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The Quadratic Residue Diffuser

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

$\displaystyle s_n = n^2 \quad ( \; \texttt{mod}\; N),$     (1)

i.e., $n^2$ is taken as the least nonnegative remainder modulo $N$, and $N$ is an odd prime number. For $N = 17$, the quadratic residue sequence reads as follows (starting with $n=0$):

\begin{displaymath}s_n = 0,1,4,9,16,8,2,15,13,13,15,2,8,16,9,4,1;0,1,\cdots \end{displaymath}

These sequences have a few properties:
  1. they are symmetric [around $n \equiv 0$ and $n \equiv (N-1)/2$];
  2. they are periodic with period $N$;
  3. surprisingly, the discrete Fourier transform $R_m$ of the exponentiated sequence
    $\displaystyle r_n = e^{\pm j2\pi s_n/N}$     (2)

    has constant magnitude
    $\displaystyle \left\vert R_m\right\vert = \left\vert \frac{1}{N} \sum^{N}_{n=1}r_ne^{-j2\pi nm}
\right\vert^2 = \frac{1}{N}.$     (3)

The quadratic residue diffuser, or Schroeder diffuser, is implemented by having periodic wells of different depths proportional to $s_n$ with period $N$ over the surface. Figure 1 shows a cross section through the diffusing surface based on the quadratic residue sequence with $N = 17$.

Figure 1: Cross section through a highly diffusing surface based on quadratic residue sequence when $N = 17$. The thin vertical lines represent rigid separators between individual wells.

picture(6216,2424)(418,-3523) (6526,-1936)(0,0)[lb] $d_n$% (5101,-3361)(0,0)[lb] $w$% (3001,-3061)(0,0)[lb] $N$%

The width of each well $w$ is determined by the design wavelength $\lambda_0 (\gg w)$, and the depths of the well $d_n$ are defined as

$\displaystyle d_n = \frac{\lambda_0}{2N}s_n,$     (4)

where $s_n$ is the quadratic residue sequence with period $N$.

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