Towards a comprehensive account of valid ambisonic transformations

The classic ambisonic transformations of amplitude scaling, rotation and mirroring are well known. Their validity can be established by algebraic analysis of transformation matrices.

Gerzon & Barton[1] used such a technique to demonstrate the validity of the first-order 'dominance' transformation.

The search for a transformation similar to dominance but applicable to higher-order signal sets is ongoing. Cotterell[2] published a numerical proof that a second-order dominance operation corresponding to Gerzon & Barton's Lorentz transformation is impossible.

Chapman[3] inverted Gerzon and Barton's algebraic method to prove that the only valid first-order transformations are amplitude scaling, rotation, mirroring and dominance.

In this paper the authors generalise this latter approach and apply it beyond first-order pantophonic matrices, proposing a generalised and extensible approach to the search for ambisonic manipulations.

1. Gerzon, Michael A. & Geoffrey J. Barton. "Ambisonic Decoders for HDTV", Audio Engineering Society Preprint, from the 92nd Convention 1992 March 24-27, Vienna.
2. Cotterell, P.S. "On the Theory of the Second-Order Soundfield Microphone", Doctor of Philosophy Thesis, University of Reading, February 2002.
3. Chapman, Michael. "New Dimensions for Ambisonics", Audio Engineering Society Convention Paper 7478, from the 124th Convention 2008 May 17-20, Amsterdam