Page 23 - Summer 2006

residue sequence is based on a prime
number. For the diffuser in Fig. 4 (left)
the prime number is 7. The depth of
the nth well is proportional to n2 modu-
lo 7, where modulo indicates the least
non-negative remainder. So the third
well has a depth proportional to 32
modulo 7 = 2. The sequence mapped
out in this case is 0,1,4,2,2,4,1 that can
be seen in Fig. 4. (The diffuser in Fig. 4
on both ends, but these are constructed to be half the width). If this quadratic residue sequence is used to construct the dif- fuser, then the diffraction or grating lobes generated all have the same energy, as shown in Fig. 6 (left).
To understand why number theory is useful, we need to turn to optical theories developed in the early 19th century by great physicists such as Fraunhofer. It is interesting to note that it is the Fraunhofer diffraction theory that has essential- ly been used by many researchers to examine the perform- ance of Schroeder diffusers. This theory was aptly named after the first person to construct a diffraction grating in 1821—Joseph von Fraunhofer. The far field “scattering” from a reflection phase grating can be directly related by a Fourier transform to the distribution of the pressure reflection coef- ficients on the front surface of the diffuser. If our desire is to generate “even” scattering, then we need a distribution of reflection coefficients that are maximally random, or to be more precise, we need a set of reflection coefficients whose autocorrelation function is a delta-function. This is what pseudo-random number sequences provide—sequences of real or complex numbers with optimal autocorrelation prop- erties. While it is possible to roll dice to generate well depths, number theory will provide sequences with better autocorre- lation properties and consequently better scattering.
There are many other number sequences apart from those based on quadratic residues that can be used. Primitive root, Chu, and Luke sequences are three examples of sequences that have been examined as acoustic diffusers. Many of these sequences were originally developed for appli- cations as diverse as astronomy, error-checking systems for computer and digital audio data, and mobile telephony. Rather incredibly, these sequences find a use in manipulating room acoustics. With roots back in 18th century mathemat- ics, it seems almost impossible that these sequences, with their strange generation algorithms and modular arithmetic, should still be of use in room acoustic engineering. As Schroeder is fond of saying, number theory is, in many ways, unreasonably useful.
Enhancements
The basic Schroeder diffuser based on number theory sequences is an ingenious invention, however, aspects of its performance are not optimal. Building on Schroeder’s initial design, several revisions have been suggested to improve its performance.
In room acoustics, designs must work over a wide band- width. A diffuser’s wells need to be narrow and deep, and this makes the device very impractical. First, the structure
becomes highly expensive to make, and second, the diffusers become very absorbing. If one uses very narrow and deep wells, it is possible to make a rather effective “absorber.” Another issue is that if the well spacing becomes too small compared to a wavelength, then the diffuser behaves as though it is a sur- face with an average admittance rather
than one with a complex spatial distribution of admittances. Inspired by chaos theory and fractals, a solution to this prob- lem has been developed.
Element roughness
To cover many octaves, the diffuser needs to have “ele- ment roughness” on different scales. Using elements of dif- ferent sizes is common in 2-way loudspeaker designs. For room acoustic diffusers, some elements need to have dimen- sions that are meters in size, and some need to have dimen- sions that are centimeters in size.
Fractals
Fractals are objects that have scaleable properties. The effect can be achieved for diffusers, as shown in Fig. 4 (right). In the surface shown, smaller diffusers are mounted within larger diffusers. The small diffusers scatter the high frequen- cies, and the larger diffusers scatter the low frequencies. This
8
type of diffuser is rather fittingly named diffractal . An exam-
ple of applying a diffractal on the rear wall of a mastering room is shown in Fig. 7.
Phase gratings
Phase gratings, whether optical or acoustical, are nor- mally periodic. For Schroeder diffusers, many periods of the device are stacked next to each other. Diffraction lobes that are designed to have the same energy are a function of peri- odicity, and therefore Schroeder’s definition of optimum dif- fusion requires the structure to be periodic. These diffraction lobes represent energy concentrated into particular direc- tions with a lack of reflected energy between. When there are many lobes this is not a problem, but this is not usually the
“Building on Schroeder’s initial design, several revisions have been suggested to improve the performance.”
has zero depth wells
Fig 7. Gateway Mastering, Portland, ME showing a fractal diffusing rear wall (Diffractal®). (Photo courtesy of Gateway Mastering & DVD).
A Brief History of Room Acoustic Diffusers 21