Diffuse Reflections in the Waveguide Mesh

In [419], Manfred Schroeder proposed the design of a
diffuse reflector based on a *quadratic residue sequence*. A
quadratic residue sequence
corresponding to a prime number
is the sequence
mod
, for all integers
. The sequence
is periodic with period
, so it is determined by
for
(*i.e.*, one period of the infinite sequence).

For example, when , the first period of the quadratic residue sequence is given by

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

has a DFT with exactly constant magnitude:

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

Figure C.37 presents a simple matlab script which demonstrates the constant-magnitude Fourier property for all odd integers from 1 to 99.

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 [281]. An article reviewing
the history of room acoustic diffusers may be found
in [94].^{C.15}

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