First and Second Order Ambisonic Decoding Equations

This document contains Ambisonic decoding matrices for a number of speaker rigs for both first and second order signals and some discussion of the mathematics behind them. The second order format used is Furse-Malham Higher Order Format which provides a full 3D second order Ambisonic implementation.

This data is used by the MN Audio Library (MNLib) Ambisonic Player, Ambisonic Command-Line Decoder and MN Ambisonic Decoder Unit.

MNLib is a set of C++ libraries and programs for audio processing and music. MNLib is designed with portable C++ code for use on Linux, Microsoft platforms and others. MNLib, its documentation and this data are Copyright 1999-2000 Richard W.E. Furse (all rights reserved).

I've set up a company specialising in the field - Blue Ripple Sound.

The Encoding

Conventions

Directions are expressed using unit vectors. Such vectors can be expressed using an Angle/Elevation representation or a Cartesian representation. These are related by x=cos(A)cos(E), y=sin(A)cos(E), z=sin(E).

Note that in both encoding formats described below, the individual channels are orthogonal but not orthonormal.

The first order Ambisonic encoding format (B-Format) uses the following channels:

The second order Ambisonic encoding (Furse-Malham Higher Order Format) uses the following channels:

Justification of the Furse-Malham Higher Order Format

The second order components are related to the formal complex set (defined in terms of associated Legrendre polynomials). Working for the standard formulation for this and taking x=cos(PHI)sin(THETA), y=sin(PHI)sin(THETA) and z=cos(THETA), the relationship between the formal set and the Cartesian set above is given by:

Interpreting the Tables

For each rig an initial table provides an overview of the rig's behaviour, including its level of support for each second order spherical harmonic. The information is presented in the following ways:

Three further tables are provided containing the gains that should be appled to an encoded soundfield to produce each channel of a speaker feed.

The first of these tables provides the scalars that should be used to reproduce the soundfield in the idealised `matching' mathematical form of the problem. This is calculated by consideration of an idealised Ambisonic Microphone at the centre of a rig with an infinite number of speakers. These values are valid for both first and second order encodings.

The second and third tables contain `controlled opposite' formulations separately for first and second order. These forms guarantee that speakers will never generate an `anti-phase' signal (this is also known as `in-phase' decoding). Also, for second order cases the speaker response has only a single maximum. Below are coloured plots of the response of a speaker from a cubic rig. The plots indicates the strength of response for one speaker for different locations of a single sound source in the encoded soundfield. Source angle is horizontal and elevation vertical, green is used to indicate a positive response and cyan is used to indicate a negative `anti-phase' response. Note the faint cyan `anti-phase' band and second maximum in the `matching' plot.

Matching	Controlled Opposites

In general I find the `controlled opposite' equations produce a larger listening area at the expense of some directional information. Compromises can be struck by choosing intermediate scalars. In some of the MNLib programs this can be done using a `fudge factor.'

Decoding for Custom Rigs

The code that generates this web page is part of the MN Ambisonic Decoder Unit. This is also used by the MNLib Ambisonic Command-Line Decoder. This will attempt to find sensible decode matrices for many shapes of speaker rig.

Mono

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Stereo

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Square

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Pentagon

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Hexagon

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Octagon1

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Octagon2

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Surround1

Rig Overview

Note that the second order rig has a high symmetry distortion rating. Such a rig is likely to provide an unstable image.

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

Not provided for this rig configuration because of unusual characteristics.

Cube

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Dodecahedron1

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Dodecahedron2

Rig Overview

Rig Decode Matrix to Reproduce Spherical Harmonics (First Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig Decode Matrix to Reproduce Spherical Harmonics (Second Order)

This rig configuration produces a strict idealised response that satisfies the Ambisonic matching equations. Generally this type of configuration produces a relatively small stable listening area. (See controlled opposites below.)

Rig `Controlled Opposites' Decode Matrix (First Order)

This rig configuration produces an `in-phase' response.

Rig `Controlled Opposites' Decode Matrix (Second Order)

This rig configuration produces an `in-phase' response and each speaker will have only a single maximum given a moving virtual sound source.

Links

• Ambisonic Player Documentation
• General 3D Audio Information
• MN Library Main Index
• VSpace Introduction

The author Richard Furse can be emailed as richard@muse.demon.co.uk.

Thanks to Philip Cotterell for providing the dodecahedral speaker geometries used and for spotting a number of slips on this page. Any further errors are entirely my fault!

"Ambisonics" is a registered trademark of Nimbus Communications International.