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Schroeder presented methods of designing concert hall ceilings that could avoid direct reflections into the audience. In 1975, he provided a way of designing highly diffusing surfaces based on binary maximum-length sequences, and showed that these periodic sequences have the property that their harmonic amplitudes are all equal [SchroederSchroeder1975]. He later extended his method and proposed surface structures that give excellent sound diffusion over larger bandwidths [SchroederSchroeder1979]. This is based on quadratic residue sequences of elementary number theory, investigated by A. M. Legendre and C. F. Gauss. These sequences are defined by

 (1)

i.e., is taken as the least nonnegative remainder modulo , and is an odd prime number. For , the quadratic residue sequence reads as follows (starting with ):

These sequences have a few properties:
1. they are symmetric [around and ];
2. they are periodic with period ;
3. surprisingly, the discrete Fourier transform of the exponentiated sequence
 (2)

has constant magnitude
 (3)

The quadratic residue diffuser, or Schroeder diffuser, is implemented by having periodic wells of different depths proportional to with period over the surface. Figure 1 shows a cross section through the diffusing surface based on the quadratic residue sequence with .

 @font picture(6216,2424)(418,-3523) (6526,-1936)(0,0)[lb] % (5101,-3361)(0,0)[lb] % (3001,-3061)(0,0)[lb] %

The width of each well is determined by the design wavelength , and the depths of the well are defined as

 (4)

where is the quadratic residue sequence with period .

Strube did empirical and numerical analyses on scattering characteristics of Schroeder's diffuser [StrubeStrube1980a,StrubeStrube1980b], and design techniques of concert halls were provided by Ando using Schroeder's diffuser [AndoAndo1985].

Next: The 2-D Digital Waveguide Up: Implementation of a Highly Previous: Introduction
Kyogu Lee 2004-05-28