Ambisonic Rotation Matrices

This document assumes a knowledge of matrices.

Interpreting the Matrices

Frame of Reference

Three nine-by-nine matrices are presented below. They correspond to rotation by a chosen angle around the X, Y and Z axes.

To explore how coordinates are moved by the matrices, consider the three-by-three submatrix surrounded by lines near the top-left of each of the full matrices. This can be applied to sound location vectors to find out how sounds are moved during rotation.

For instance, consider a sound initially located at <x,y,z> in the soundfield. During rotation around the Z axis when the angle is A, the location will change to <cos(A)x-sin(A)y,sin(A)x+cos(A)y,z>. This corresponds to horizontal rotation in an anticlockwise direction. The full matrix will rotate an entire soundfield horizontally.

Note that the matrices can be multiplied together to form compound rotation transformations.

First Order Rotations

First order rotation can be applied to a conventional B-Format encoded signal using WXYZ channel ordering. For this, only the four-by-four submatrices at the top left of the full matrices should be used.

Second Order Rotations

Second order rotation can be applied to a Furse-Malham Higher Order Format signal. The full matrices should be used. If the additional tenth W' channel is present this, like W, should pass through the transformation untouched.

The Matrices