Fwd: [PD] basic DSP stuff

[Mathieu Bouchard]

On Sat, 12 Nov 2005, Charles Henry wrote:

> The complex numbers do not necessarily come from "conservation of
> energy"

Indeed, there are other ways to explain conservation of energy, basically 
by invoking kinetic energy; and there are other ways to justify complex 
numbers in wave phenomena.

> the complex numbers are the eigenvalues of your basic oscillator x'' +
> w^2*x = 0

Let me take this apart:

those complex numbers are eigenvalues because eigenvalues are obtained by
factoring the characteristic polynomial of the matrix.

Factorization of elements of R[L] (polynomials with Real coefficients with
a single variable called L) can yield irreductible elements of degree 2,
that is, L*L + positive constant.

The reason for introducing complex numbers is that they make factorization
smoother by allowing all polynomials to be factored down to terms of
degree 1. And then "L*L + positive constant = 0" means "L*L = negative
constant", so the only way to find L here is to invent a number whose
square is a negative constant.

Inventing extra numbers is allowed as long as they stay consistent with 
the number system they are based on. So the Complex numbers are called an 
Extension of the Real numbers because + - * / on Complexes are intuitive 
extensions of those same operations on Reals.

Indeed, playing with Complexes feels like playing with a very limited 
version of polynomials on Reals, so you can do it with 8th grade algebra.

____________________________________________________________________
Mathieu Bouchard - tél:+1.514.383.3801 - http://artengine.ca/matju
Freelance Digital Arts Engineer, Montréal QC Canada