[sdiy] Looking for help with Bessel functions and FM

[Tim Stinchcombe]

> I have a pet project that I'd like to get started on. I would 
> like to analyze an FM synthesizer patch and calculate the 
> theoretical output spectrum that would exist any given moment 
> when playing one note using that patch. What I would like to 
> eventually do with that ability is translate an FM synth 
> patch into an additive synth patch.
> 
> I think Bessel functions are something I need to learn about 
> for this project, but I'm having trouble finding enough 
> information, especially in a format that I can understand. 
> Does anyone know of any good FM modulation and/or Bessel 
> function tutorials?

Glen,

A tricky one! There is a huge gulf between a simplistic explanation that you
may find in a very few places, and the real nitty-gritty of what is really
involved. To start I would suggest looking at some of the papers out there
on this subject, to get a feel for what is involved, then perhaps start
checking references they contain, etc. A few references that I know of (and
undoubtedly if you really get into it, there will be many, many more
incredibly academic papers around):

Chowning's original FM paper:
'The Synthesis of Complex Audio Spectra by Means of Frequency Modulation',
Chowning JM, J of AES, vol 21, no. 7, 1973, pp526-534	

Bernie Hutchins (Electronotes):
'The Frequency Modulation Spectrum of an Exponential Voltage-Controlled
Oscillator', Hutchins BA, J of AES, vol 23, no. 3 1975, pp200-206
(From a quick search of the Electronotes indices I don't think there is much
in them, but I could be wrong - I haven't had much chance to check out my
copies in depth for anything, let alone FM!)

'Understanding FM Implementations: A Call for Common Standards', Holm F,
Computer Music Journal, vol 16 no. 1, 1992, pp34-42

The rather large Roads book (>1200 pages) has a small section on FM and
mentions Bessel functions, but is probably rather superficial - however it
contains a wealth of references, some of which will probably be quite meaty:
The Computer Music Tutorial, Curtis Roads, 1996 MIT, ISBN 0262680823

The Chowning and Bristow book 'FM Theory & Applications by Musicians for
Musicians', Yamaha 1986, ISBN 4636174828, is quite simplistic, but may help
put things into the musical perspective.

After that you will need to check out some pretty heavy maths books - a
thorough understanding of FM at the level you suggest is definitely not for
the faint hearted! It's an area I'd like to get into myself, but it will
have to wait until my current studies have finished. So good luck, and keep
us posted on progress!

Tim
__________________________________________________________
Tim Stinchcombe 

Cheltenham, Glos, UK
email: tim102 at tstinchcombe.freeserve.co.uk