Notes on Spherical Harmonic Equations
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, up to 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, up to 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 ux=cos(E).cos(A), uy=cos(E).sin(A), and uz=sin(E).As the three terms are not independent (ux2 + uy2 + uz2 =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 (ux, uy and uz) 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:YACN(conv2) = αACN(conv2)conv1.YACN(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:BACN = YACN × input_signalBoth B and Y can be expressed in terms of degree (l) and order (m) (see glossary), that is Blm and Ylm. The latter is used here to be consistent with mathematical texts on spherical harmonics.(The, here, deprecated Ym,nσ 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: BACN and YACN. (With ACN = l(l+1)+m, see glossary.) (A superscript may be added for the ‘,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.
Copyright:This page copyright © 2008 The Ambisonics Association.Images copyright © 2008 Michael Chapman. Used with permission.
Published: November 2008. Amended: November 2008.
