NULL CENTERED FRACTAL ACOUSTIC DIFFUSER

DIFFUSEUR ACOUSTIQUE FRACTAL A POINT CENTRAL NUL

English 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.

Claims

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE PROPERTY
OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:
Claim 1. A new type of quadratic residue based acoustical diffuser based on a
unique arrangement
of the quadratic residue sequence.

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.
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)

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.
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).


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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)

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(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

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)

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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.

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.


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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)

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=

3 0 2 6 1 9 8 9 1 6 2 0
2 4 8 3 0 10 0 3 8 4 2
9 6 8 1 7 4 3 4 7 1 8 6
4 1 3 7 2 10 9 10 2 7 3 1


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1 9 0 4 10 7 6 7 10 4 0 9
0 8 10 3 9 6 5 6 9 3 10 8
1 9 0 4 10 7 6 7 10 4 0 9
4 1 3 7 2 10 9 10 2 7 3 1
9 6 8 1 7 4 3 4 7 1 8 6
2 4 8 3 0 10 0 3 8 4 2
3 0 2 6 1 9 8 9 1 6 2 0

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
4 6 3 2 3 6 4
6 1 5 4 5 1 6
3 5 2 1 2 5 3
2 4 1 0 1 4 2
3 5 2 1 2 5 3
6 1 5 4 5 1 6
4 6 3 2 3 6 4
(t)1.
(M')=3 5 2 1 2 5 3


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0 4 3 4 0 5
0 2 6 5 6 3 0
4 6 3 2 3 6 4
3 5 2 1 2 5 3
4 6 3 2 3 6 4
0 2 6 5 6 3 0
5 0 4 3 4 0 5
(t)2.
(M')=4 6 3 2 3 6 4
6 1 5 4 5 1 6
1 3 0 6 0 3 1
5 0 4 3 4 0 5
4 6 3 2 3 6 4
5 0 4 3 4 0 5
1 3 0 6 0 3 1
6 1 5 4 5 1 6
(t)3.

(M)=5 0 4 3 4 0 5
0 2 6 5 6 3 0
2 4 1 0 1 4 2
6 1 5 4 5 1 6
5 0 4 3 4 0 5
6 1 5 4 5 1 6
2 4 1 0 1 4 2
0 2 6 5 6 3 0
(t)4.
(M')=6 1 5 4 5 1 6


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1 3 0 6 0 3 1
3 5 2 1 2 5 3
0 2 6 5 6 3 0
6 1 5 4 5 1 6
0 2 6 5 6 3 0
3 5 2 1 2 5 3
1 3 0 6 0 3 1
(t)5.
(M')= 0 2 6 5 6 3 0
2 4 1 0 1 4 2
4 6 3 2 3 6 4
1 3 0 6 0 3 1
0 2 6 5 6 3 0
1 3 0 6 0 3 1
4 6 3 2 3 6 4
2 4 1 0 1 4 2
(t) 6.
(M)= 1 3 0 6 0 3 1
3 5 2 1 2 5 3
0 4 3 4 0 5
2 4 1 0 1 4 2
1 3 0 6 0 3 1
2 4 1 0 1 4 2
5 0 4 3 4 0 5
3 5 2 1 2 5 3

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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
4 6 3 2 3 6 4
6 1 5 4 5 1 6
3 5 2 1 2 5 3
2 4 1 0 1 4 2
3 5 2 1 2 5 3
6 1 5 4 5 1 6
4 6 3 2 3 6 4

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

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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.

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

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.


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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.


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Description

CA 02753792 2011-09-28
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.

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CA 02753792 2011-09-28
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

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CA 02753792 2011-09-28

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
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CA 02753792 2011-09-28

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.

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CA 02753792 2011-09-28

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.

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