Reverberation prediction using the Image Source Method

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1 Reverberation prediction using Image Source Method DESC9115 Lab Report 2 Alex Tu Introduction The image source model technique (ISM) is a commonly utilised tool in acoustics and signal processing industry which has a wide range of applications in acoustic engineering source separation, reverberation prediction, acoustic source localisation, speech intelligibility and enhancement and many more. Different approaches to ISM produces different room impulse responses (RIRs) some of which are more indicative of a real time measured impulse response for same room. The image source method allows calculation of sound behaviour using ray propagation assumptions, travelling directly from source to receiver but also indirectly, reflecting specularly. It can n be extrapolated as sound reaching receiver from two sources, second source being image source behind mirror. The image source position is calculated from source position and reflector position and angle. The straight line distance between image source and receiver contains information required to model actual reflected sound path. Each reflective surface itself produces an image source and a second order image source is produced by combined reflection off two surfaces giving us four effective sources instead of one. This can be extended to a desired number of image sources, greater number of image sources greater degree of accuracy in reverberation time estimate calculation. Function A detailed explanation of function code is as follows: function [h]=my_ism(fs,mic,n,wallrefl,rm,src) Generates a room impulse response(rir) calculation by means of image method. This code is a modification of a prior existing code which calculates an RIR with assumption of all reflection coefficients from all enclosing boundaries are equal. This modified code allows input of individual reflection coefficients for each boundary to more accurately calculate RIR and implements a phase inversion for each sound reflection off each boundary which results in more accurate representation of true RIRs recorded in real world acoustic environments. % Input Arguments: % fs = sampling rate. % mic = row vector giving x, y, z coordinates of microphone position (m). % n = This function will calculate (2*n+1)^3 virtual sources, n being number of image sources to + or direction of original source. % wallrefl = row vector [x1,x2,y1,y2,z1,z2] reflection coefficients for walls, for this function only positive values of r (0<r<1) are assumed. % rm = row vector giving x, y, z dimensions of room (m). % src = row vector giving x, y, z coordinates of sound source (m). An example of input arguments is as follows: % fs=44100; % mic=[ ]; % n=128; % wallrefl=[ ]; % rm=[ ];

2 % src=[ ]; All distances and coordinate positions are in metres. The output is scaled such that largest value of absolute value of output vector is equal to one. nn=-n:1:n; The index for sequence: a row vector from 'n'th virtual source to 'n'th virtual source wheree 0 is real source. Part of equation for coordinate position (x, y or z) Part of equation for coordinate position (x, y or z) Figure 1 Equation for x coordinate for a virtual source. The function code replaces 'i'th position with nnn for 3D calculation. The rms argument represents component inside [] and srcss is first number. Y and Z coordinates found using same equation. xi=srcs*src(1)+rms*rm(1)-mic(1); Distance between 'i'th virtual source and mic position in x axis. This is equation shown in Figure 1 above. yj=srcs*src(2)+rms*rm(2)-mic(2); Distance between 'j'th virtual source and mic position in y axis. This is equation shown in Figure 1 above. zk=srcs*src(3)+rms*rm(3)-mic(3); Distance between 'k'th virtual source and mic position in z axis. This is equation shown in Figure 1 above. [i,j,k] =meshgrid( (xi,yj,zk); Converts vectors to 3D matrices for calculation of 3D virtual sources. d=sqrt( (i.^2+j.^2+ +k.^2); time=round(fs*d/343)+1; Straight line distance calculation to each virtual source using Pythagoras' orem. Time domain IR function of each virtual source. Figure 2 t is time, d ijk is distance found via Pythagoras' Theorem above and c is speed of sound, for this function default at 343m/s Figure 3 The unit impulse response function for each virtual source [e,f,g] =meshgrid( (nn, nn, nn); wallrefl = -abs(wallrefl); Converts vectors to 3D matrices for calculation of 3D virtual sources. Implementation of phase inversion as per Lehmannn & Johansson's ISM structure. rms=nn *(-1).^nn; srcs=(-1).^(nn); Combined reflection x axis: rx=wallrefl(1).^( (abs(0.5.*e *(-1).^e)).*wallrefl(2).^(abs(0.5.*e *( (-1).^e)); Figure 2 Equation to calculate combined reflection coefficient of surfaces in x axis. The combined reflection coefficients for Y and Z axes are also calculated using this equation Combined reflection y axis: ry=wallrefl(3).^( (abs(0.5.*f *(-1).^f)).*wallrefl(4).^(abs(0.5.*f *( (-1).^f));

3 Combined reflection z axis: rz=wallrefl(5).^( (abs(0.5.*g *(-1).^g)).*wallrefl(6).^(abs(0.5.*g *( (-1).^g)); c=rx.*ry.*rz; ee=c./d; Total reflection coefficient for every virtual source Total magnitude of each echo, as distance sound travels from source to receiver is proportional to 1/ /d total magnitude of each echo is equal to combined total reflection coefficient multiplied by 1/d. h=full( (sparse(time(:),1,ee(:))); Impulse Response: Total magnitude multiplied by unit impulse response function and sum over all indices. h=h/max(abs(h)); Figure 3 Equation for calculation of impulse response Scale outpu Discussion The ISM structure used in my function my_ism.m is based on work of McGovern, S. whichh itself is based on Allen & Berkley "Image method for efficiently simulating small roomm acoustics' method of obtaining RIRs. This model produces an impulse response practically identical to Allen & Berkley method barring a uniform differencee in magnitude and tail errors due to omission of some virtual sources which is overcome by finding a longer impulse response and truncating tail end. This model is useful as calculations and coding are relatively simple compared to many or ISM structures and hence computation time and processing capacity are much lower. The trade off is accuracy of measurement and function outlined for this paper is a modification of McGovern's code which allows for six user defined surface reflection coefficients instead of a singular value for all six surfaces and implementation of a phase inversion to each sound reflection on room surfacess which results in a much more realistic impulse response shape than Allen & Berkley method. Anor ISM structure by Lehmannn and Johansson using fractional delays to represent exact non integer time delays has a much more complex function code giving greater accuracy in generated RIR but for greater reverberation times and virtual sources computation time and processing power required becomes quite large. The following figures are generated using input arguments shown in function explanation above. The RIR generated by McGovern's code structure shown in Figure 6 shows only positive values of RIR (Allen & Berkley's method shows in practicality near identical results with negligible Figure 6 RIR generated by function Version Copyright 2003 Stephen G. McGovern Figure 7 Reverse Integrated Schroeder Decay Curve of McGovern's RIR

4 negative values that as a whole does not reflect a real RIR) which is problematic as a real life recorded RIR has a much more balanced contribution from each image source at receiver/microphone hence has more balanced positive and negative values. The Schroeder decay curve of this RIR (Figure 7) shows a slightly staggered decay time which is a rough approximation of expected decay time in a real room. The function my_ism.m generates a RIR shown in Figure 8 below which exhibits a shape much more alike to real time measured RIRs with positive and negative values, showing balanced contribution of each image source at receiver. The Schroeder decay curve (Figure 9) for this RIR shows a similar curve shape to McGovern RIR with a near identical decay up to 5.96ms but a reduced level decay rate, a 3.2dB difference at 23.65ms. The rate of decay is near constant in McGovern model up to 40ms but for my_ism.m decay rate from 23.65ms to 34.06ms is much greater than McGovern's and holds near level at 42.15dB up to 52.54ms n falls rapidly to 60dB at 62.27ms. The McGovern RIR from 40ms to 52.54ms falls approx. 2.4dB n falls to around 60dB at 57.47ms. The more accurate but slower Lehmann and Johansson method generates RIR and Figure 8 RIR generated by function my_ism.m Figure 9 Reverse Integrated Schroeder Decay Curve of my_ism.m generated RIR decay curve shown in Figure 10 and 11 respectively, as can be seen impulse response shape matches closely with my_ism.m with greater energy and datapoints at impulse point due to use of fractional delays to account for non integer time delays. The early and late reflections all match closely in time, though a different scaling factor was used in Lehmann and Johansson method resulting in different amplitude scale. The Schroeder decay curve is nearly identical across all time and level. Figure 10 RIR generated by Lehmann and Johansson's ISM structure Figure 11 Reverse Integrated Schroeder Decay Curve for Lehmann and Johansson's RIR Furr Development As it currently stands function my_ism.m generates an impulse response that greatly improves on Allen and Berkley method in producing a more realistic RIR with nearly equal computation time and utilises a vastly simpler, more concise code and noticeably shorter computation time compared to Lehmann and Johansson method.

5 Refinement to each image source impulse to include non integer time delays could be achieved by introducing a sinc function to each image source impulse, though added computation time this operation would require is untested and may not be worth benefit in accuracy gained. References Allen, J and Berkley, D. 'Image Method for efficiently simulating small room acoustics'. The Journal of Acoustical Society of America, Vol 65, No.4, pp , 1978 McGovern, S. 'A Model for Room Acoustics', audio.com/research/rir/rir.html, Code used as basis: audio.com/research/rir/rir.m Lehmann, E and Johansson, A. 'Diffuse Reverberation Model for Efficient Image Source Simulation of Room Impulse Responses,' IEEE Transactions on Audio, Speech and Language Processing, Vol. 18, No.6, pp , August Lehmann, E and Johansson, A. 'Prediction of Energy Decay in Room Impulse Responses Simulated with an Image Source Model', Journal of Acoustical Society of America, Vol. 124, nr.1, pp , July 2008.

Room Acoustics. CMSC 828D / Spring 2006 Lecture 20

Room Acoustics. CMSC 828D / Spring 2006 Lecture 20 Room Acoustics CMSC 828D / Spring 2006 Lecture 20 Lecture Plan Room acoustics basics Structure of room impulse response Characterization of room acoustics Modeling of reverberant response Basics All our