DIFFUSER - The quest for the best acoustic diffuser, pioneers and studies

The diffuser he created, which perfectly met his expectations, convinced the acousticians of his time and later the RPG company, which commercialized it in 1983.

However, my experiments showed that this method, while effective in theory, is unfortunately practically unfeasible in terms of materials because it implies creating diffusers with cells created by walls of zero thickness, which is perfectly impossible in reality.

Nevertheless, I observed that creating cells with thin walls, even if not zero, yielded remarkable results, but it required the use of costly materials like thin metal sheets when designing a broadband diffuser.

Furthermore, my analyses revealed that the distribution method based on quadratic residues was not necessarily the best from a temporal perspective, as the number of different cell depths (wells) was much lower than the total number of cells needed to compose the structure, and, most importantly, it was entirely possible to dispense with this formula to achieve results that were just as good, or even better in spatial aspects (dispersions).

This led me to understand the interest of Peter D'Antonio, John H. Konnert, and Trevor Cox in searching for and finding other sequences that particularly improve these issues. Thus, these three figures, whom I consider my mentors, have filed numerous conceptualization patents based, among other things, on primitive roots and coprime numbers (coprime numbers). These innovations, without questioning the need to start with a prime number and a sequence yielding a "discrete Fourier transform" with uniform response (i.e., uniform efficiency across the diffuser's frequency range), continued to be considered promising.

My mentors. On the left, Peter D'Antonio, born in New York in 1941. Specialized in a wide range of scientific disciplines, including spectroscopy, X-ray and electron diffraction, electron microscopy, software development, and architectural acoustics. Dr. D'Antonio is the founder of RPG Diffusor Systems, Inc. 1983-2017.

Next is John H. Konnert, who is relatively low-profile as he works for the research lab of the United States Navy and the United States Marine Corps in Washington, but he has shared numerous patents with Peter D'Antonio and Trevor Cox on behalf of RPG. He has recently been involved in the development of new AFM techniques for studying the molecular growth mechanisms of protein crystals (to give an idea of his expertise...).

Third, Trevor Cox, a lecturer at the University of Salford and a research specialist in architectural acoustics, signal processing, and perception. He has also served as the president of the Institute of Acoustics (IOA) and received the prestigious Tyndall IOA award as well as its award for promoting acoustics to the public.

And finally, Bogic Petrovic, also known as Boggy, an electrical engineering genius at the University of Belgrade, founder and owner of "MyRoom Acoustics," became a figure in independent studio acoustic research due to his particularly innovative "control room" designs demonstrating the effectiveness of hybrid diffusers combining 1D diffusion through MLS sequences and absorption based on Helmholtz resonator principles. He passed away in September 2019, and I would like to express my gratitude and appreciation for his many constructive and benevolent contributions.

Not understanding why Peter D'Antonio and Trevor Cox seemed to persist (rightly so?) in continuing on a theoretically unfeasible basis that generated industrial design problems. This required conceptualizing a diffuser based on a prime number to define the number of lines for 1D diffusers and columns for 2D ones using Quadratic Residue Diffusor (QRD) formulas. Or defining the total number of cells -1 using formulas based on primitive roots (PRD). So, I made an effort to start from scratch, finding distribution formulas free from these constraints, aiming to achieve results at least similar to the best QRD and PRD diffusers that we could create simply, thus continuing in the path taken by our late, dear Bogic Petrovic.

Furthermore, my own research and developments successfully led to two types of sequences that could break free from the previously mentioned constraints, but they still required the use of prime numbers to solve the distribution of cell heights. This further amplified my fascination with these types of numbers. So, it seems that Schroeder, D'Antonio, Cox, and Konnert were partly right to continue down this path, as was Bogic Petrovic, who did not consider it fundamental for the distribution of the "checkerboard" (cell height arrangement).

Searching for Formulas to Patent vs. Brute Force Research for the Best Diffuser

After voraciously digesting all the studies and writings of my mentors, I grew increasingly skeptical of the obstinacy of some in systematically rejecting brute force research methods and proposing only new research paths based on already patented formulas...

Would RPG, the undisputed pioneer of acoustic diffusion, adopt a strategy of protecting its patents to the detriment of developing new ideas? This may be an exaggeration, but let's not blame them. What company, once well-established and serene, does not adopt this strategy?

Know that the QRD formula (Quadratic Residue Diffuser) used in the "Classic Schroeder" diffusers is originally invented by mathematician Karl Frederick Gauss (died in 1855) and is free to all. Therefore, you'll understand why from Manfred Schroeder to today, we continue to build and sell this type of diffuser even though it may not necessarily be the best (patience, I will prove it later), although it is still very effective.

Then comes the PRD (Primitive Root Diffuser) based on primitive roots, patented by Peter D'Antonio, John H. Konnert, and Trevor Cox under the RPG company, generally equivalent to QRD when the designs are made with real pit cells and far superior when the designs are based on many pseudo-stick cells. This, of course, troubles some businesses and possibly engenders some bad faith among others.

Next, the LSD (Lüke Sequence Diffuser) and the PWRD (Power Residue Diffuser), two variants of PRD, and thus dependent on its patent. According to my simulations, the PWRD is currently the best-known sequence (but slightly inferior to my own sequence currently under patent) when multiple units of diffusers are positioned side by side (in period).

Finally, the MLS (Maximum Length Sequence) based on a pseudo-random binary sequence that clearly questions the previous sequences by freeing itself from almost all the constraints mentioned earlier. It provides quite satisfactory results, as demonstrated by the creations of Bogic Petrovic and others that I have personally made for professional studios.

However, do not misunderstand me. I do not question the proven effectiveness of these sequences, especially those based on the PRD sequence when the right value of variation for the primitive root is chosen, and the QRD sequence when the designs are based on real pit cells and a symmetrical diffuse field is desired. But I aim to empower the brute force method to identify the best diffusers we can create and attempt to establish a distribution formula based solely on the analysis of the best diffusers obtained through computer simulation. This is contrary to the usual procedure of taking an existing number theory or patented formula and making a slight modification to compare the results and judge the interest of this update.

The brute force method is generally used in cryptanalysis to find a password or key. Its principle is to test all possible combinations according to defined criteria. In the application of finding the best diffuser, it "suffices" to define the criteria for the best diffuser and compile them into a program that will test all possible height combinations to determine the sequences that best meet these criteria.

A Random Sequence for Testing the Brute Force Attack Algorithm Gave Surprising Results

For this purpose, I chose to work directly on 2D diffusers and establish an equivalent number of columns and rows to generate a square checkerboard. This was to prove that it is not necessary to work on the basis of a prime number in reality.

So, I started with a checkerboard of 22 columns and 22 rows, for a total of 484 cells or sticks. I defined the maximum height of the cells as 16 cm to match exactly the same maximum height as the other diffuser models I had created for my previous comparative studies.

Additionally, with the aim of achieving an energy decay as linear as possible over time, I first chose to divide the 16 cm by 484 to define the necessary step for obtaining the 484 different stick heights, which is approximately 0.033 cm.

Finally, to improve this height distribution by incorporating the list of the first 484 prime numbers, starting from the assumption that the 484th (which is 3301) should have a height of 16 cm. This allowed me to define the height values in centimeters of the other heights using a simple cross product:

Prime number #1 (2) = (2 x 16 cm) / Prime number #484 (3301) = 0.0097 cm
Prime number #2 (3) = (3 x 16 cm) / Prime number #484 (3301) = 0.0145 cm
Prime number #3 (5) = (5 x 16 cm) / Prime number #484 (3301) = 0.0242 cm
Prime number #4 (7) = (7 x 16 cm) / Prime number #484 (3301) = 0.0339 cm
etc...
Prime number #45 (197) = (197 x 16 cm) / Prime number #484 (3301) = 0.9549 cm
etc...
Prime number #102 (557) = (557 x 16 cm) / Prime number #484 (3301) = 2.6998 cm
etc...
Prime number #202 (1231) = (1231 x 16 cm) / Prime number #484 (3301) = 5.9667 cm
etc...

So, in order to distribute the different heights on the "checkerboard" as randomly as possible to generate diffusers for comparison by brute force, I chose to use the "Mersenne Twister" pseudo-random number generator, a fairly recent and well-reputed generator designed by Makoto Matsumoto and Takuji Nishimura in 1997.

The results of brute force analysis of BEM simulations for diffusers generated by the "Mersenne Twister" spoke for themselves. The diffusers based on this generator, with a checkerboard of 22 columns and 22 rows, achieved similar diffusion and absorption results as QRD, PRD, LSD, and PWRD of the same size, with nearly identical cell counts (give or take a column). Sometimes, they even outperformed them based on the generated sequence... by "chance." Therefore, Petrovic was right to free himself from the constraints of the QRD and PRD sequences and focus mainly on the MLS (pseudo-random binary sequence) sequences for designing the lateral diffusers in his control room.

So, I isolated the best models with real pit designs in one group and the best stick designs in another group to try to identify their similarities and understand the reasons for the superiority of these diffusers.

Without reaching the podium, QRD, PRD, LSD, and PWRD sequences were present in the group of the best diffusers with real pit cells. However, their performances in stick designs did not even allow them to enter the top 10 (out of 60) of pseudo-stick cell diffusers generated by the "Mersenne Twister," except for the PWRD, which managed to sneak in.

Why Did the QRD, PRD, and LSD Sequences Not Reach the Podium?

The main reason for the "dethroning" of the QRD, PRD, and LSD sequences is that the BEM simulation prediction algorithms I used are quite recent and were developed to take into account all the physical walls of the diffuser, unlike most of those used by diffuser manufacturers to establish efficiency curves for their acoustic products.

As a result, the QRD and PRD, based on a theory involving cells with walls of negligible thickness, inherently question the usefulness of using them for two reasons:

Either because the cells have no walls independent of each other, which is the case for stick designs, and thus produce diffractions entirely different from those predicted by the theoretical model.

Or, on the contrary, because the thickness of the walls is too great to be perceived as "negligible" by the sound wave hitting the diffuser. This is much less of a problem for the application of such sequences, but it remains very problematic for models with a large number of small cells. In practice, a model with 10 cm wide cells with 5 mm walls (i.e., 5% of the cell width) will be much less affected than a model with 3 cm wide cells with 3 mm walls (i.e., 10% of the cell width).

However, we will see that the prediction methods aimed at defining diffusion and absorption efficiency coefficients are far from reliable. Moreover, they regularly contradict each other, are continually the subject of studies, and seriously call into question the efficiency curves produced by acoustic diffuser manufacturers.

In this document, for purely didactic and resource reasons, I chose to use only 2D simulations by "Wave Tank" so that you can easily visualize the acoustic phenomena associated with passive diffusion devices (diffusers), rather than presenting you with efficiency curves obtained from averaging coefficients, which ultimately are very unrepresentative unless you master the algorithm that created them.

The wave tank method is irrefutable for visualizing and understanding wave phenomena on a single plane (2D). The summary of my work, originally based on two-plane diffuser models (2D), will therefore be transposed to one-plane diffuser models to allow me to present reliable animations and graphics.

Since this is utterly impossible, ISO 17497-1 and ISO 17497-2 (formerly AES-4id) try to define "feasible" methods to establish a reasonable number of measurements for studying diffusion and dispersion. ISO 1 aims to define dispersion coefficients that reduce the number of measurements to about 400 in a 200-cubic-meter volume using a scaled-down model of the diffuser. It is (very) far from perfect. ISO 2 (formerly AES-4id) is more convincing but still far from perfect, requiring no fewer than 40,000 measurements in an anechoic chamber of over ... 1,200 cubic meters.

To be honest, few acoustic diffuser manufacturers can afford to rent the services of such laboratories... Consequently, these devices (especially AES) are much more useful for verifying the accuracy of a predictive model of acoustic software that will predict the behavior of manufacturers' diffusers. However, the complexity of the problem is such that, even if we are capable of creating more or less reliable software to predict the behavior of an object (diffuser) at a specific frequency, what is the point of establishing averages across a frequency range?

If you get 100% diffusion for a 1,000 Hz frequency for an acoustic device and 0% diffusion for a 1,200 Hz frequency, can you say that the device has 50% diffusion from 1,000 Hz to 1,200 Hz? If you think so, then consider whether it is useful to average the amplitudes of frequencies in music to appreciate it. If you think not, are you willing to visualize and appreciate each of the 4,000 simulations of diffusion or dispersion of a diffuser with efficiency thresholds at 1,000 Hz and 5,000 Hz one by one? Personally, I can affirm that I have done this in the course of my research, and it is particularly laborious, and I have no plans to repeat it anytime soon.