Ambisonic data exchange formats

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

Y

ℓ

m

(

θ

,

ϕ

)

=

N

ℓ

|

m

|

P

ℓ

|

m

|

(

sin

⁡

ϕ

)

⋅

{

sin

⁡

(

−

m

θ

)

if

m

<

0

,

cos

⁡

(

m

θ

)

if

m

≥

0

,

{\displaystyle Y_{\ell }^{m}(\theta ,\phi )=N_{\ell }^{|m|}P_{\ell }^{|m|}(\sin \phi )\cdot {\begin{cases}{\sin(-m\theta )}&{\text{if }}m<0,\\{\cos(m\theta )}&{\text{if }}m\geq 0,\end{cases}}}

where

Y

{\displaystyle Y}

denotes a spherical harmonic of degree

ℓ

{\displaystyle \ell }

and index

m

{\displaystyle m}

with a range of

−

ℓ

≤

m

≤

+

ℓ

{\displaystyle -\ell \leq m\leq +\ell }

.

(Note that if

m

=

0

{\displaystyle m=0}

, then

cos

⁡

m

θ

=

cos

⁡

0

=

1

{\displaystyle \cos m\theta =\cos 0=1}

.)

N

{\displaystyle N}

is a normalisation factor (see below), and

P

ℓ

m

{\displaystyle P_{\ell }^{m}}

is the associated Legendre polynomial of degree

ℓ

{\displaystyle \ell }

and order

m

{\displaystyle m}

.

The azimuth angle

θ

{\displaystyle \theta }

is zero straight ahead and increases counter-clockwise. The elevation angle

ϕ

{\displaystyle \phi }

is zero on the horizontal plane and positive in the upper hemisphere.