An Alternative Ambisonics Formulation:

Modal Source Strength Matching and the Effect of Spatial Aliasing

Franz ZotterHannes PombergerMatthias Frank

Institute of Electronic Music and AcousticsUniversity of Music and Performing Arts, Graz

Classical Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation)

Superposition of a weighted continuum of plane wave directions at one point is capable of reproducing any incident field

Angular band-limitation by representing plane-wave directions (delta functions) in spherical harmonics truncating order

Discretization of plane-wave directions by loudspeaker positions

Mode-matching of a variable (finite-order) desired field with band-limited finite-order discrete loudspeaker fields (Decoder)

Angular band-limitation

DiscretizationComparison (Decoding)

ls1 ls2

ls3ls4

http://iem.at/Members/zotter

Classical Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation)

Superposition of a weighted continuum of plane wave directions at one point is capable of reproducing any incident field

Angular band-limitation by representing plane-wave directions (delta functions) in spherical harmonics truncating order

Discretization of plane-wave directions by loudspeaker positions

Mode-matching of a variable (finite-order) desired field with band-limited finite-order discrete loudspeaker fields (Decoder)

Angular band-limitation

DiscretizationComparison (Decoding)

ls1 ls2

ls3ls4

http://iem.at/Members/zotter

Classical Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation)

Teaching problems: Non-temporal spectral representation in spherical harmonics How to imagine sound pressure of plane-wave directions encoded

into spherical harmonics ? Decoding and angular band-limitation

[thanks to Markus Noisternig for his teaching experience]

Angular band-limitation

DiscretizationComparison (Decoding)

ls1 ls2

ls3ls4

http://iem.at/Members/zotter

Classical Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation)

What is spatial aliasing and what is the sweet-spot ? Do loudspeakers create plane waves ? What about ill-posed frequencies in mode-matching ?

Angular band-limitation

DiscretizationComparison (Decoding)

ls1 ls2

ls3ls4

http://iem.at/Members/zotter

Compact Spherical Loudspeaker Arrays for Radiation Synthesis

(or: Motivation for reformulation of Ambisonics ? )

Boundary condition of sound particle velocity (surface) is capable of reproducing any radiating field

Angular band-limitation by encoding loudspeaker apertures into finite-order spherical harmonics series

Discretization of the boundary by loudspeakers Mode-matching of a variable finite-order desired pattern at the surface with band-limited

finite-order loudspeaker fields (Decoder) Radial focusing of radiation pattern feasible (SH: base-solutions of wave-equation) High orders are suppressed in radiation [Zotter07], the model is accurate [Meyer00,

Pollow07] !

[Zotter, Sontacchi, Höldrich: Modeling a Spherical Loudspeaker System as Multipole Source, DAGA 2007]

smoothing of sound radiation

spherical base solutions of the wave-equation

irradiation: radiation:

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Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Does radiation synthesis work similarly inwards ?

Is a boundary condition the correct way of describing Ambisonics ?

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Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Does radiation synthesis work similarly inwards ?

Is a boundary condition the correct way of describing Ambisonics ?

What about spherically distributed point sources ?- no obstacle/reflection- fair model of loudspeakers

there are existing models ![J. Ahrens, S. Spors, AES 2008; simple-source approach][F. Fazi et al, AES 2008; single-layer potential]

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Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

spherically distributed point sources

phi … source strength distribution Continuous source strength is capable of reproducing any irradiating field

(unlike boundary condition: radiation only !)

Discretization of sphere by loudspeaker placement

Angular band-limitation by encoding the discretized source strength (delta-function) into spherical harmonics

Modal source-strength matching of the loudspeaker fields with variable desired field (Decoder)

Radial focusing of reproduced source strength feasible (SH: base-solutions of wave-equation)

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Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Infinite-order source distributionof one source

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Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Angular band-limitation: finite resolution source strength

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Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Angular band-limitation: finite resolution source strength

r

r0

http://iem.at/Members/zotter

Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Angular band-limitation: finite resolution source strength

r

r0

12

34

http://iem.at/Members/zotter

Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Discretizationspatial Aliasing

r

r0

r

r0

http://iem.at/Members/zotter

Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

Discretizationspatial Aliasing

truncation

r

r0

http://iem.at/Members/zotter

Alternative Ambisonics Formulation(Large Spherical Loudspeaker Arrays for Irradiation Synthesis)

http://iem.at/Members/zotter

http://iem.at/Members/zotter

Distance Coding (Virutal Resizing of Playback Facility)

Questions?http://iem.at/Members/zotter

First Ambisonics Symposium 25th – 27th June 2009http://ambisonics.iem.at/symposium2009