Prefiltering the wave field synthesis operators - anti-aliasing and - PowerPoint PPT Presentation

Prefiltering the wave field synthesis operators - anti-aliasing and source directivity Gergely Firtha, P´ eter Fiala firtha@hit.bme.hu Budapest University of Technology and Economics ISMA2012 September 17, 2012

Introduction Theory Research Applications Conclusion Outline ◮ Introduction ◮ Motivation ◮ Theory of sound field reproduction ◮ Wave field synthesis (WFS) ◮ Spectral division method (SDM) ◮ Research activity ◮ Comparison fo WFS and SDM driving functions ◮ Effects of linear filtering of the synthesis operators ◮ Applications of linear filtering: ◮ new method for synthesizing directive sources ◮ proper filter design to avoid spatial aliasing ◮ Conclusion G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Motivation Synthesis of a Problem with stereophonic techniques: virtual monopole ◮ Perfect surround sound only at the sweet spot Aim of sound field reconstruction: ◮ Physically recreating virtual wave fields, y → [m] wave fronts with properly driven densely spaced loudspeakers Applications: ◮ Entertainment (Cinemas, sound enhancement in theatres) ◮ Active noise control x → [m] G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Motivation Phenomena, described as filtering problems physically arise from the theory of sound field reconstruction Effects of linear filtering of the loudspeaker driving functions have not been investigated Objectives: ◮ Examine the effects of filtering on the reconstructed sound field ◮ Give a utilizeable physical interpretation for linear filtering ◮ Utilize the presented technique for improved synthesis of directional sources ◮ Find optimal filter design for anti-aliasing filtering G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Theory of sound field reconstruction z y Virtual source Reference line Listener y s y ref x : Elements of secondary source distribution General sound field reconstruction problem: � ∞ ! P ( x , ω ) synth = Q ( x 0 , ω ) G ( x | x 0 , ω ) d x = P ( x , ω ) virtual −∞ G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Wave field synthesis ◮ Derivation of driving functions x in spatial domain V n ( x ) ◮ Rayleigh integrals: Sound field of a sound source via boundary conditions, in case of linear boundary ↓ Primary Rayleigh I. contains explicitly source the driving functions ◮ Approximations → perfect y synthesis only on reference line � ∞ P ( x , ω ) primary = − j ωρ 0 V n ( x 0 , ω ) G ( x | x 0 , ω ) d x � �� � −∞ Q ( x 0 ,ω ) G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Spectral Division Method ◮ Derivation of driving functions in spectral domain F ◮ Fourier-transform along x -axis: x − → k x = k sin ϕ ◮ The general reconstruction integral, written along the reference line represents a convolution along the synthesis line: � ∞ P ( x , y ref ) = Q ( x 0 ) G 3 D ( x − x 0 , y ref ) d x = Q ( x ) ∗ G 3 D ( x , y ref ) −∞ ◮ If spectra of the virtual sound field and secondary monopole known on reference line: ˜ Q ( k x ) = F ( P ( x , y ref )) P ( k x , y ref ) ˜ F ( G ( x , y ref )) = ˜ G ( k x , y ref ) G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Comparison of WFS and SDM driving functions Result: in the far-field of monopole ( kr ≫ 1) WFS ≈ SDM F − 1 ( Q SDM ( k x , ω )) ≈ Q WFS ( x , ω ) = Q ( x ) G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Linear filtering of the driving functions Linear filter is defined with it impulse response h ( x ) ( h ( x )) or transfer function (˜ H ( k x )) x Conclusions: ◮ Filtering Q ( x ) ≡ synthesis with extended secondary sources Virtual source Secondary source distribution G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Linear filtering of the driving functions Linear filter is defined with it impulse response h ( x ) ( h ( x )) or transfer function (˜ H ( k x )) x Conclusions: ◮ Filtering Q ( x ) ≡ synthesis with extended secondary sources ◮ Filtering Q ( x ) ≡ synthesis of the field of an extended virtual source Virtual source Secondary source distribution G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Linear filtering of the driving functions Linear filter is defined with it impulse response h ( x ) ( h ( x )) or transfer function (˜ H ( k x )) x Conclusions: ◮ Filtering Q ( x ) ≡ synthesis with extended secondary sources ◮ Filtering Q ( x ) ≡ synthesis of the field of an extended virtual source ◮ Virtual source extension and secondary source extension are interchangeable ◮ Extension of virtual, or secondary source Virtual source elements is defined by the filter impulse Secondary response source distribution G. Firtha Prefiltering the wave field synthesis operators

0.4 300 0.8 1 30 210 60 240 90 270 120 150 0.2 330 180 0 -k k -1 0 1 0 0.6 Introduction Theory Research Applications Conclusion Application: Synthesizing directive sources D ( ϕ ) ◮ Spatial extension (normal velocity) of the virtual source is given → impulse response of the filter is defined explicitly V n ( x S ) ∝ h ( x ) ◮ In the far-field the extended source seems as a directive point source: k x = k sin ϕ ˜ H ( k x ) P ( x ) = D ( ϕ ) G ( x | x s ) ∝ ˜ V n ( k x ) G ( x | x s ) ◮ Directivity function of the virtual source is given → transfer function of the filter is defined explicitly: k x = k sin ϕ → ˜ D ( ϕ ) − − − − − − H ( k x ) k x G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Example: synthesizing a quadrupole ◮ Directivity function is given: D ( ϕ ) = cos (2 ϕ ) � � ◮ Transfer function of the linear filter: ˜ 2 arcsin k x H ( k x ) = cos k ◮ Impulse response of the filter: h ( x ) = δ ( x ) − 2 δ ′′ ( x ) k 2 ◮ The proposed (lineary filtered) driving functions: Q quad ( x ) = h ( x ) ∗ Q mono ( x ) = Q mono ( x ) − 2 k 2 Q ′′ mono ( x ) ◮ Traditional WFS driving functions for a virtual quadrupole: Q trad quad ( x ) = D ( ϕ ) Q mono ( x ) G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Example: synthesizing a quadrupole Original Synthesis using Synthesis using field traditional WFS proposed method 1 3 3 3 0 . 5 2 2 2 x [m] 0 1 1 1 − 0 . 5 reference reference line line 0 0 0 − 1 0 1 1 . 5 0 1 1 . 5 0 1 1 . 5 [Pa] y [m] G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Example: synthesizing a quadrupole Cross−section along the reference line 2 Original field Field using linear filtering Field using traditional WFS 1.5 1 Pressure −> [Pa] 0.5 0 −0.5 −1 0 0.5 1 1.5 2 2.5 3 x −> [m] ◮ Traditional synthesis: great amplitude and phase errors ◮ Proposed method: only slight amplitude errors G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Effects of spatial sampling Spectrum of sampled driving function m ] k ← [ rad Spectrum of Green-function − 2 π 2 π − k x , Nyq 0 k x , Nyq dx dx k x → [ rad m ] ◮ Continuous source distribution → discrete loudspeakers ≡ spatial sampling of the secondary distribution ◮ Mathematical model: applying sampled driving function G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Avoiding spatial aliasing ◮ Effect of aliasing: undesired plane wave components Spatial-bandlimiting is needed ◮ Reproduction filter: ◮ Spatial low pass filtering by the secondary source distribution ◮ Directive source elements act as spatial low pass filter ◮ Anti-aliasing filter: ◮ Pre-filtering of Q ( x ) is needed ◮ Proper low-pass filter design is possible G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Avoiding spatial aliasing ◮ Assuming ideal stop band attenuation: The virtual source extension is defined by the resultant of anti-aliasing and reconstruction filters ◮ Filter design considerations for a virtual monopole: ◮ The resultant impulse response should be a Dirac-delta ◮ Narrow impulse response → low stop-band attenuation ◮ Sharp transition on cut-off k x → wide virtual source ◮ Compromise is needed G. Firtha Prefiltering the wave field synthesis operators

Introduction Theory Research Applications Conclusion Examples for anti-aliasing filtering Aim: synthesize a virtual monopole on extended area, applying discrete sources Ideal LP filter Hann filter Chebyshev filter x ← [m] y → [ m ] h ( x ) x x x G. Firtha Prefiltering the wave field synthesis operators