Ambisonic Channels
 
Notes on Spherical Harmonic Equations

Index page
Glossary

There are various possible ways of presenting equations for spherical harmonics. These notes try and explain the choices made here. Firstly as the interest is in ambisonic audio only, real (as against complex) equations are presented. There are plenty of sources of the complex variants in mathematical texts and on the Web (well at least for lower degrees).

Punctuation

is excessive. Especially the use of dots to represent multiplication. Also some brackets are rather gratuitous. It is felt that the user can easily remove any unnecessary punctuation. They have been left in, as the author uses a script that reads the published equations into a program, which then performs some elementary checks on them.
Editing some hundreds of equations by hand, after automated checking, for the moment was deemed an unnecessary step.

Notation

Whilst θ (theta) and φ (phi) are used in graphical representations of equations, for azimuth and elevation, in HTML (i.e. much of this page) the more cumbersome, but both easier to type and more certain to be rendered, A and E are used.

Simplification

Most sources simplify some of the results, upto third order. This convention has been followed here.
(E.g. The geometric parts of Y₅ and Y₇ can be calculated as sin(E).cos(E).sin(A) and sin(E).cos(E).cos(A) respectively. But as 2.sin(E).cos(E) = sin(2E), these are usually simplified to use sin(2E).)

Direction cosines

As a work in progress (at the time of writing, upto sixth degree) equations are presented in two alternative formats.

• The first is the classical, using geometric functions. This is the form produced when generating from the associated Legendre Function.
• In terms of direction cosines, where u_x=cos(E).cos(A), u_y=cos(E).sin(A), and u_z=sin(E).
  As the three terms are not independent (u_x² + u_y² + u_z² =1) there is no unique way of writing these. Their utility is described below.

For computation direction cosines have the utility that once the three values (u_x, u_y and u_z) have been calculated then all other values can be calculated without having to call geometric functions (only addition, subtraction and multiplication are required). Programs using such an approach are much faster. (The numerical parts of the equations do involve square roots, but the numerical part is a constant. Such constants can be pre-calculated and in the software or calculated once on installation, or when the program is called.)

Normalisation

It is not normal to show normalisation (normalization). Most texts use one normalisation throughout and thus it is implicit. It does need stating when discussing conversion from one to normalisation to another. Daniel's thesis (equation 3.19, modified) defines:
Y_ACN^(conv2) = α_ACN^(conv2)conv1.Y_ACN^(conv1)
where conv1 and conv2 are the conventions used (e.g. FuMa, N3D, etc., etc.).
In the equations on the pages here the first Y has the convention shown. This is gratuitous as it has already been stated that the equations refer to N3D.

B and Y, superscripts and subscripts

B is the value of a B-format signal, and for a point source is given by:
B_ACN = Y_ACN × input_signal
Both B and Y can be expressed in terms of degree (l) and order (m) (see glossary), that is B_l^m and Y_l^m. The latter is used here to be consistent with mathematical texts on spherical harmonics.
(The, here, deprecated Y_m,n^σ style will be seen in some texts. See the glossary section on deprecated usages of m, etc.!)
For greater simplicity a notation with only one subscript, and no superscript is introduced here: B_ACN and Y_ACN. (With ACN = l(l+1)+m, see glossary.) (A superscript may be added for the &lsquo,convention’ (.e.g. N3D, FuMa), but will rarely be necessary.)

Approximation

Calculating spherical harmonics above the first few orders involves handling integers beyond the capabilities of most simple software. A purist approach has been taken here and the results kept as integers. For practical purposes floating point approximations of the numbers in the equations would suffice.

Errors

For discussion of this material and/or to report any errors please join the (low volume) discussion list on this site.

Index page
Glossary

Copyright:
This page copyright © 2008 The Ambisonics Association.
Images copyright © 2008 Michael Chapman. Used with permission.