Diffuser Design

Diffuser Design Prof. Trevor Cox 0.0 Aims To introduce some of the concepts behind some modern diffusers. 1.0 Learning Outcomes Students will be able to: Design one dimensional dimensional diffusers based on quadratic residue seq uences. Qualitatively describe how Schroeder diffusers scatter sound. Describe the advantages and disadvantages of such diffusers. Contrast possible definitions for optimum diffusion. Outline how optimization techniques can improve on Schroeder’s design. 2.0 Introduction In the last two decades, the increasingly widespread use of diffusing elements in studio spaces has been the most important development in the design of small critical listening environments. Diffusers also find applications in places such as concert halls1, listening liste ning rooms2, teleconferencing suites, stage shells, road side noise barriers and aircraft engine liners. (In the last two cases the absorption abilities of the surfaces are really being exploited).

The catalyst for widespread use of diffusers came from Schroeder’s designs in the 1970s 3,4 . Consequently, Consequently, we will for much mu ch of these lectures concentrate on this design. But remember this is 30 year old technology! 3.0 A Suggested Definition A diffuser is a surface which disperses sound so that it is reflected evenly in all directions whatever the angle of incidence. 4.0 Schroeder Diffusers inge nious ous new type type of diffuser In the 1970s, 1970s, Schro Sc hroeder eder 5,6 devised an ingeni generically termed Schroeder diffusers. The most common of these is the Quadratic Residue R esidue Diffuser (QRD™,7). The one dimensional form of a QRD consists of a series of wells as shown in Figure Figure 1.

O ne-dimensi dimension onal al Figure 1 A OneThe one dimensional diffusers cause scattering in one plane. In the QRD other direction, the extruded nature of the surface makes it behave like a plane surface. Because of this it is normal to just consider a cross section through the diffuser, Figure 2, which contains the plane of maximum diffusion. There are schemes for producing Schroeder diffusers which work in more than one plane 4, but we won’t consider them here. 4.1 How does a Schroeder diffuser work? Consider a mid-frequency plane wave incident onto the QRD. We get plane wave propagatio propagation n within the wells (in (i n the y-direction). If we assume the surface is rigid, the plane wave is reflected from the bottom of the well and eventually re-radiates into the space with no loss of energy. So the pressure at some point external to the diffuser Figure 2 Cross section of is an interference between the radiating waves from each well. All N=7 QRD

Diffuser design, Trevor Cox 2004-5, (c) University of Salford, www.acoustics.salford.ac.uk

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these wells have the same magnitude but a different phase because of the phase change due to the time it takes the sound wave to go down and up each well. So the polar distribution of the scattering is determined by the choice of well depths. Schroeder showed that by choosing a Quadratic Residue Sequence, the scattering from the surface equates to his definition of optimum diffusion. In Figure 3 an example of the Figure 3 Scattering from 2 periods scattering from a QRD are given calculated by Schroeder’s of N=17 QRD at design frequency theory. of 1000Hz. Well width 4.7cm. Scale 20dB/division. Schroeder defined optimum diffusion as being a surface whose Fourier Transform of the surface reflection coefficients is flat - see later. This equates to each lobe of the scattering having the same level as can be seen in Figure 3. Note, this is not even scattering into all directions. We shall return to this issue later. 4.2 Detailed Construction of a QRD 4.2.1 Low frequency limit Three limits to consider:

1. From the QRD design equations 8. The QRD behaves with maximum diffusion at it’s design frequency. This design frequency will be denoted f 0, and the corresponding design wavelength, 0. It will also created maximum diffusion at multiples of the design frequency, nf 0 where n is an integer. The design frequency is often taken to be the low frequency limit of the diffuser. 2. From diffraction theory. Alternatively, we can calculate a low frequency limit using rules of diffraction - see box. If the wells of a diffuser are very shallow compared to the wavelength, then the diffuser’s surface profile will not be seen by the well (rule 1 above). Or if the diffuser is very narrow, again it will not be seen. The low frequency cut-off of the diffuser is more often determined by the maximum well depth. /2 dmax. In fact we tend to find the low frequency cut-off is an octave or so below this. 3. From the period width (periodic diffusers only) Grating lobes are useful because they generate non-specular propagating sound (which is scattering). So unless the period width is greater than the wavelength, then we don’t have any grating lobes The first step in QRD design is to choose the design frequency. This might be limited by the maximum depth achievable in a room. 4.2.2 High frequency limit From the QRD design equations. We require plane wave propagation in the wells - see 4.1 above. We would expect plane wave propagation to start breaking down when /2 w, where w is the well width. This determines the high frequency limit. In realit y, although the design theory of the QRD is no longer obeyed when plane wave propagation breaks down, QRDs will still scatter sound pretty efficiently.

So we want a QRD with the narrowest wells possible to get the widest frequency range. What limits the narrowness of the diffusers is (i) difficulty in manufacture (ii) absorption. As the diffusers


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become more narrow, then the viscous boundary layer becomes significant compared to the well width and the absorption increases - see later. Practical well widths are at least an 2.5cm, and usually around 5cm. Useful rules of diffraction When a sound wave hits a rough surface, the effects on the sound wave will depend on the relative size of the surface roughness and the wavelength of the sound. 1. At low frequency, when the wavelength of the sound is much larger than the dimensions of the surface roughness ( /2 > d) then the wave ‘sees’ the surface as a flat plane surface and specular reflection results. 2. When the wavelength of the sound is similar to the surface roughness ( d) then the resultant scattering is complex wave interference. The simplest model is that every point on the surface act as a point source and radiates sound back into the room. The resultant pressure distribution depends on the relative phase and magnitude of all the waves received. 3. At high frequencies ( < d) the scattering can be calculated by considering the surface to be a series of smaller plane surfaces. 4.2.3 Depth sequence The well depth sequence is determined from the quadratic residue sequence. (bring in MLS). The quadratic residue sequence is a mathematical sequence based on a prime number, N. The nth term in the sequence is given by n 2 modulo N. Where modulo is sometimes written as mod and means “the remainder after dividing by”. eg: 8 mod 13 = 8; 27 mod 13 = 1; Exercise Complete the table below for the N=13 sequence. The sequence is periodic, how many terms are in one period? n n2 n 2 mod N

0 0

1 1

2 4

3 9

4 16

5 25

6 36

7 49

8 64

9 10 11 12 13 14 15 16 17 18 81 100 121 144 169 196 225 256 289 324

4.2.4 Well depths For a quadratic residue sequence, sn, the well depths are given by the following equation: (1)

Exercise A) A QRD is to be constructed based on N=7. The maximum well depth allowable is 150mm. Calculate the well depth sequence and the design frequency.

B) The quadratic residue sequences for a series of low order prime numbers are given in Table 1, you have already calculated the sequences for N=7 and N=13. A QRD is to be constructed with a maximum well depth of 150mm. Which sequence will give the best low frequency diffusion? C) Figures 4 and 5 below show how the scattering from a QRD varies as the number of periods changes and also as the prime number in the sequence changes.


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A room designer wants to put QRDs on the rear wall of a listening room. The designer can: (i) Use a single very large sequence so one period of the QRD covers the whole wall. (ii) Use many periods of a QRD with a smaller number of wells in the sequence. In light of Figures 4 and 5, which is the best option to get the best diffusion? What non-acoustic factors might affect the choice.

Figure 4 Scattering from N=17 QRD, same diffuser as Figure 2. The lines have a different number of periods and have been displaced vertically for clarity. From bottom to top: 1,2,4,8 and 16 periods.

Figure 5 Scattering from 2 periods of QRDs with different prime number generators. From bottom to top N=7, 17, 37, 89. Dimensions same as Figure 1.

Note the sequences are symmetric around the zero well and n=(N-1)/2 point, reducing manufacturing costs. There are methods for combining sequences of QRDs to minimize the lobing effects 9,10 Work has been carried out to reduce the effects of using multiple QRD periods 11,12. This is done using a QRD and its inverse. We need to put them on the wall in such a way that there are no repetitions to prevent lobing. This is best done by arranging according to a pseudo-random sequence like MLS (actually the best sequence is a Barker code 1 1 1 1 1 -1 -1 1 1 -1 1 -1). When we have a 1 in the sequence we use the N=7 QRD, when we have a -1 we use the inverse. This is known as spread spectrum after it’s use in radar technology. Other methods also exist.

29

0

1

4

9 16 25

Table Residue 1. Some Sequences Quadratic

7 20

6 23 13

5 28 24 22 22 24 28

5 13 23

6 20

7 25 16

9

4

1


5

4.3 Limitations of Schroeder diffusers We have seen above some of the limitations of Schroeder diffusers: (i) Scattering from surface does not obey the simp le Fourier Theory in all cases. (ii) Repeated periodic arrays lead to sharply defined lobes and uneven scattering. (iii) High order N value needed for even scattering - expensive to manufacture. (iv) The above scattering distributions are only true in the far field, in real applications sources and receivers are in the near field (v) Fins have to be narrow so that they don’t cause significant scattering (assumed infinitely thin in above equation) (vi) All loses are ignored - even though in reality absorption is significant at low frequencies (but this can actually be an advantage in small room applications). (vii) Do you like the look? (viii) Optimum diffusion should mean even scattering in all direction, not just the diffraction directions. (ix) Can be expensive to build. (x) Optimum diffusion only occurs at the design frequency (or a multiple of the design frequency). At frequencies in between the scattering is not optimum.

But, these diffusers have a big advantage of having design equations that can be formulated on a pocket calculator. And it should be noted that they have been successful in real applications. 4.4 So how can we improve on Schroeder Diffusers? What we do is to iteratively try different well depth sequences until to find a depth sequence with better diffusion. Just trying random depth sequences would be extremely time consuming as there are a very large number of possible combinations, so we exploit optimization processes. The process to produce optimum Schroeder-style diffusers is based on an iter ative process13: Optimization process 1. A diffuser was constructed with the well depths determined by a random set of coefficients (dn where n=1,2,3, ... N). 2. The scattered pressure from the diffuser was calculated using the Kirchhoff solution method 3. A parameter characterizing the degree of diffusion was calculated from the scattered pressure distribution 4. The surface shape was altered by changing the well depths according to standard techniques which search for a minimum in a variable, in this case diffusion parameter. Table 2. Standard deviation diffusion 5. Steps (2) to (4) were repeated until a parameters for Figures 4 and 5. Lines minimum in the diffusion parameter was numbered from bottom to top. found indicating optimum diffusion. Diffusion Parameter Detailed notes: line Figure 4 Figure 5 2. The Kirchhoff solution is a more accurate method for obtaining the scattering from 1 6.36 16.2 surfaces than the Fraunhofer method given 2 16.5 16.5 above. There are even more exact formulations for solving diffuser scattering based on 3 24.3 20.5 Boundary Element Methods (BEMs). But for large surfaces BEM models are too slow to be 4 23.8 19.6 useful as during an optimization many thousands 5 26.2 of predictions are needed.


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3. We need a cost parameter which evaluates how good a diffuser a surface is - this is the diffusion parameter, . There have been many suggestions for diffusion parameters, but for optimization techniques the most successful have been those based on the standard deviation function. A standard deviation is taken of the n scattered sound pressure levels on a polar distribution:

(2)

where L p,i is the sound pressure level for the i th measurement. If all sound pressure levels in the polar distribution are the same then the standard deviation is zero (optimum diffusion). For nonoptimum diffusion the standard deviation increases. Take for example the polar distributions shown in Figures 4 and 5. Table 2 gives the standard deviation for each of the lines. We can see that it is monitoring the quality of the scattering. 4. There are many standard methods for searching a function to find minima - so called optimization techniques14. For this work a simplex routine is used for robustness, but it isn’t the fastest. Results: Figure 6 shows the comparison between the Figure 6 Scattering from surfaces at 105 0 Hz. scattering from an N=7 QRD and an optimized N=7 QRD welled diffuser. We can clearly see that the Optimized Welled Diffuser scattering is more even from the optimized surface - we have a better diffuser. One of the advantages of using optimization techniques is that we are no longer stuck with using the Schroeder shape of wells divided by thin fins. One of the problems with this construction is it can have significant low frequency absorption15,16,17,18. This comes from two effects: 1/4 wave resonance of the wells and viscous losses as sound energy has to pass between the wells around the edge of the fins. In addition the fins are expensive to manufacture. By removing the fins we get a stepped diffuser (diagram). This can be optimized to have good diffusion, and this is also shown in Figure 6.

Optimized Stepped Diffuser

Other surfaces such as curved surfaces and fractals have been successfully optimized19,20. This are more likely to satisfy the visual requirements of architects. For QRDs with a higher prime number, the optimization can still improve on the Schroeder design, but the gains are not so large. The disadvantage of this method is that it is time consuming, a few hours to design small surfaces, a few days for entire walls. It doesn’t have the elegance of the simple design equations of Schroeder diffusers, this is design by brute force.


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Tutorial Questions

1) The rear wall of the Salford University listening room is to be covere d with a one-dimensional QRD. The QRD is to have a maximum depth of 200mm, a well width of 4cm and is to be based on the N=7 sequence. (i) Why is diffusion applied to the rear wall of the listening room? (ii) Calculate the depth sequence of the diffuser. (iii) Estimate the bandwidth of the diffuser? (iv) At what frequencies within the audible frequency range would you expect the diffuser to satisfy Schroeder’s optimum criteria? (v) The acoustic designer has suggested using a single N=89 diffuser instead of many periods of the N=7 sequence. What are the advantages and disadvantaged of such an approach? 2) A diffuser is required for a studio to have a bandwidth of 300-5kHz. The diffuser is required to cover a wall 4m wide. Design an appropriate QRD for the space. Why might it be difficult in reality to yield such a wide bandwidth? 3) (A challenging question) Ten N=5 QRDs are constructed with a well width of 5cm and a maximum depth of 272mm. The predicted scattering from the QRD is

Figure 7. Diffusion parameter for 10 periods of N=5 QRD (5x10) and for a plane surface the same size (plane)

Figure 6 Scattering from 10xN=5 QRD and plane surface at 2500 Hz

evaluated using the standard deviation as shown below in Figure 7. Note at 2500 Hz the performance of the QRD is similar to a plane surface (so not very good). Figure 8 shows the polar distribution at that frequency. Why are there frequencies at which the QRD performance is poor? What solutions can you suggest? 2) Hint Comp a (iv) 486, 972, 1 (iii) f 0 = 486 H 1) ( ii) 0,50, 200, Answers:

Diffuser design, Trevor Cox 2004-5, (c) University of Salford, www.acoustics.salford.ac.uk Bibl iograp hy (papers not-referenced)

Angus, J.A.S. “Usin g Modu lated Phase R eflection Gratings to

Schroeder, M.R., F ractals, Chaos, Power Laws: Min utes from an

Achieve Specific Diffusion Characteristics ”. Presented at th e 99th

Infinite Paradise, W.H. Freeman & Co. (19 91).

Audio Engin eering Society Convention, Preprint 4117, (October 1995).

Marshall, A.H. & Hyde, J .R., “Some Practical Cons iderations in the

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use of Quad ratic Residue D iffusing Surfaces”, P roc. 10th International Congress on Acoustics, (1980).

Angus, J., “Sou nd Diffusors using Reactive Absorption Gratings”,

D’An toni o, P. “Pe rformance Acoust ics: The Impo rtance of diffus ing surfaces and th e variable acoustics modular performance shell -

3953, (February 1995).

VAMPS”. Proc 91st Audio.Eng.Soc. Preprint 3118 (B-2). (O ctober

Reflection”. Acustica 81. 364-369 . (1995).

1991). D’Antonio, P. & Konnert, J., “The RFZ/RPG Approach to Control

Feldman, E. “A reflection grating that nullifies the specular reflection: A cone of silence”. J .Acoust.Soc.Am. 98(1) 623-634 (1995).

Room Monitoring”, Proc. A udio Engineering Society Co nvention,

ten Kate, W., “On The Bandwidth Of Diffusors Based Upon The

Preprint 2157 (I-6), (October 1984).

Quadratic R esidue Sequence”. J.Acoust.Soc.Am. 98 (5) 2575-2579 (1995).

Strube, H.W., “Diffraction by a plane, locally reacting, scattering surface”, J. Acoust. Soc. Am., 67 (2), 460-469 (February 1980).

Presen ted at the 98th Audio Engineering Society Conven tion, P reprint Schroeder, M.R. “Phase Gratings with Su ppressed Specular

Commins, D.E., Au letta, N. & Suner, B., “D iffusion and Absorption

Strube, H.W ., “Scattering of a plane wave by a Schroed er diffusor: A mode-matching approach”, J. Acou st. Soc. Am., 67 (2), 453-459

of Quadratic Residue Diffusors”, Proc. of the Institute of Acoustics,

(February 1980).

Angus, J.S.A., and Mcmanmon,C.I., "Orthogonal Sequence

Strube, H.W., “M ore on the diffraction theory of Schroeder diffusors”,

Modulated P hase Reflection Gratings for W ideband Diffusion",

J. Acoust. Soc. Am., 70 (2), 633-635 (August 1981).

Audio E ngineering Society 100th C onvention, 11-14 May 1996,

Gen-hua , D. & And o, Y., “Generalized Ana lysis of Sound Scattering by Diffusing Walls”, Acustica, 53, 296 -301 (1983).

Copenh agen, Denma rk, preprint #4249.

Cox, T.J. Optimization of Curved Surfaces for Auditoria. International Conference in Acoustics. Trondheim. (Jun e 1995).

Diffusers Using Orthogonal Modu lated Sequ ences", Aud io Engineering Society 103rd Convention, 26-29 September 1997, New

D’Antonio, P., “P erformance Evaluation of Optimized Diffusors”, J.

York, US A, preprint #4640.

Acoust. So c. Am. 97 , No. 5, Pt. 1, 2937 -2941 (May 19 95).

De Jong, B.A. and van den Berg, P.M . “Theoretical design of optimum planar so und diffusers ”. J.Acoust.Soc.Am. 95. 297-305.

D’Antonio, P., “Th e QRD Diffractal: A New O ne- and TwoDimensional Fractal Sound Diffusor”, J. Audio Eng. Soc. 40, No. 3, 117-129 (March 1992). Angus, J., “Large Area Diffusors using M odulated Phase Reflection

10, Part 2, 223-232 (1988)

Angus, J.A.S and Simpson, A.,"Wideband Two Dimension al

(1994). Angus, J.A.S., “Alternative d iffusor sequences”, Proceedings of the Institute of Acoustics 14 , Part 5, 193-202 (O ctober 1992).

Gratings”, Presen ted at the 98 th Aud io Engineering Society Convention, Preprint 395 4 (D4), (February 1995).

References 1.Marsh all, A.H., “A spects of the acoustical design and pro perties of Christchu rch Town H all”, J. Sound Vib.,62 , 181-194 (19 79). 2. D’Antonio, P an d Cox, T.J. Two Decades of Room D iffusors. Part 1: Applications and Design. J.Audio.Eng.Soc. (In press). 3. Schroeder,

M.R., “Diffuse Sound Reflection by Maximum Length Sequences”, J. Acoust. Soc. Am. 57 , No. 1, 149-150 (January 1975). 4. Schroeder, M.R., “Binaural Dissimilarity and Optimum Ceilings for Concert Halls: More Lateral Sound Diffusion”, J. Acoust. Soc. Am. 65 , 958-963 (1979). 5. Schroeder, M.R., “Diffuse Sound Reflection by Maximum Length Sequences”, J. Acoust. Soc. Am. 57 , No. 1, 149-150 (January 1975). 6. Schroeder, M.R., “Binaural Dissimilarity and Optimum Ceilings for Concert Halls: More Lateral Sound Diffusion”, J. Acoust. Soc. Am. 65 , 958-963 (1979). 7. QRD is a registered tradem ark o f RPG Diffusor Systems Inc., U SA. 8. D’An tonio, P. & Konn ert, J.H., "The Reflection P hase Gra ting Di ffusor: Design The ory and Application", J . Audio Eng. S oc. 32, No. 4 , 228-238 (April 1984). 9.D’Anto nio, P., "The QR D Diffractal: A N ew One - and T wo-Dimensional Fractal Soun d Diffusor", J. Audio E ng. So c. 40, No. 3, 117-129 (March 199 2). 10. An gus, J., "Large Area D iffusors using Modulated P hase Reflection Gratings", Presented at th e 98th A udio Engineering Society Convention, Preprint 395 4 (D4), (February 1995). 11.D’An tonio, P., "The QR D Diffractal: A N ew One - and Two-Dimensional F ractal Sou nd Diffusor", J . Audio Eng. Soc. 40, No. 3, 117-129 (March 199 2). 12. An gus, J., "Large Area D iffusors using Modulated P hase Reflection Gratings", Presented at th e 98th A udio Engineering Society Convention, Preprint 395 4 (D4), (February 1995). 13.T J Cox. Optimization of profiled diffusers. J.Acoust.Soc.Am. 97(5) 29 28-2941 (1 995) 14.W. H. Press et al. "Numerical Recipes, the Art of Scientific Computing" Cambridge University Press. 289-292. (198 9). 15.Fujiwara, K. & M iyajima, T., “Absorption Characteristics of a Practically Constructed Schroede r Diffu sor of Qradratic-Residue Type”, Applied Acoustics 35 , 149-15 2 (1992 ). 16.Fujiware, K. “A study of the sound absorption of a quaratic-residue type diffusor”. Acustica. 81. 370-378. (19 95). 17.Kuttruff, H., “Sound Ab sorption by Pseudostochastic Diffusors (Schroeder Diffusors)”, Applied Acoustics 42 , 215-231 (1044). 18.Mechel, F.P. “The Wide-angle diffusor - a wide-angle absorber?”. Acustica 81. 379 -401 (1995). 19.Cox, T.J. D esigning curved diffusors for performance spaces. J .Audio.E ng.Soc. 44(5) 354-364 (1996). 20.Cox, T.J and D’A ntonio, P. Fractal Sound Diffusors",103rd Convention of the Audio E ngineering Society, Preprint 4578, P aper K-7, New York (September1997).

Diffuser Design

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