In , 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:C.10
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.
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 . An article reviewing the history of room acoustic diffusers may be found in .