CA2753792A1 - Null centered fractal acoustic diffuser - Google Patents

Abstract

The Null Centred Fractal Acoustical Diffuser is a device for optimizing sound reproduction within enclosed spaces. The invention is manifested as a single, or a series of acoustical tiles based on a unique arrangement of the quadratic residue sequence. A two dimensional null centred sound diffuser is described by a two dimensional pattern of peaks and wells, which is characterized through application of quadratic residue sequence theory, with the sequence null point located in the centre of the diffuser. When tiles are combined with other self-similar tiles, the acoustical performance of the tile array is extended to embrace properties of a fractal array. The fractal-array diffuser as described in this document does not permit the buildup of diffraction-lobes, thereby eliminating a well known deficiency of repeating periodic arrays.

Description

NULL CENTERED FRACTAL ACOUSTIC DIFFUSER

This invention provides an improved acoustical diffuser for improving sound reproduction in enclosed or semi-enclosed spaces.

BACKGROUND OF THE INVENTION
REFERENCES CITED

U.S. PATENT DOCUMENTS
5,401,921 3/1995 D'Antonio et al.
Des. 291,601 8/1987 D'Antonio et al.
Des. 306,764 3/1990 D'Antonio et al.

Unless a room has been purpose-built for ideal sound reproduction, it will normally exhibit undesirable acoustic properties that reduce the apparent fidelity of sound.
For example, flat walls that are directly in the sound source path, or within the indirect sound source path, will reflect nearly all audible frequencies of sound coherently. The reflective surface acts as a secondary source of sound, delayed in time, interfering with the sound energy arriving directly from the source at the listening position. This results in a series of undesirable audible acoustic manifestations such as comb filtering, flutter-echo , resonance and uneven distribution of sound.

The use of acoustic absorption is generally considered a low cost countermeasure to acoustic problems, however absorption simply tends to mask or diminish the acoustical energy within a room within a certain frequency range rather that to address the problem at it's source. Absorption applied in significant quantities will remove nearly all reverberant energy from a room, making the room sound 'dead' and thus creating an entirely new set of acoustical problems. A quantitative study by Schroeder' in 1979 proves that acoustical diffusion provides a superior listening experience over a highly acoustically absorptive room. Hence, the objective is not to remove acoustical energy from a room, but rather, to modify it with specially designed acoustical diffusers.
Coherent-reflection is not wanted. A non-coherent reflection, generally known as diffusion is far 1 M.R. Schroeder, "Binaural dissimilarity and optimum ceilings for concert halls: More lateral sound diffusion", Journal of the Acoustical Society of America, 65(4), April 1979.

more desirable.

The ideal diffuse-reflection is non-coherent in time and space. Spatial diffusion spreads the sound energy of a reflection into a larger volume of space, lowering the amount of energy returned to any given point. Temporal diffusion spreads reflected energy in time, smoothing the overall dissipation of a reflected sound field, and discouraging time-related effects such as comb-filtering.

The poly-cylindrical acoustical diffuser dating to 1942' provides good diffusion in space, removing most flutter-echo effects, but is still coherent in time, leading to comb-filtering at the listening position. This can be somewhat minimized by varying the sizes of the cylindrical diffusers in a room, but the effect remains.

Most modern diffusers are numerically based, a concept introduced by Schroeder in 19753.
Numerical diffusers utilize pseudo-random sequences; determinate sequences which resemble large random groupings to a precise degree. The proper application of number-theoretic designs results in surfaces with predictable characteristics, which can be controlled and tuned, offering good diffusion in both the temporal and the spatial realm. Most designs are derived from Maximum Length Sequences(MLS), Quadratic Residue, or Primitive Root sequences. The current invention is of the Quadratic Residue type, though in a hitherto unseen embodiment.

Generally, well performing diffusers are made to follow a series of peaks and wells, with a mathematical formula guiding the ratio of peak to well height in a sequence.
For example, U.S.
Pat. No. 4,821,839 and U.S. Pat. No. 4,964,486 describe a one dimensional diffuser that utilizes quadratic residue number theory to dictate the well depth in one dimension.
This differs from the presented invention due to the one dimensional design, and the lack of a canter based null point in the sequence of peaks and valleys. Another mathematical approach is to use primitive root number theory to dictate the pattern of peaks and wells such as in U.S. Pat. No.
5,401,921. In the latter case a two dimensional diffuser is presented, however this design uses the primitive root number sequence which differs from the presented invention and also does not use a centre null point.

2 Polycylindrical Diffusers in Room Acoustic Design, Journal of the Acoustical Society of America, John E Volkmann January 1942.

3 M.R. Schroeder, "Diffuse sound refection by maximum-length sequences", Journal of the Acoustical Society of America 57(1), 149-50 1975 The null-centred fractal acoustical diffuser manifests a unique arrangement of the peaks and wells with a quadratic residue sequence null point in the centre of the diffuser which differs from existing designs. Secondly, by arranging a series of self-similar tiles a fractal diffuser may be created.
SUMMARY OF THE INVENTION

Acoustical diffusers of the embodiment described are constructed as a square or rectangular series of tiles that are manifested as a series of peaks and valleys. The sequence of peaks and valleys follow a unique application of quadratic-residue number theory known as null-centred fractal. Tiles may be used individually, or in groups that consist of specific modulations of the fundamental quadratic-residue null centred sequence. Tiles may be made of a variety of materials, so long as the material chosen is acoustically reflective at the desired audio wavelengths.
When tiles are arranged in groups as specified, the tiles form a fractal-array.

In the drawings, which form a part of this specification, Fig. A illustrates a component of a single tile, and the basic features of each component.
Fig. B is an isometric projection of Fig A.
Fig. C illustrates how tile components are assembled into a complete tile.
Fig.D illustrates a complete tile, in isometric projection.
Fig.E illustrates a conventional approach to creating a diffusive array. A
single component is shown for clarity.
Fig. F,G,H,I illustrate components of a fractal-array.
Fig. J illustrates a conventional diffusive array.
Fig. K illustrates a fractal-array.

DETAILED DESCRIPTION OF THE INVENTION
An acoustical tile of the embodiment is constructed as a series of linear components, each of which follows a unique manifestation of quadratic-residue number theory as described in the following claims. In Fig. A, a single component is illustrated. The unique application of quadratic-residue theory is manifested as a series of peaks and valleys, the height of which are determined by the mathematical results cited in the following claims. Because the mathematics resolve to zero, a 'base plate' must be made to mechanically support the individual columns. This is shown as'B' in Fig. A. The thickness of this baseplate is given as the linear distance of'A' to 'C', and may vary according to the materials and mechanical strength properties necessary for mounting a tile onto a surface. Hence, the thickness of'B' may vary substantially without affecting the acoustical performance of the device.

The features 'D', 'E', 'F' .. 'N' denote linear distances from C, which are determined by the mathematical results of the claims.

In Fig. B, an isometric projection of component Fig. A is shown. '0' denotes the thickness of the component, which is equivalent to the linear side length of features 'D' .. W.
Hence, each top surface of features 'D' . . 'N' are square.

In Fig. C, an assembly of components are constructed into a complete tile.
Each component of the tile 'A', 'B', 'C', 'D', 'E', 'F' is a unique modulation of the null-centred quadratic-residue sequence result. Hence, each component may be said to be 'self-similar', albeit not identical. The tile exhibits bilateral symmetry across the centre component 'F".

In Fig D, an isometric projection of the complete tile as shown in Fig. C is illustrated. The tile components may be manufactured individually and combined, or the tile may be manufactured as a single component.

Fig E illustrates a 'conventional' diffuser component. The purpose of this illustration is to show how acoustical diffusers are constructed using conventional technology. For clarity, only a single element of a complete array is shown. This component contains peaks and valleys determined by non-null centred quadratic-residue theory. This is consistent with US Patents 5,401,921 3/1995 D'Antonio et al, Des. 291,6018/1987 D'Antonio et al and Des. 306,764 3/1990 D'Antonio et al.
The numbers presented (0,1,4,2,2,4,1) illustrate a conventional quadratic-residue result permutation as applied to acoustical diffusers.

When these components are assembled conventionally, they are typically placed side-by-side in a fashion as illustrated in Fig J. This arrangement has a specific limitation:
The acoustical array will generate diffraction-lobes, which are detrimental to ideal sound reproduction in an enclosed, or semi-enclosed space. This occurs because each element in the array is identical. Schroeder describes the manifestation and undesirable properties of such diffraction-lobes in his paper M.R.
Schroeder, "Binaural dissimilarity and optimum ceilings for concert halls:
More lateral sound diffusion ", Journal of the Acoustical Society ofAmerica, 65(4), April 1979.

By applying a unique application of the quadratic-residue theory, described as null-centred, a series of unique components may be constructed that manifest excellent acoustical diffusion when used singularly, however exhibit fractal-array properties when used in a specific arrangement as described in the claims. Specifically, such an array does not generate undesirable diffraction-lobes.
In Figures 'F' . . 'I', four examples of diffuser components that embody the invention are shown.
These components when assembled into an array as shown in Fig. K, may be considered to be a fractal-array, and thus free of undesirable diffraction-lobes.

Claims (11)

1. Claim 1. A new type of quadratic residue based acoustical diffuser based on a unique arrangement of the quadratic residue sequence.
2. Claim 2. Of claim 1, groupings of self similar tiles placed adjacent to each other may be considered to form a Fractal array. 'Fractal' may be defined as self-similar at different scales of magnitude.
3. Claim 3. Of claim 1 and 2, said arrangement of the quadratic residue sequence shall begin at a given prime number (p) plus one, divided by two.
   
   (n) = (p+1)/2 The result will be an integer. The sequence is begun at this integer, and incremented through successive integers (p) number of times.
   
   Example 2)a. (p) = 7 (7+1)/2=4 result (n) = (4,5,6,7,8,9,10) (n) squared, modulus (7) = (2,4,1,0,1,4,2) Example 2)b. (p) = 11 (11+1)/2=6 result (n) = (6,7,8,9,10,11,12,13,14,15,16) (n) squared, modulus (11) = (3,5,9,4,1,0,1,4,9,5,3)
4. Claim 4. Of claim 3, it may be seen that the prime number (p) appears at the mid-point in the sequence, and that the corresponding reside is null, or zero at the mid point of the sequence.
5. Claim 5. Of claim 4, a two dimensional array measuring (p) x (p) may be created which, when extended across a planar surface, will create a periodic array. A given null-centred quadratic residue sequence, or a modulated permutation thereof, is laid in an x/y configuration with the (x) axis designated (m) and the (y) axis designated (n).
   
   To generate the array, (m) is added to (n), and the residue, modulus (p) is taken as the value at the coordinate point defined by (m) and (n).
   
   (m+n) mod P
   Example 5)a.
   
   (p) = 11 from Example 2)b, the sequence derived is (m) = (3,5,9,4,1,0,1,4,9,5,3) (n) is given as successive members of the quadratic sequence (m). To generate the array, (m) is added to (n), and the residue, modulus (p) is taken as the value of the coordinate at point (m),(n).
   The permutation (m') is given as (m') = (m + n) mod (p) if (m) =(3,5,9,4,1,0,1,4,9,5,3) and the first permutation (n) is the first value of the sequence (m) _ 3, then (m') = ((3,5,9,4,1,0,1,4,9,5,3) + (3)) mod (11) therefore (m') = (6,8,1,7,4,3,4,7,1,8,6) the next successive value of (m) = (n), and provides the next sequence permutation (m") = ((3,5,9,4,1,0,1,4,9,5,3) + (5)) mod (11) therefore (m") = (8,10,3,9,6,5,6,9,3,10,8) When expressed as a two dimensional array, the following solution is found for prime number (p).
   Example 5)b.
   
   For a given prime number (p) = 11, (m) =(3,5,9,4,1,0,1,4,9,5,3) Successive values of (n) shall therefore be (3,5,9,4,1,0,1,4,9,5,3) (m') = (m + n (1)) mod (p) (rn") = (m + n(2)) mod (p) (m"') = (m + (n(3)) mod (p) . . . . and so forth.
   The array sequence therefore resolves to n=
   3 6 8 1 7 4 3 4 7 1 8 6 (m') (A) 8 10 3 9 6 5 6 9 3 10 8 (m") (B) 9 1 3 7 2 10 9 10 2 7 3 1 (m"') (C) 4 7 9 2 8 5 4 5 8 2 9 7 (m"") (D) 1 4 6 10 5 2 1 2 5 10 6 4 (m""') (E) 0 3 5 9 4 1 0 1 4 9 5 3 (m""") (F) 1 4 6 10 5 2 1 2 5 10 6 4 (m"""') (E) 4 7 9 2 8 5 4 5 8 2 9 7 (m"""") (D) 9 1 3 7 2 10 9 10 2 7 3 1 (m""""') (C) 5 8 10 3 9 6 5 6 9 3 10 8 (m""""") (B) 3 6 8 1 7 4 3 4 7 1 8 6 (m"""""') (A) Table A
6. Claim 6. Of claim 5; From the given array, a set of well depths or inversely, column heights may be determined and used to construct acoustic diffusers of unique appearance and application.
   
   Example 6) Fig A may be described as a'Column Block'. Considering FigA as an example, the sequence (r n') may be manifested as a series of column heights above a surface (C). Because the column heights may include zero, a'base plate' (B) must be created for mechanical support of the columns. The base plate (B) can be made any thickness required for mechanical or esthetic reasons without affecting the acoustical performance of the diffuser.
   
   In this case, the sequence (m') is given as (m') = (6,8,1,7,4,3,4,7,1,8,6) a series of column heights may be constructed to be equal to, or a factor of the members of (m').
   The column heights are measured from the surface (C) to the top of each column as shown in Fig A. Columns D, E..N are mapped to the members of (m') sequentially as shown below.
   
   Column: D E F G H I J K L M N
   Column Height: 6 8 1 7 4 3 4 7 1 8 6 Fig B shows an isometric projection of the column block from Fig A. The thickness of the block (0) is equal to the linear length of any individual column. For example, column (E) is square, and each side of this square is equal to the thickness of the plate given as (0).
   
   By combining column blocks as shown in Fig A and Fig B, a complete diffuser may be created.
   Column blocks must follow an array sequence as shown in example 5b) or an array sequence permutation as defined in Claim (7) and Claim (8). For this example, the array sequence shown in example 5b) shall be used.
   
   A column block for each array sequence (m' to m"""""') is made and these are combined to create a single acoustical diffuser. The column block array sequences shown in Table A
   exhibit symmetry around the middle sequence (F). For example, the first and last sequence are equivalent, and both are therefore equal and arbitrarily named (A). The assembled acoustical diffuser will also exhibit this same symmetry. Fig C therefore shows how column blocks (A) - (F) are combined to create an acoustical diffuser.
   
   Fig D shows a three dimensional view of the completed acoustical diffuser.
   Notable features are the null in the centre of the diffuser, and the symmetry about the X and Y
   axes.
7. Claim 7. Of claim 5) and claim 6) a series of self-similar permutations may be constructed of the array sequence by adding an integer (q) to each reside in the sequence, and a new residue modulus may be calculated. Successive integer values for (q) are permitted.
   
   This will allow a large variety of acoustical tiles to be constructed with different appearance, however sharing similar acoustical diffusion properties. The choice of integer (q) dictates which permutation is created.
   
   Each element in the sequence permutation (M) is given by:
   (M) = (m + q)mod (p) Example 7)a (p) = 11, (m) = (3,5,9,4,1,0,1,4,9,5,3) (q) = successive integers = (1,2,3,4 . . . .
   M' = (m + q(1))mod (p) M' = (m + 1)mod(p) M' = (4,6,10,5,2,1,2,5,10,6,4) M" = (m + q(2)) mod (p) M" = (m + 2) mod (p) M" = (5,7,0,6,3,2,3,6,0,7,5)
8. Claim 8. Of claim 5) to 7), self-similar permutations may be used to construct arrays exhibiting self-similar properties. These arrays are based on integer modulations (q) which are defined to be all integer numbers.
   
   Example 8)a (p) = 11, (q) = 5, (m) = (3,5,9,4,1,0,1,4,9,5,3), (n) = successive members of (m) M'= (m) + (q) mod (p) (M' ) = (8,10,3,9,6,5,6,9,3,10,8) n=
9. 9 6 8 1 7 4 3 4 7 1 8 6 Claim 9. Of claim 8, arrays combined with self-similar arrays with differing integer modulations (q) may be used to create a Fractal Array. The term 'Fractal Array' may be defined as self-similar at different scales of magnitude. An array of acoustical tiles as shown in the following example may be described as a null-centred fractal-array.
   
   Example 9)a For simplicity, a prime 7 array and it's self-similar permutations are used.
   However, any prime number equal to or greater than 3 may be used.
   
   For prime number(p) = 7, there are seven permutations permitted. These will be designated (t)0 through (t)6 (t) 0.
   (M')= 2 4 1 0 1 4 2 (t)1.
   (M')=3 5 2 1 2 5 3 (t)2.
   (M')=4 6 3 2 3 6 4 (t)3.
   
   (M)=5 0 4 3 4 0 5 (t)4.
   (M')=6 1 5 4 5 1 6 (t)5.
   (M')= 0 2 6 5 6 3 0 (t) 6.
   (M)= 1 3 0 6 0 3 1 The above unit-arrays are positioned in accordance with the integer values as defined by the initial array chosen as the geometric centre-point.
   
   Initial Array = (t)0 Therefore, the Fractal Array is given as;
   (t)0 fractal array =
   
   t4 t6 t3 t2 t3 t6 t4 t6 t1 t5 t4 t5 t1 t6 t3 t5 t2 t1 t2 t5 t3 t2 t4 t1 t0 t1 t4 t2 t3 t5 t2 t1 t2 t5 t3 t6 t1 t5 t4 t5 t1 t6 t4 t6 t3 t2 t3 t6 t4 This fractal-array consists of a 7 x 7 array of 49 tiles total. Tiles (t)0 to t(6) comprise the fractal-array with the arrangement of tiles as shown.
   
   Example 9)b Of claim 8 and 9 and example 9)a, a fractal-array of higher orders or lower orders of prime numbers may be created than given in example 9)a. For this example, the prime number 11 is chosen to illustrate a larger fractal array, although all prime numbers may be used.
   
   For prime number (p) = 11, a null-centered tile of 16.5" x 16.5" is arbitrarily chosen. This will require (p) squared individual tiles in total (121) chosen from the fractal set as generated and arranged in accordance with claims 1 through 9, and will result in a square array with side length of 181.5" per side.
10. Claim 10 A fractal-array need not be square. Fractal-array expansion may be accomplished in one or two planar dimensions permitting non-square arrays to be constructed. This may be also described as a null-centred fractal-array
11. Claim 11 The null-centred fractal-array made of null centred diffusers as described in this document does not permit the buildup of diffraction-lobes, thereby eliminating a well known deficiency of repeating periodic arrays such as Quadratic Residue, Primitive Root or Maximum-length Sequences.' When treating large acoustical spaces, it is common practice to group numerical diffusers together in a periodic-array, utilizing the periodic nature of the sequences involved to create larger diffusive surfaces. For example, in FIG E, a short diffuser sequence is created using a non-null centred prime 7 modulus (0,1,4,2,2,4,1). This diffuser is grouped into a larger sequence using identical instances of FIG E. The resulting large scale diffuser is shown in FIG J. This is the 'standard' way of building a large scale diffuser.
    
    This otherwise practical solution suffers an inherent shortcoming: The repeating elements in a periodic-array are made up of identical elements, and therefore produce coherent rays of sound energy at design frequencies, commonly referred to as diffraction lobes. These lobes are detrimental to diffusion performance. A large-scale diffuser may be designed using a longer sequence of numbers, based on a larger prime number, thus eliminating the lobes, but difficulties in the construction of these designs have prevented widespread use.
    
    The Null Centred Fractal Acoustic Diffuser as presented in this invention does not exhibit repeating elements, as the individual elements are arranged in accordance with the quadratic-residue sequence at different scales over the entire array. This eliminates diffraction-lobes and improves 1 M.R. Schroeder, "Binaural dissimilarity and optimum ceilings for concert halls: More lateral sound diffusion", Journal of the Acoustical Society of America, 65(4), April 1979.
    
    overall performance, and escapes the limitations of prior diffuser array designs.
    Example 11) A series of null-centred fractal acoustical diffusers are used to create a fractal-array. For illustration simplicity, only the first sequence in each array is illustrated. In this case, a series of small acoustical diffusers are combined to form the large-scale fractal array. FIG
    F,G,H and I show each sequence individually. The prime 7 sequences used as the basis for these diffusers as calculated from Claims 1- 8 are given as follows.
    
    FIG F 4,6,3,2,3,6,4 FIG G 6,1,5,4,5,1,6 FIG H 3,5,2,1,2,5,3 FIG I 2,4,1,0,1,4,2 When combined these form a fractal-array as shown in FIG K.
    
    The fractal-array as described has all the advantages of a longer prime sequence, but construction demands are greatly reduced. Furthermore, a large array can provide these advantages at scales in which practical solutions are unattainable with larger primes and longer sequences.