B-Format Sequence for Ambisonic Channels


The order in which to present ambisonic channels has been the subject of various approaches. These pages present a standardised method, where each channel is given a unique number out of a set of consecutive integers, starting with 0 (zero).
Whilst ‘order’ is probably a more natural English usage than ‘sequence’, the latter term is used to avoid confusion with the ambisonic order (see below).

Basis

Ambisonics uses a spherical harmonic decomposition of the soundfield. The channels are grouped according to the ambisonic order. A zero order representation consists of solely one channel and is equivalent to a mono (though 360° (actually, omnidirectional)) rendition of the sound.
Each succeeding order (l), in three dimensions (periphony) adds 2l + 1 channels. So first order is 4 channels (1+3), second order is 9 channels (4+5), etc. (So the total number of channels is the order plus one, squared.)
For two dimensional representations (pantophony) then each extra order always adds two channels. (So the total number of channels is twice the order, plus one.)

Relation between channels and spherical harmonics

Each channel is related to a spherical harmonic. The notation for the harmonics varies between authors. Here we use Ylm, with l the order and m the range within that order. The range values are always such that: -m ≤ l ≤ +m.
Daniel [1] uses m for order, and n for the numerical part of the range, with a third value σ indicating the sign of the range. That is σ is either +1 or -1.
Malham [2] uses a similar notation, but uses ς instead of σ (both are the Greek letter sigma). (These two authors' notations vary also, slightly in that Daniel uses sub-/super-scripting, whilst Malham expresses the values linearly.)
The present system reduces the number of sub- and super-scripts and makes explicit that there is only one value for range = 0 (no ‘+0’ with ‘-0’ !).

Sequencing

The system for sequencing is taken from Green [3] (who also uses the l, m notation copied here), that is:

channel number = l ( l + 1) + m
which gives a series of consecutive unique numbers.

Concordance

Tables for translation between these channel numbers, the related spherical harmonics and the channel letters that have been used for lower order (0 to 3) ambisonic files is given of succeeding pages:

(The tables refer to the related spherical harmonic Y. Strictly, perhaps, we should have used B (with the same sub- and super-script) to refer to each B-channel. However ‘B-format’ is used both generically to refer to signal sets in general and for specific examples of such sets. (I.e. a set of signals with Furse-Malham weightings.) The question of what weightings should be applied, what normalization should be applied, is irrelevant for channel naming/numbering and is a subject left for a more general discussion of file formats. To avoid any ambiguity between B-format signal sets in general and specific examples of these then Y has been used.)

Additional Reading:
It is intended to release the final standard under an appropriate licence (that allows use and copying, but which prohibits changes (that would confuse the result with the original) and requires attribution).For the moment these pages are copyright © 2008 Michael Chapman. All rights reserved.