Data exchange formats for Ambisonics have undergone radical changes since the early days of four-track magnetic tape. Researchers working on very high-order systems found no straightforward way to extend the traditional formats to suit their needs. Furthermore, there was no widely accepted formulation of spherical harmonics for acoustics, so one was borrowed from chemistry, quantum mechanics, computer graphics, or other fields, each of which had subtly different conventions. This led to an unfortunate proliferation of mutually incompatible ad hoc formats and much head-scratching.

This page attempts to document the different existing formats, their rationales and history, for the terminally curious and those unfortunate enough to have to deal with them in detail.

Most modern applications use ACN and SN3D, although traditional first order is still common.

Spherical harmonics in Ambisonics A common formulation for spherical harmonics in the context of Ambisonics is

where denotes a spherical harmonic of degree and index with a range of .

(Note that if , then .)

is a normalisation factor (see below), and is the associated Legendre polynomial of degree and order . The azimuth angle is zero straight ahead and increases counter-clockwise. The elevation angle is zero on the horizontal plane and positive in the upper hemisphere.

Unfortunately, the "Ambisonic order" is called the degree in mathematical parlance, which uses order for the "Ambisonic index" .

Relationship of spherical harmonics and B-format signals

For a source signal in direction , the Ambisonic components are given by

If we span a direction vector from the origin towards the source until it intersects the respective spherical harmonic, the length of this vector is the coefficient that gets multiplied with the source signal. Repeat for all spherical harmonics up to the desired Ambisonic order.

Prerequisites for successful data exchange

For successful exchange of Ambisonic material, some software requires the sender and receiver have to agree on the ordering of the components, their normalisation or weighting, and the relative polarity of the harmonics.

Since it is possible to omit parts of the spherical harmonic multipole expansion for content that has non-uniform, direction-dependent resolution (known as mixed-order), it might also be necessary to define how to deal with missing components.

In the case of transmission "by wire", be it an actual digital multichannel link or any number of virtual patchcords within an audio processing engine, these properties must be explicitly matched on both ends, since there is usually no provision for metadata exchange and parameter negotiation. In the case of files, some flexibility might be possible, depending on the file format and the expressiveness of its metadata set.

However, in practice, just two formats are in widespread use. The first is Furse-Malham higher-order format, which is an extension of traditional B-Format, and the more modern SN3D, in ACN channel order. In neither case is there any ambiguity about ordering, normalisation, weighting or polarity and it is rare to see cases with missing components. A third format is in limited use: N3D, also in ACN channel order.

Component ordering

The traditional B-format () only concerned itself with zeroth and first Ambisonic order. Because of a strong correspondence between the spherical harmonics and microphone polar patterns, and the fact that those polar patterns have clearly defined directions, it seemed natural to order and name the components in the same way as the axes of a right-hand coordinate system.

For higher orders, this precedent becomes awkward, because spherical harmonics are most intuitively arranged in symmetric fashion around the single z-rotationally symmetric member m=0 of each order, with the horizontal sine terms m<0 to the left, and the cosine terms m>0 to the right (see illustration).

Furse-Malham

In Furse-Malham higher-order format, an extension of traditional B-format up to third order, orders 2 () and 3 () begin with their z-rotationally symmetric member and then jump outward right and left (see table), with the horizontal components at the end. Higher-order extensions are trivially defined, but are not used.

SID 0 231 578641012141513119 In his seminal 2001 thesis,Jérôme Daniel, Représentation de champs acoustiques, application à la transmission et à la reproduction de scènes sonores complexes dans un contexte multimédia, Paris 2001, p.151 Daniel used a three-index nomenclature for the spherical harmonics, which corresponds to in the notation used here.sgn(x) is the Sign function. He implied yet another channel ordering, subsequently developed into an explicit proposal called SID for Single Index DesignationJérôme Daniel, Spatial Sound Encoding Including Near Field Effect: Introducing Distance Coding Filters and a Viable, New Ambisonic Format, 23rd AES Conference, Copenhagen 2003, p. 13 which was adopted by a number of researchers. This scheme is compatible with first-order B-format, and continues to traverse the higher spherical harmonics in the same fashion, with the z-rotationally symmetric component at the end, going through the horizontal components first. It is, however, incompatible with Furse-Malham. SID ordering is not in widespread use.

ACN 0 123 456789101112131415 For future higher-order systems, adoption of the Ambisonic Channel Number (ACN)Michael Chapman et al., A standard for interchange of Ambisonic signal sets, Ambisonics Symposium, Graz 2009 has reached wide consensus. It is determined algorithmically as .

ACN is used widely with SN3D and N3D, below.

More simply:

FuMa = WXYZ | RSTUV | KLMNOPQ

ACN = WYZX | VTRSU | QOMKLNP

SID = WXYZ | UVSTR | PQNOLMK

SID used in iem_ambi in Puredata.

Normalisation For successful reconstruction of the sound field, it is important to agree on a normalisation method for the spherical harmonic components. The following approaches are common:

maxN The maxN scheme by Daniel normalizes each single component to never exceed a gain of 1.0 for a panned monophonic source. Malham states that "[w]hilst this approach is not rigorously "correct" in mathematical terms, it has significant engineering advantages in that it restricts the maximum levels a panned mono source will generate in some of the higher-order channels." This property is particularly interesting for fixed-point digital interfaces. The maxN weights may be determined by visual inspection up to the third order; above this value the maxima of each polynomial need to be determined explicitly.

MaxN is used in the Furse-Malham format (with the exception of a -3dB correction factor for , which makes it directly compatible with traditional B-Format). Otherwise, it is not in widespread use.

SN3D SN3D stands for Schmidt semi-normalisation and is commonly used in geology and magnetics. The weighting coefficients areNachbar, Zotter, Deleflie, and Sontacchi (2011) lc, p.3, eq(3)

Originally introduced into Ambisonic use by Daniel, he notes: "High degree of generality - the encoding coefficients are recursively computable, and the first-order components are unity vectors in their respective directions of incidence".Daniel (2001), lc, p.156, translated from French "Grande généricité: calcul récursif des coefficients d'encodage, les composantes d'ordre 1 étant celles du vecteur incidence (unitaire) ."

With SN3D, unlike N3D, no component will ever exceed the peak value of the 0th order component for single point sources. This scheme has been adopted by the proposed AmbiX format.

SN3D (in the ACN channel order) is in widespread use and a common choice in new software development.

In the Ambix specification paper the term is replaced with .

N3D N3D or full three-D normalisation is the most obvious approach to normalisation. Daniel describes it as follows: "Orthonormal basis for 3D decomposition. Simple relationship to SN3D [..]. Ensures equal power of the encoded components in the case of a perfectly diffuse 3D field. [..] Obvious significance for solving decoding problems [..] (3D reconstruction)."Daniel (2001) lc, p.156, translated from French "Base orthonormée pour la décomposition 3D. Relation simple á SN3D (facteur ). Assure une puissance égale des composantes encodées dans le cas d'un champ parfaitement diffus 3D (intérêt dans le domaine analogique). Intérêt évident pour la résolution (en 3.3) des problèmes de décodage (restitution 3D)."

The relation to SN3D isDaniel (2001), lc, p.150 eq(3.9)

This normalisation is standard in physics and mathematics and is supported by some Ambisonic software packages. It is used in MPEG-H. However, SN3D is now much more common.

As N3D and SN3D differ only by scaling factors, care is needed when working with both, as it may not be obvious on first listening if an error has been made, particularly on a system with a small number of speakers.

N2D / SN2D Additionally, two schemes exist which consider only the horizontal components. This has practical advantages for fixed-point media in the common situation where sources are concentrated on the horizontal plane, but the normalisation is somewhat arbitrary and its assumptions do not hold for strongly diffuse soundfields and sound scenes with strong elevated sources. Since Ambisonics is meant to be isotropic and the 2D schemes definitely are not, their use is discouraged.

Polarity A third complication arises from the quantum mechanical formulation of spherical harmonics, which was adopted by some Ambisonics researchers. It includes a factor of , a convention called Condon–Shortley phase, which will invert the relative polarity of every other component within a given Ambisonic order. The term can be folded both into the formulation of the associated Legendre polynomials or the normalisation coefficient, so it may not always be obvious.

MATLAB and GNU Octave both include Condon–Shortley phase in its legendre(ℓ,X) functions, but undo it by applying the factor again in the Schmidt semi-normalized form legendre(ℓ,X,'sch').MathWorks documentation: legendreGNU Octave documentation: legendre

Wolfram Language also includes C-S phase in its legendreP(ℓ,X) implementation,Wolfram language documentation: LegendreP and retains it in SphericalHarmonicY[ℓ,m,θ,φ], which is fully normalized.Wolfram language documentation: SphericalHarmonicY Note that this function returns complex values and uses the physics convention for spherical coordinates where is the zenith angle (angle from the positive Z-axis) and is the azimuth (counter-clockwise angle around the positive Z-axis).

The presence of Condon–Shortley phase in parts of the signal chain usually manifests itself in erratic panning behaviour and increasing apparent source width when going to higher orders, which can be somewhat difficult to diagnose and much harder to eliminate. Hence, its use is strongly discouraged in the context of Ambisonics.

None of the ambisonic exchange formats described above use Condon–Shortley phase. Polarity is generally only a concern when trying to reconcile theoretical formulations of the spherical harmonics from other academic disciplines.

Reference table of layouts and normalisations

The following table gives an overview of all Ambisonic formats published so far. For Furse-Malham (and traditional B-format), sort by FuMa column and multiply the spherical harmonic by the maxN*MaxN* (starred) denotes maxN normalisation with the additional -3dB correction factor for W. factor. For SID according to Daniel, sort by SID and apply the relevant normalisation factor as specified in the file metadata. For basic AmbiX, sort by ACN and use the SN3D factor.

Conversion factors can be applied either to the Ambisonic components or the spherical harmonics .

The data is taken from Chapman (2008).Michael Chapman, Ambisonic channel sequence (proposed standard)