slide1 n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Padé and Symmetrized Padé Approximants in Data Processing PowerPoint Presentation
Download Presentation
Padé and Symmetrized Padé Approximants in Data Processing

Loading in 2 Seconds...

play fullscreen
1 / 9

Padé and Symmetrized Padé Approximants in Data Processing - PowerPoint PPT Presentation


  • 148 Views
  • Uploaded on

Padé and Symmetrized Padé Approximants in Data Processing. Jean-Daniel Fournier (1) (2) member of the CNRS fournier@obs-nice.fr. Bénédicte Dujardin (1) (2) (3) (4) Ph.D. student dujardin@obs-nice.fr. Talk presented by J.-D. Fournier at the GWDAW 2004.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Padé and Symmetrized Padé Approximants in Data Processing' - bill


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

Padé and Symmetrized Padé Approximants

in Data Processing

Jean-Daniel Fournier (1)(2)

member of the CNRS

fournier@obs-nice.fr

Bénédicte Dujardin (1) (2)(3)(4)

Ph.D. student

dujardin@obs-nice.fr

Talk presented by J.-D. Fournier

at the

GWDAW 2004

(1) member of ARTEMIS, a department of the “Observatoire de la Côte d ’Azur “, affiliated with the CNRS

(2) postal address : Observatoire de Nice, BP 4229, 06304 NICE Cedex 4, FRANCE

(3)expected date of defense : spring 2005

(4)teaching assistant at the “ Université de Nice Sophia Antipolis“

slide2

Padé and Symmetrized Padé Approximants in Data Processing

  • 1.Analytic continuation and Fourier tools
  • 2.“Natural“ spectra, resonances or worse ;
  • rationality and quasi rationality
  • 3.Rational Approximation :
  • AR, Szegö, ARMA, interpolation, Hermite…Padé, MPPA
  • 4.The energy spectrum : Taylor series at the origin.
  • The outer and inner part of the spectrum
  • 5.The Padé approximant of the spectrum.
  • Alternative definition : the symmetrized Padé approximant
  • 6.Success and pitfall
  • 7.Test on simulated data and “real“ data from a Virgo E.Run
slide3

Analytic Continuation and Fourier Tools I.

Fourier Transform

Inverse Fourier Transform

Modulus squared

analytic continuation

to the complex

plane

Correlation

P1 and P2 hold

Fourier Transform

Inverse Fourier Transform

slide4

Analytic Continuation and Fourier Tools II.

Fourier Transform

P1 reads

Conformal change of variable

P2 guaranteed by the conformal mapping

Inverse Fourier Transform

P3 see next slide

Fourier Transform

Inverse Fourier Transform

slide5

Invariance Property of the Spectrum

P3 : C(k) = C(- k) implies the symmetry of the two parts of this formula

slide6

“ Natural“ Spectrain the physical sense

Violin Modes

Mathematical Tools for Noise Analysis

E. Cuoco & A. Vicere 1994

meromorphic function

Infinite number of simple poles in the variable f, located on a cone in the complex

f domain + one simple pole at the origin.

Filtered thermal noise

A thermal noise linearly filtered by the machine will produce rational spectra

Opto. Thermal couplings in mirrors

Vinet 2001

It is reasonable to look for meromorphic or rational approximants

of spectra even in the presence of branch points, in view of the

good behavior of Padé Approximants.This is what ARMA models

do ; we will do it the other way around

slide7

“Natural“ form of the Spectrumin the mathematical sense

Finite number of poles or zeros in a finite domain

This must be true for the unit disk D1, which contains A poles and B zeros

Through (I), the same is true for the domain outside the unit disk

…(Th)

slide9

Symmetrized Padé Approximant II.

Lesson from the numerical simulations to come :

as compared eg to AR, the robustness and practical

value of the method dwell on the more careful

representation of the poles -the zeros of Q and Q+

- which has been obtained using the classical

Padé Approximants