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Physical Audio Signal Processing
    Digital Waveguide Theory
       Digital Waveguide Mesh
          Diffuse Reflections in the Waveguide Mesh


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Diffuse Reflections in the Waveguide Mesh

In [416], Manfred Schroeder proposed the design of a diffuse reflector based on a quadratic residue sequence. A quadratic residue sequence $ a_p(n)$ corresponding to a prime number $ p$ is the sequence $ n^2$ mod $ p$, for all integers $ n$. The sequence is periodic with period $ p$, so it is determined by $ a_p(n)$ for $ n=0,1,2,\ldots,p-1$ (i.e., one period of the infinite sequence).

For example, when $ p=7$, the first period of the quadratic residue sequence is given by

\begin{eqnarray*} a_7 &=& [0^2,1^2,2^2,3^2,4^2,5^2,6^2] \quad (\mbox{mod }7)\\ &=& [0, 1, 4, 2, 2, 4, 1] \end{eqnarray*}

An amazing property of these sequences is that their Fourier transforms have precisely constant magnitudes. That is, the sequence

$\displaystyle c_p(n) \isdef e^{j\frac{2\pi}{p} a_p(n)} $

has a DFT with exactly constant magnitude:C.11

$\displaystyle \vert C_p(\omega_k)\vert \isdef \vert\dft _k(c_p)\vert \isdef \l... ...^{p-1} c_p(n) e^{-j2\pi nk/p}\right\vert = \sqrt{p}, \quad \forall k\in[0,p-1] $

This property can be used to give highly diffuse reflections for incident plane waves.

Figure C.35 presents a simple matlab script which demonstrates the constant-magnitude Fourier property for all odd integers from 1 to 99.

Figure C.35: Matlab script for demonstrating the Fourier property of an odd-length quadratic residue sequence.
  
function [c] = qrsfp(Ns)
%QRSFP Quadratic Residue Sequence Fourier Property demo
  if (nargin<1)
     Ns = 1:2:99; % Test all odd integers from 1 to 99
  end
  for N=Ns
    a = mod([0:N-1].^2,N);
    c = zeros(N-1,N);
    CM = zeros(N-1,N);
    c = exp(j*2*pi*a/N);
    CM = abs(fft(c))*sqrt(1/N);
    if (abs(max(CM)-1)>1E-10) || (abs(min(CM)-1)>1E-10)
       warn(sprintf("Failure for N=%d",N));
    end
  end
  r = exp(2i*pi*[0:100]/100); % a circle
  plot(real(r), imag(r),"k"); hold on;
  plot(c,"-*k"); % plot sequence in complex plane
end

Quadratic residue diffusers have been applied as boundaries of a 2D digital waveguide mesh in [279]. An article reviewing the history of room acoustic diffusers may be found in [94].

Previous: The Lossy 2D Mesh
Next: FDNs as Digital Waveguide Networks

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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