Figure 1 A One-dimensional QRD Figure 2 Cross section of N=7 QRD 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: C Design one dimensional diffusers based on quadratic residue sequences. C Qualitatively describe how Schroeder diffusers scatter sound. C Describe the advant ages and disadvantages of such di ffusers. C Contrast possible definitions for optimum diffusion. C 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 halls 1 , listening rooms 2 , 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, we will for much 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 In the 1970s, Schroeder 5,6 devised an ingenious new type of diffuser generically termed Schroeder diffusers. The most common of these is the Quadratic Residue Diffuser (QRD™ ,7 ). The one dimensional form of a QRD consists of a series of wells as shown in Figure 1. The one dimensional diffusers cause scattering in one plane. In the 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 propagation within the wells (in 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 is an interference between the radiating waves from each well. All
Transcript
Figure 1 A One-dimensionalQRD
Figure 2 Cross section ofN=7 QRD
Diffuser DesignProf. Trevor Cox
0.0 AimsTo introduce some of the concepts behind some modern diffusers.
1.0 Learning OutcomesStudents will be able to:C Design one dimensional diffusers based on quadratic residue sequences.C Qualitatively describe how Schroeder diffusers scatter sound.C Describe the advantages and disadvantages of such diffusers.C Contrast possible definitions for optimum diffusion.C Outline how optimization techniques can improve on Schroeder’s design.
2.0 IntroductionIn the last two decades, the increasingly widespread use of diffusing elements in studio spaces hasbeen the most important development in the design of small critical listening environments.Diffusers also find applications in places such as concert halls1, listening rooms2, teleconferencingsuites, stage shells, road side noise barriers and aircraft engine liners. (In the last two cases theabsorption abilities of the surfaces are really being exploited).
The catalyst for widespread use of diffusers came from Schroeder’sdesigns in the 1970s3,4 . Consequently, we will for much of theselectures concentrate on this design. But remember this is 30 yearold technology!
3.0 A Suggested DefinitionA diffuser is a surface which disperses sound so that it is reflectedevenly in all directions whatever the angle of incidence.
4.0 Schroeder DiffusersIn the 1970s, Schroeder5,6 devised an ingenious new type of diffusergenerically termed Schroeder diffusers. The most common of theseis the Quadratic Residue Diffuser (QRD™ ,7). The one dimensionalform of a QRD consists of a series of wells as shown in Figure 1.
The one dimensional diffusers cause scattering in one plane. In theother direction, the extruded nature of the surface makes it behavelike a plane surface. Because of this it is normal to just consider a cross section through thediffuser, Figure 2, which contains the plane of maximum diffusion.
There are schemes for producing Schroeder diffusers which work in more than one plane4, but wewon’t consider them here.
4.1 How does a Schroeder diffuser work?Consider a mid-frequency plane wave incident onto the QRD. We getplane wave propagation within the wells (in the y-direction). If weassume the surface is rigid, the plane wave is reflected from thebottom of the well and eventually re-radiates into the space with noloss of energy. So the pressure at some point external to the diffuseris an interference between the radiating waves from each well. All
Diffuser design, Trevor Cox 2004-5, (c) University of Salford, www.acoustics.salford.ac.uk 2
Figure 3 Scattering from 2 periodsof N=17 QRD at design frequencyof 1000Hz. Well width 4.7cm.Scale 20dB/division.
these wells have the same magnitude but a different phasebecause of the phase change due to the time it takes the soundwave to go down and up each well. So the polar distribution ofthe scattering is determined by the choice of well depths.Schroeder showed that by choosing a Quadratic ResidueSequence, the scattering from the surface equates to hisdefinition of optimum diffusion. In Figure 3 an example of thescattering from a QRD are given calculated by Schroeder’stheory.
Schroeder defined optimum diffusion as being a surfacewhose Fourier Transform of the surface reflection coefficientsis flat - see later. This equates to each lobe of the scattering having the same level as can be seen inFigure 3. Note, this is not even scattering into all directions. We shall return to this issue later.
4.2 Detailed Construction of a QRD4.2.1 Low frequency limitThree limits to consider:
1. From the QRD design equations8.The QRD behaves with maximum diffusion at it’s design frequency. This design frequency will bedenoted f0, and the corresponding design wavelength, 80. It will also created maximum diffusion atmultiples of the design frequency, nf0 where n is an integer. The design frequency is often taken tobe 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 thewells of a diffuser are very shallow compared to the wavelength, then the diffuser’s surface profilewill not be seen by the well (rule 1 above). Or if the diffuser is very narrow, again it will not beseen. The low frequency cut-off of the diffuser is more often determined by the maximum welldepth. 8/2 . 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 isscattering). So unless the period width is greater than the wavelength, then we don’t have anygrating lobes
The first step in QRD design is to choose the design frequency. This might be limited by themaximum depth achievable in a room.
4.2.2 High frequency limitLFrom the QRD design equations.We require plane wave propagation in the wells - see 4.1 above. We would expect plane wavepropagation to start breaking down when 8/2 . w, where w is the well width. This determines thehigh frequency limit. In reality, although the design theory of the QRD is no longer obeyed whenplane 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 limitsthe 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 wellwidth and the absorption increases - see later. Practical well widths are at least an 2.5cm, andusually around 5cm.
4.2.3 Depth sequenceThe well depth sequence is determined from the quadratic residue sequence. (bring in MLS). Thequadratic residue sequence is a mathematical sequence based on a prime number, N. The nth termin the sequence is given by n2 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;
ExerciseComplete the table below for the N=13 sequence. The sequence is periodic, how many terms are inone period?
4.2.4 Well depthsFor a quadratic residue sequence, sn, the well depths are given by the following equation:
ExerciseA) 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 amaximum 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 periodschanges and also as the prime number in the sequence changes.
Useful rules of diffractionWhen a sound wave hits a rough surface, the effects on the sound wave will depend on therelative 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 (8/2 > d) then the wave ‘sees’ the surface as a flat plane surfaceand specular reflection results.
2. When the wavelength of the sound is similar to the surface roughness (8 . d) then theresultant scattering is complex wave interference. The simplest model is that every pointon the surface act as a point source and radiates sound back into the room. The resultantpressure distribution depends on the relative phase and magnitude of all the wavesreceived.
3. At high frequencies (8 < d) the scattering can be calculated by considering the surface tobe a series of smaller plane surfaces.
Diffuser design, Trevor Cox 2004-5, (c) University of Salford, www.acoustics.salford.ac.uk 4
Figure 5 Scattering from 2 periods of QRDs withdifferent prime number generators. From bottomto top N=7, 17, 37, 89. Dimensions same asFigure 1.
Figure 4 Scattering from N=17 QRD, samediffuser as Figure 2. The lines have a differentnumber of periods and have been displacedvertically for clarity. From bottom to top: 1,2,4,8and 16 periods.
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-acousticfactors might affect the choice.
Note the sequences are symmetric around the zero well and n=(N-1)/2 point, reducingmanufacturing costs.There are methods for combining sequences of QRDs to minimize the lobing effects9,10
L Work has been carried out to reduce the effects of using multiple QRD periods11,12. This is doneusing a QRD and its inverse. We need to put them on the wall in such a way that there are norepetitions to prevent lobing. This is best done by arranging according to a pseudo-randomsequence like MLS (actually the best sequence is a Barker code 1 1 1 1 1 -1 -1 1 1 -1 1 -1). Whenwe have a 1 in the sequence we use the N=7 QRD, when we have a -1 we use the inverse. This isknown as spread spectrum after it’s use in radar technology. Other methods also exist.
Diffuser design, Trevor Cox 2004-5, (c) University of Salford, www.acoustics.salford.ac.uk 5
4.3 Limitations of Schroeder diffusersWe have seen above some of the limitations of Schroeder diffusers:(i) Scattering from surface does not obey the simple 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 andreceivers are in the near field(v) Fins have to be narrow so that they don’t cause significant scattering (assumed infinitely thin inabove equation)(vi) All loses are ignored - even though in reality absorption is significant at low frequencies (butthis 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 diffractiondirections.(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 apocket 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 withbetter diffusion. Just trying random depth sequences would be extremely time consuming as thereare a very large number of possible combinations, so we exploit optimization processes. Theprocess to produce optimum Schroeder-style diffusers is based on an iterative process13:Optimization process1. 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 method3. A parameter characterizing the degree of diffusion was calculated from the scattered
pressure distribution4. The surface shape was altered by changing the well depths according to standard techniques
which search for a minimum in avariable, in this case diffusion parameter.
5. Steps (2) to (4) were repeated until aminimum in the diffusion parameter wasfound indicating optimum diffusion.
Detailed notes:2. The Kirchhoff solution is a more accuratemethod for obtaining the scattering fromsurfaces than the Fraunhofer method givenabove. There are even more exact formulationsfor solving diffuser scattering based onBoundary Element Methods (BEMs). But forlarge surfaces BEM models are too slow to beuseful as during an optimization many thousandsof predictions are needed.
Table 2. Standard deviation diffusionparameters for Figures 4 and 5. Linesnumbered from bottom to top.
Diffusion Parameter
line Figure 4 Figure 5
1 6.36 16.2
2 16.5 16.5
3 24.3 20.5
4 23.8 19.6
5 26.2 -
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3. We need a cost parameter which evaluates how good a diffuser a surface is - this is the diffusionparameter, ,. There have been many suggestions for diffusion parameters, but for optimizationtechniques the most successful have been those based on the standard deviation function. Astandard deviation is taken of the n scattered sound pressure levels on a polar distribution:
where Lp,i is the sound pressure level for the ith measurement. If all sound pressure levels in thepolar distribution are the same then the standard deviation is zero (optimum diffusion). For non-optimum diffusion the standard deviation increases. Take for example the polar distributions shownin Figures 4 and 5. Table 2 gives the standard deviation for each of the lines. We can see that it ismonitoring the quality of the scattering.
4. There are many standard methods for searching a function to find minima - so calledoptimization techniques14. For this work a simplex routine is used for robustness, but it isn’t thefastest.Results:Figure 6 shows the comparison between thescattering from an N=7 QRD and an optimizedwelled diffuser. We can clearly see that thescattering is more even from the optimizedsurface - we have a better diffuser.
One of the advantages of using optimizationtechniques is that we are no longer stuck withusing the Schroeder shape of wells divided bythin fins. One of the problems with thisconstruction is it can have significant lowfrequency absorption15,16,17,18. This comes fromtwo effects: 1/4 wave resonance of the wells andviscous losses as sound energy has to passbetween the wells around the edge of the fins. Inaddition the fins are expensive to manufacture.By removing the fins we get a stepped diffuser(diagram). This can be optimized to have gooddiffusion, and this is also shown in Figure 6.
Other surfaces such as curved surfaces and fractals have been successfully optimized19,20. This aremore 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 ofSchroeder diffusers, this is design by brute force.
Figure 6 Scattering from surfaces at 1050 Hz. N=7 QRD
Optimized Welled Diffuser
Optimized Stepped Diffuser
Diffuser design, Trevor Cox 2004-5, (c) University of Salford, www.acoustics.salford.ac.uk 7
Figure 7. Diffusion parameter for 10 periodsof N=5 QRD (5x10) and for a plane surfacethe same size (plane)
Figure 6 Scattering from 10xN=5 QRD and planesurface at 2500 Hz
Tutorial Questions
1) The rear wall of the Salford University listening room is to be covered with a one-dimensionalQRD. The QRD is to have a maximum depth of 200mm, a well width of 4cm and is to be based onthe 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 satisfySchroeder’s optimum criteria?(v) The acoustic designer has suggested using a single N=89 diffuser instead of many periods of theN=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 tocover a wall 4m wide. Design an appropriate QRD for the space. Why might it be difficult in realityto 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. Thepredicted scattering from the QRD is
evaluated using the standard deviation as shown below in Figure 7. Note at 2500 Hz theperformance of the QRD is similar to a plane surface (so not very good). Figure 8 shows the polardistribution at that frequency. Why are there frequencies at which the QRD performance is poor?What solutions can you suggest?
Answers:
1) (ii) 0,50,200,100,100,200,50mm
(iii) f0 = 486 Hz. Upper frequency limit = 4250Hz.
(iv) 486, 972, 1458, 1944, 2430, 2916, 2402, 2916 Hz.
2) Hint Comp are the waveleng th at 2500 Hz to surface dimensio ns.
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