Fwd: [PD] basic DSP stuff
Charles Henry
czhenry at gmail.com
Sun Nov 13 23:30:02 CET 2005
> > 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.
And in this case, we are talking about the functions in a Hilbert
space, with eigenvalues of the differential equation
x'' + w^2*x: The eigenfunctions are just e^-iwx and e^iwx, w is the
eigenvalue. Our "matrix" in this case is the Hilbert space L2. It's
not like you can take the determinate of a set of functions, the
linear operator can be factored:
x'' + w^2*x=(D^2 + w^2)x=(D + w*i)(D - w*i)x, where D is the
differential operator,
Your eigenfunctions are the solutions to : (D + w*i) and (D - w*i)
We can interchange the orders of (D + w*i) and (D - w*i)
so, the set of eigenfunctions is the solutions to either (D + w*i)=0
or (D - w*i)=0
It's the same result you have for a matrix, except you have different
linear operators, like matrix multiplication instead of the D's.
Then, you find the eigenvalues of your matrix operations like,
Ax-bI =0, where I is the identiry operator
The Fourier transform is an operation which can be performed using NxN
matrix multiplication on a sampled signal of length N, although in the
Hilbert space, we use the inner product on L2. It's no surprise that
the matrix F, for the Fourier transform, X(f)=Fx(t), can be
diagonalized, because it is unitary, and all of its eigenvalues are 1.
>
> 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
>
--
Charles Zachary Henry
anti.dazed.med
Med student who needs a Mickey's
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