivity index of that one increase, as expected from a microphone with similar pattern?

I did not calculate the directivity indices. I also do not know if the theoretical cardioid directivity will result in the best sound. My guess is that we want to minimize the sound energy radiated to the back while not having a too narrow dispersion towards the front. However, the optimization algorithm I used was to get close to the theoretical cardioid by adjusting the rear driver delay and attenuation (by minimizing the total sum of the weighted squared errors between the simulations and theoretical cardioid, with the error of the front angles receiving heavier weights). Here I included the polar plots of the optimized cardioid and monopole with the theoretical cardioid shown as reference, to show the amounts of deviations.

 

[Curvature]

Ha! I thought just rerunning my cardioid model to higher frequencies would be simple. Not so. I found out that using fixed delay and fixed attenuation for the rear driver of the cardioid did not work well at all at higher frequencies. As frequency increases, the sound radiation by each driver progressively becomes directional, and the initial assumption that the each driver radiates omnidirectionally starts to break down. So I spent some time "optimizing" the delay and attenuation with frequency, but nothing like the sophistication of P.L.'s method (DIY Active Cardioid Implementation Using Global Optimization Algorithm).

Here are the free field simulation results after optimization from 40 Hz to 300 Hz. My algorithm basically turned off the rear driver at frequencies > ~225 Hz as the directional response of the front driver is sufficiently "directional".
View attachment 505359

Here are the half space simulation results of the three configurations to specifically look at SBIR. It is unlikely one would want to integrate a non-cardioid sub frequencies significantly above ~120 Hz. However, this can be representative of placing a full range speaker against the front wall. The monopole should be representative of a front firing speaker, and the bipole should be representative of a speaker with symmetrically side mounted woofers. The cardioid does seem to have better control of SBIR.

Note: The polar plots (on the right) are normalized. The on-axis (90°) value is alway 1.

View attachment 505360

View attachment 505362
View attachment 505361

These remind me of the halfspace simulations and measurements used to assess acoustic panel radiation patterns.

Have you ever tried to decompose the polar patterns into spherical harmonics? Or their like in 2D. I don't know enough of the theory to follow this idea to the end, but intuitively it seems to be a good way to analyze and describe room acoustic phenomena.

 

[NTK]

These remind me of the halfspace simulations and measurements used to assess acoustic panel radiation patterns.

Have you ever tried to decompose the polar patterns into spherical harmonics? Or their like in 2D. I don't know enough of the theory to follow this idea to the end, but intuitively it seems to be a good way to analyze and describe room acoustic phenomena.

No, I haven't tried. But if I were to do it, I'll probably first convert my models to axisymmetric cylindrical 2-D (i.e. modeling the subwoofer as a cylinder), which should be a better approximation of the true 3-D case than the 2-D planar models. (I did my models in 2-D planar because my next steps were modeling 2-D rooms, and 2-D cylindrical models will not work there.)

I'd like to get started on the pyroomacoustics first though.

 

[Arindal]

I also do not know if the theoretical cardioid directivity will result in the best sound.

My guess would be, this depends on the room geometry and direction of room modes. Idealized super-cardioids usually have their nulls at +-135deg which in theory does not look like the best measure to not excite one-dimensional modes. If the 180deg attenuation is sufficiently high, though, I would expect the differences to be negligible.

the optimization algorithm I used was to get close to the theoretical cardioid by adjusting the rear driver delay and attenuation

So delay and attenuation are variables of the frequency, or fixed? Did you try an allpass phaseshift instead of a delay?

In the simulation you have added, the super-cardioid pattern seems to be established only at higher frequencies, so not relevant to room mode issues.

 

[NTK]

So delay and attenuation are variables of the frequency, or fixed? Did you try an allpass phaseshift instead of a delay?

The delay and attenuation for this particular configuration are as follows. If these are implemented in real life, they will probably best done using FIR filters.