On the Modeling of Rectangular Geometries in Room Acoustic Simulations

Enzo De Sena, Niccolò Antonello, Marc Moonen, Toon van Waterschoot
KU Leuven, ESAT--STADIUS
Stadius Center for Dynamical Systems, Signal Processing and Data Analytics,
Kasteelpark Arenberg 10, 3001 Leuven, Belgium

This paper is concerned with an acoustical phenomenon called sweeping echo, which manifests itself in a room impulse response as a distinctive, continuous pitch increase. In this paper, it is shown that sweeping echoes are present (although to greatly varying degrees) in all perfectly rectangular rooms. The theoretical analysis is based on the rigid-wall image solution of the wave equation. Sweeping echoes are found to be caused by the orderly time-alignment of high-order reflections arriving from directions close to the three axial directions. While sweeping echoes have been previously observed in real rooms with a geometry very similar to the rectangular model (e.g. a squash court), they are not perceived in commonly encountered rooms. Room acoustic simulators such as the image method (IM) and finite-difference time-domain (FDTD) correctly predict the presence of this phenomenon, which means that rectangular geometries should be used with caution when the objective is to model commonly encountered rooms. Small out-of-square asymmetries in the room geometry are shown to reduce the phenomenon significantly. Randomization of the image sources' position is shown to remove sweeping echoes without the need to model an asymmetrical geometry explicitly. Finally, the performance of three speech and audio processing algorithms is shown to be sensitive to strong sweeping echoes, thus highlighting the need to avoid their occurrence.

Code - Randomized Image Method

function h=rim(mi, so, ro, be, Np, Nr, Tw, Fc)
% mi (microphone), so (source) and ro (room) are three-dimensional column vectors.
% Np: samples of the RIR.
% Nr: no. of random samples (Nr=0 for original IM).
% Tw: samples of low-pass filter, Fc: cut-off freq.
% All quantities above are in sample periods.
% be: matrix of refl. coeff. [x1,y1,z1;x2,y2,z2]
h=zeros(Np,1); ps=perm([0,1],[0,1],[0,1]);
Rps=repmat(so,[1,8])+(2.*ps-1).*repmat(mi,[1,8]);
or=floor(Np./(ro.*2))+1;
rs=perm(-or(1):or(1),-or(2):or(2),-or(3):or(3));
for i=1:size(rs'); r=rs(:, i);
 for j=1:8; p=ps(:,j); Rp=Rps(:,j);
  d=norm(2*ro.*r+Rp)+1+Nr*(2*rand-1);
  if round(d)>Np || round(d)<1; continue; end
  am=be(1,:)'.^abs(r+p).*be(2,:)'.^abs(r);
  if Tw==0; n=round(d); else
    n=(max(ceil(d-Tw/2),1):min(floor(d+Tw/2),Np))';
  end
  s=(1+cos(2*pi*(n-d)/Tw)).*sinc(Fc*(n-d))/2;
  s(isnan(s))=1; h(n)=h(n)+s*prod(am)/(4*pi*(d-1));
end; end;
function res=perm(varargin)
[res{1:nargin}]=ndgrid(varargin{1:nargin});
res=reshape(cat(nargin+1,res{:}),[],nargin)';

Figure 2a - Highly regular setup with Image Method

Room dimensions: 4 m, 4 m, 4 m
Source position: 1 m, 2 m, 2 m
Microphone position: 2 m, 1.5 m, 1 m

Room Impulse Response (RIR):

RIR convolved with bongos1:

h=rim([2;1.5;1]/343*4E4, [1;2;2]/343*4E4, [4;4;4]/343*4E4, 0.93.*ones(2,3), 4E4, 0, 40, 0.9);

Figure 11a - Highly regular setup with Randomized Image Method

Room dimensions: 4 m, 4 m, 4 m
Source position: 1 m, 2 m, 2 m
Microphone position: 2 m, 1.5 m, 1 m

Room Impulse Response (RIR):

RIR convolved with bongos1:

h=rim([2;1.5;1]/343*4E4, [1;2;2]/343*4E4, [4;4;4]/343*4E4, 0.93.*ones(2,3), 4E4, 0.08/343*4E4, 40, 0.9);

Figure 2c - Allen and Berkley setup with Image Method

Room dimensions: 3.43 m, 5.145 m, 4.2875 m
Source position: 1.28625 m, 4.2875 m, 1.715 m
Microphone position: 2.14375 m, 0.42875 m, 2.5725 m

Room Impulse Response (RIR):

RIR convolved with bongos1:

h=rim([5;1;6]*343/800/343*4E4, [3;10;4]*343/800/343*4E4, [8;12;10]*343/800/343*4E4,...
  0.93.*ones(2,3), 4E4, 0, 40, 0.9);

Figure 2d - Irregular setup with Image Method

Room dimensions: 4.1 m, 4.2 m, 4.3 m
Source position: 1.4 m, 2.5 m, 2.6 m
Microphone position: 2.7 m, 1.8 m, 1.9 m

Room Impulse Response (RIR):

RIR convolved with bongos1:

h=rim([2.7;1.8;1.9]/343*4E4, [1.4;2.5;2.6]/343*4E4, [4.1;4.2;4.3]/343*4E4,...
  0.93.*ones(2,3), 4E4, 0, 40, 0.9);

Figure 2b - Highly regular setup with FDTD

Room dimensions: 4 m, 4 m, 4 m
Source position: 1 m, 2 m, 2 m
Microphone position: 2 m, 1.5 m, 1 m

Room Impulse Response (RIR):

RIR convolved with bongos1:

RIR after low-pass filter with 9 kHz cutoff:

RIR convolved with bongos1 after low-pass filter with 9 kHz cutoff:

Figure 10b - Slightly distorted cube and highly regular setup with FDTD


Positions of the 8 angles:
1 -0.020000 -0.020000 0.0000000
2 4.0200000 -0.020000 0.0000000
3 4.0200000 4.0200000 0.0000000
4 -0.020000 4.0200000 0.0000000
5 0.0200000 0.0200000 4.0200000
6 3.9800000 0.0200000 3.9800000
7 3.9800000 3.9800000 3.9800000
8 0.0200000 3.9800000 4.0200000
Source position: 1 m, 2 m, 2 m
Microphone position: 2 m, 1.5 m, 1 m

Room Impulse Response (RIR):

RIR convolved with bongos1:

RIR after low-pass filter with 9 kHz cutoff:

RIR convolved with bongos1 after low-pass filter with 9 kHz cutoff:

Acknowledgments

1. First 5 seconds of track #26 of Bang & Olufsen CD "Music for Archimedes". All rights of this anechoic sample belong to B&O. The authors would like to thank Søren Bech and B&O for allowing reproduction here.

This research work was carried out at the ESAT Laboratory of KU Leuven, in the frame of (i) the FP7-PEOPLE Marie Curie Initial Training Network "Dereverberation and Reverberation of Audio, Music, and Speech (DREAMS)", funded by the European Commission under Grant Agreement no. 316969, (ii) KU Leuven Research Council CoE PFV/10/002 (OPTEC), (iii) Interuniversity Attractive Poles Programme initiated by the Belgian Science Policy Office IUAP P7/19 Dynamical systems control and opti- mization (DYSCO) 2012-2017, (iv) and was supported by a Postdoctoral Fellowship of the Research Foundation Flanders (FWO-Vlaanderen). The scientific responsibility is assumed by its authors.