### 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 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

^{C.11}

Figure C.35 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 [279]. An article reviewing the history of room acoustic diffusers may be found in [94].

**Next Section:**

Lossless Scattering

**Previous Section:**

The Lossy 2D Mesh