Sound Field Reproduction Peter Goss

Outline What is sound field reproduction? Free-field theory and simulation results Reverberant theory Implementation issues How to simulate reverberation Simulation results Future directions

Sound Field reproduction A sound field is the variation in pressure in a region Reproduction mimics an original sound field (say that of a point source) with a different array of speakers Malham 1998 Trueman 2000

Long distance: point  plane (1kHz below) 10m 20m 40m

Applications of Spatialisation Entertainment (surround sound) Virtual reality Teleconferencing Strickland 1997 Nintendo.com 2007

Other Systems: Dolby Used in most home theatre setups and cinemas Described by 5.1, etc. Speakers have to be in designated positions Shockwavesound 2008

Other Systems: WFS Wave field synthesis uses Huygen’s principle Requires a very large number of speakers Has been implemented in a cinema: IOSONO Dellers 2007

Ambisonics Similar to the technique used here Early versions only take first order expansion Generally assumes sources radiate plane waves Can’t have speaker close to listener

Binaural techniques Rely on pyscho- acoustics Control sound at each ear (usually with headphones) Specialization of sound due to two things – time delay + HRTF HRTF unique to each person

Why only 2D? 2D requires much less speakers Simpler to setup Theory is very close

MATHS WARNING! Disclaimer: the speaker does not take any legal responsibility for physical pain induced due to the following mathematics

SFR Theory Any sound field (steady state) must satisfy Helmholtz eq. Sol. in 2D is: Each unique sound field has a unique set of coefficients If we look at ‘incoming’ waves, then the Ys disappear

Bessel Functions Useful property is high-pass nature Can ignore higher terms

SFR Theory Any sound field (steady state) must satisfy Helmholtz eq. Sol. in 2D is: Each unique sound field has a unique set of coefficients If we look at ‘incoming’ waves, then the Ys disappear We can truncate

Setup Speakers arranged around a circular reproduction area: Speakers do not have to be in a circular arrangement We want to minimize reproduction error: Bethelem & Abhayapale 2005

Theory Pressure at each point is sum of pressure due to each speaker Assuming each speaker is an ideal point source: the transfer function is: H 0 (2) is the Hankel function of the second kind, n=0

Transfer function theory H can be expanded in the same way as the field: So:

Theory cont. The reproduction error is:

Matrix form:

Finally: This has its known global minimum at: So, by varying βd, we vary the reproduced sound field… theoretically To check, simulation code from David Excell’s Thesis was modified

Summary of method: 1.Define region (pick maximum radius - R) 2.Choose N by rounding up kR 3.Need at least L = 2N+1 speakers 4.Position speakers around region 5.Calculate transfer function coefficients for each speaker 6.Calculate desired coefficients 7.Use magical formula:

Simulation Results: Reproduction error: 2.45%Reproduction error: 6.82% Reprod. Radius = 0.3m, f = 1000Hz, speakers in circle at 2m, 0.3m -> N=6 -> 13 speakers

Simulation Results: Reproduction error: 8.35%Reproduction error: 0.58% Point sourcesPlane wave sources

Error as a function of radius

How reverberation affects the method The transfer function for each speaker is now unknown Will depend on room geometry, speaker position and speaker type (directivity etc.) Need some way of measuring these transfer functions

Reverberation theory Transfer functions are expanded as before: If these transfer coefficients, α, are measured then method is same as before

Measuring the transfer functions: Operate just 1 speaker at a time The field at each microphone can be given by their harmonic expansion: Microphones placed at the edge of the area

Measuring the transfer functions: Again representing in matrix form: Using the Moore- Penrose pseudo- inverse (†) to solve for α(l)

Summary of method: 1.Define region (pick maximum radius - R) 2.Choose N by rounding up kR 3.Need at least L = 2N+1 speakers 4.Position speakers around region 5.Measure transfer function coefficients for each speaker: * 6.Calculate desired coefficients 7.Use magical formula:

5.Measure transfer function coefficients for each speaker: 1.Operate just 1 speaker at a time 2.Place microphones at edge of region 3.Calculate D matrix based on (expected) positions of microphones 4.Calculate best coefficients using another magical formula:

Possible Issues in Implementing: Impracticality in ring of microphones in centre of room Inaccuracy in positions of speakers Inaccuracy in positions of microphones Microphones having different responses

How to simulate Reverberation Image-Source method: In optics, an image source can be placed at equal distance from a reflector: Same principle, except losing some energy (α) + 4 walls – so many image sources

Image-Source Method:

Image Source method data: num reflecB num sourcestotal N Using Number of sources quickly increases: Distance increase + weight decreases, so may cancel out number of sources

Effect of reverberation

Simulation setup: Used same settings as paper for comparison

Results!! Code done myself – with equations from Betlehem & Abhayapala + Allen & Berkely’s paper Simulating 5 reflections Reproduction good, but why 5 reflections? Varying the number of reflections get very odd results: Reproduction error: 8.35%

Plot of error against number of reflections considered:

Simulation setup:

Changing to central position in room:

Corner of room:

Speaker position uncertainty Vary position of speakers from known positions by an amount ~N(0,sigma^2), where sigma is varied Shouldn’t have effect at all, because the system does not need speaker positions Initial results showed no clear trend Repeating with same setting showed large variability in error So, running with a set sd of 5cm, and varying number of reflections, repeating each value 50 times

50 trials of each number of reflections: (note: log scale!) Reproduction Error (%) Number of reflections

Examining the outliers: Each outlier examined deviated only at edge Strongly suggests that it’s the higher basis functions that are wrong Weights given are very high – so suggesting it is the measuring of transfer functions Perhaps increasing N would help, or increasing the microphones error: 58189%

Comparison Reproduction Error (%)

Mean comparison: Reproduction Error (%) Number of reflections

Median Comparison Reproduction Error (%) Number of reflections

More work to be done: Not clear what this effect is due to – may just be a program error One theory is certain positions have very high N+1, N+2 coefficients, which cancel out the effect of the small bessels This needs further testing One possibility in implementing is ignoring higher orders due to large time delay – possibility to be looked at

Showing general effect of speaker pos uncertainty Choosing 10 reflections (with extra N) and varying uncertainty:

Means:

Microphone variability Varied microphone weight again by normal dist. – assuming zero bias Showed nice clear results, below is a 95% confidence interval for the mean:

Plans for future: Extend to Quasi-2D Look at unexpected speaker position results in more detail Effect of microphone position uncertainty Attempt to implement this system