Representation of signals as series of orthogonal functions

The Bessel function J n(x), n2N, called the Bessel function of the rst kind of order n, is de ned by the absolutely convergent in nite series J n(x) = xn X m 0 ( 21)mxm 22m+nm(n+ m) for all x2R: (1) It satis es the Bessel di erential equation x2 J00 n (x) + xJ0 n (x) + (x2 n2)J n(x) = 0: (2) The Bessel functions most relevant to this course are J

Radial Separation

Figure 23 1: The zeroth spherical Bessel function { this gives the radial wavefunction for a free particle in spherical coordinates (for ‘= 0) Spherical Bessel Functions We quoted the result above, the di erential equation (23 4) has solu-tions that look like u ‘(r) = rj ‘(kr) ( nite at the origin) But how

Representation of signals as series of orthogonal functions

Series expansions involving Bessel functions : Neumann series Fourier-Bessel series (can be generalized to real indexes Jν) A reference book about Bessel functions (800 pages) α np is the nth positive root of J p Convenient for expansions on [-∞,+∞] Convenient for expansions on [0,1] with boundary condition atx=1

LECTURE 5: Fluid jets 51 The shape of a falling ﬂuid jet - MIT

Eliminating Z(r) and P(r) yields a diﬀerential equation for R(r): r2 d2R dr2 +r dR dr − 1+(kr)2 R = 0 (18) This corresponds to modiﬁed Bessel Equation of order 1, whose solutions may be written in terms of the modiﬁed Bessel functions of the ﬁrst and second kind, respectively, I 1(kr) and K 1(kr) We note that K

Calcul stochastique et modèles de diffusions

12 7 Carré de processus de Bessel 258 12 8 Dépendance en la condition initiale 259 12 9 Équation différentielle stochastique de Tanaka 261 CHAPITRE 13 • DIFFUSIONS ET OPÉRATEURS AUX DÉRIVÉES PARTIELLES, EXERCICES 13 1 Compléments de cours 265 13 2 Passages successifs de barrières pour un mouvement brownien réel 267

Course notes

where no convergence condition is imposed, and de ne asymptoticity by the following De nition 1 2 A function f is asymptotic to the formal series f~ as tt 0 (once more, the approach of t 0 may have to be restricted to a generally complex curve) if f(t) XM k=0 f k(t) = o(f M(t)) (8M2N or 8M6 M 0 2N)(1 3)

Differential Equations I

A diﬀerential equation (de) is an equation involving a function and its deriva-tives Diﬀerential equations are called partial diﬀerential equations (pde) or or-dinary diﬀerential equations (ode) according to whether or not they contain partial derivatives The order of a diﬀerential equation is the highest order derivative occurring

Differential Equations

Chapter 0 A short mathematical review A basic understanding of calculus is required to undertake a study of differential equations This zero chapter presents a short review

Electrodynamique II S´erie 1 - Boston University: Physics

La fonction de Green joue le rˆole d’une fonction d’inﬂuence : 4πG(x − x′) d´etermine le potentiel au point x duˆ a une unit´e de charge ponctuelle plac´ee au point x′ Exercice 2 : Fonction de Green de l’´equation d’onde `a trois plus une dimensions (voir cours)

[PDF] multiple de 13 entre 1 et 1000

[PDF] multiple de 12

[PDF] multiple de 19

[PDF] fonction de bessel j0

[PDF] table de 13

[PDF] fonction de bessel pdf

[PDF] fonction de bessel modifiée

[PDF] introduction ? la microéconomie varian pdf

[PDF] cours microeconomie 1 pdf

[PDF] cours de microéconomie licence 1 pdf

[PDF] corrélation multiple

[PDF] correlation multiple r

[PDF] exercice fonction cout de production

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1/66

Representation of signals

as series of orthogonal functions

Eric Aristidi

Laboratoire Lagrange - UMR 7293 UNS/CNRS/OCA

1. Fourier Series

2. Legendre polynomials

3. Spherical harmonics

4. Bessel functions

Ecole BasMatIBases mathématiques pour l'instrumentation et le traitement du signal en astronomie

Nice - Porquerolles, 1 - 5 Juin 2015

1 2/66

Fourier Series

Théorie analytique de la chaleur, 1822

Solutions to the heat equation (diffusion PDE) as trigonometric series 3/66

For a signal with periodT

-N max N max frequencyn/T Approching a periodic signal by a sum of trigonometric functions

Fourier Series

4/66

Partial Fourier seriesIndividual terms

Frequencies: 0 , ±1/Tn=1

5/66

Partial Fourier seriesIndividual terms

Frequencies :0 , ±1/T , ±2/T, ±3/Tn=1

n=3 n=2 6/66

Partial Fourier seriesIndividual terms

7/66

Partial Fourier seriesIndividual terms

8/66

Partial Fourier seriesIndividual terms

9/66

Two representations of the signal

Ensemble of discrete sampled values{f(t

k )}Ensemble of Fourier coefficients{c n 10/66

Fourier Series

Complex form :

for a signal f(t)of periodT

Even functions : sum of cosines

Odd functions : sum of sines

General case ; real form :

periodT/n 11/66 uniform convergence

Calculating the coefficientsc

n weighting function " Key » relation : nul ifpn

Then :

Calculate the integral

12/66

Fourier series and scalar product

Vectors (3D)

Functions

Orthogonality :Orthonormal base :

Decomposition :with

(bilinear, for u and v)

Scalar product :

2 vectors...1 number

Norm :

(positive) fandgare orthogonal iif

Orthonormal base :

Decomposition :

= Fourier series

Scalar product :

Norm :

(suited to Fourier series) 13/66

Fourier series and Differential Equations

Harmonic oscillator equation

2 independent solutions are

andwith

Any solution is the superposition

Fourier base functions are associated with harmonic differential equation(trigonometric series were introduced from Fourier work on PDE heat equation)

14/66

Essential ideas on Fourier Series

Periodic signal

Can be expressed as a series of

trigonometric base functions (exp, cos, sin) Base functions are orthonormal, with respect to an appropriate scalar product Coefs. of the series calculated as a scalar product between the signal and each base function Base functions are solutions of a differential Eq. 15/66

Representation of signals

as series of orthogonal functions

Eric Aristidi

Laboratoire Lagrange - UMR 7293 UNS/CNRS/OCA

1. Fourier Series

2. Legendre polynomials

3. Spherical harmonics

4. Bessel functions

Ecole BasMatIBases mathématiques pour l'instrumentation et le traitement du signal en astronomie

Nice - Porquerolles, 1 - 5 Juin 2015

16/66

Legendre Polynomials

Introduced by A.M. Legendre, 1784, " Recherches sur la figure des planètes », mém. de l'académie royale des sciences de Paris xyz R r'

Gravitational potential at

R mass M with:

Introducing

and

Generating function of Legendre Polynomials

O 17/66

Taylor expansion atr=0:with

Degree :n

Recurrence relation :

Parity :n

ndistinct roots in the interval [1-,1] 18/66

Fourier-Legendre series

Scalar product :

Orthogonality :

Not orthonormal !

Coefficient determination :

Fourier-Legendre series (for a functionf(x)square-summable on [-1,1]) : (suited to Legendre polynomials) 19/66

Partial Fourier-Legendre seriesIndividual term

Signal :

a=0.3c 0 P 0 (x)

Example of Fourier-Legendre reconstruction

20/66

Partial Fourier-Legendre seriesIndividual terms

c 0 P 0 (x) andc 2 P 2 (x) (c 1 =0 for an even signal) 21/66

Partial Fourier-Legendre seriesIndividual terms

c 0 P 0 (x) c 4 P 4 (x) c 2 P 2 (x) 22/66

Partial Fourier-Legendre seriesIndividual terms

c 0 P 0 (x) c 4 P 4 (x) c 2 P 2 (x)c 6 P 6 (x) 23/66

Partial Fourier-Legendre seriesIndividual terms

c 0 P 0 (x) c 4 P 4 (x) c 2 P 2 (x)c 6 P 6 (x) c 8 P 8 (x) 24/66

For an even signal : all oddc

n vanish In this example, 8 coefficients seem enough to reconstruct the signal 25/66

Error on the reconstruction :

Euclidean distance

26/66

Another example : f(x)=tan(x)

Comparison with Fourier series

-11

The signal is periodized (period 2)

Advantage to Fourier-Legendre (for this case)

signal N max =1 N max =3 27/66
Example in physics : gravitational potential of a uniform bar P rL/2 r'

Gravitational

potential : =linear mass density Using the generating function of Legendre polynomials:

One finds easily :

with -L/2 This is a Fourier-Legendre expansion of the potential (a.k.a. multipolar expansion) 28/66

Fourier-Legendre series of the potential

r=0.75 L 29/66
Polar plotIndividual terms Potential (partial sums)Polar plot c 0 P 0 (cos c 2 P 2 (cos c 4 P 4 (cos 30/66

Essential ideas on Fourier-Legendre Series

Signal defined on interval [-1,1]

Can be expressed as a series of

Legendre polynomials (base functions)

Legendre poly's are orthogonal, with respect to an appropriate scalar product Coefs. of the series calculated as a scalar product between the signal and each polynom Polynoms are solutions of Legendre differential Eq.Legendre polynomials are defined by Taylor expansion of a characteristic function 31/66

Representation of signals

as series of orthogonal functions

1. Fourier Series

2. Legendre polynomials

3. Spherical harmonics

4. Bessel functions

Ecole BasMatIBases mathématiques pour l'instrumentation et le traitement du signal en astronomie

Nice - Porquerolles, 1 - 5 Juin 2015

32/66

Towards the Spherical Harmonics :

associated Legendre functions :P lm

Definition :

Orthogonality (same scalar product asP

n )ĺgeneralisation of Legendre polynomials ifm=0:

Fixedm, differentl:Fixedl, differentm:(for m>0)

33/66
l=2: 5 "polynoms"l=1: 3 "polynoms"l=0: 1 polynom

For a givenl

there are 2l+1 possible P lm 34/66

Spherical Harmonics

Back to the Newtonial potential :

If azimuthal symmetry, the potential is a

function of r andand can be developped as a Fourier-Legendre series :What if no azimuthal symmetry ?

The potential depends on the 3

spherical coordinates(r,). Laplace (1785) showed that a similar series expansion canbemade: P r r' function of r alonefunction ofalone 35/66

Spherical Harmonics

To find the functionsC

lm (r)andY lm say that the potential obey Laplace's equation (Solutions are "harmonicfunctions») andfind:Spherical

Harmonics

The potential at the surface of a sphere (fixed r) is a 2D functionf( )which can be developped as a series of spherical harmonics :l: degree, m: order; -l m l 36/66

Orthogonality and series expansion

Scalar product :

Orthogonality :

orthonormal base

Series expansion for a functionf(

)on a sphere :

Coefficient determination :

(suited to Spherical harmonics) 37/66

Comparison : 1D (polar) and 2D (spherical)

Functionf(

)with values on a circler=C te

2periodic in

Fourier expansionFunctionf(

)with values on a spherer=C te

2periodic in and

Spherical harmonic expansionIn the plane : polar coordinates(r, )In space : spherical coordinates(r, 38/66

First spherical harmonic

(uniform) zzz x y 39/66
l=1 m=-1 m=0m=1 indep. of: azimuthal symmetry 40/66

Different representation

l=1, m=0 l=1, m=1 41/66
l=2 m=-2 m=-1m=0 for positive m, use : 42/66

Symmetries of the spherical harmonics

m=0: azimuthal symmetry

Parity in

cos( l+m even : symmetry planez=0l+modd: anti-symmetry planez=0 Spherical harmonic expansion becomes a Fourier-Legendre series ofcos(

Adapted for series expansion of functions

having a symmetry planeAdapted for series expansion of functions having a anti-symmetry plane 43/66

HarmonicY

lm mperiods in directionterme im n - mnode lines (Y lm =0) in direction

Large mor (l-m)

= small details 44/66

Application to wide-field imagery : WMAP images

Source : B. Terzic,

http://www.nicadd.niu.edu/~bterzic/

Decomposition of the CMB signal on theY

Plot of coefficients |c

l 2 (averaged overm), i.e. "angular power spectrum» 45/66
Cumulative image of the WMAP sky as incresing l numbers are summed (For each l, all orders m are accumulated)

Application to wide-field imagery : WMAP images

46/66
Cumulative image of the WMAP sky as incresing l numbers are summed (For each l, all orders m are accumulated)

Application to wide-field imagery : WMAP images

47/66

Essential ideas on Spherical Harmonic expansion

Signal depending on of 2 angular

spherical coord.

Can be expressed as a series of

spherical Harmonicsquotesdbs_dbs8.pdfusesText_14

[PDF] Representation of signals as series of orthogonal functions

Representation of signals

Eric Aristidi

Laboratoire Lagrange - UMR 7293 UNS/CNRS/OCA

1. Fourier Series

2. Legendre polynomials

3. Spherical harmonics

4. Bessel functions

Nice - Porquerolles, 1 - 5 Juin 2015

Fourier Series

Théorie analytique de la chaleur, 1822

For a signal with periodT

Fourier Series

Partial Fourier seriesIndividual terms

Frequencies: 0 , ±1/Tn=1

Partial Fourier seriesIndividual terms

Frequencies :0 , ±1/T , ±2/T, ±3/Tn=1

Partial Fourier seriesIndividual terms

Partial Fourier seriesIndividual terms

Partial Fourier seriesIndividual terms

Two representations of the signal

Ensemble of discrete sampled values{f(t

Fourier Series

Complex form :

Even functions : sum of cosines

Odd functions : sum of sines

General case ; real form :

Calculating the coefficientsc

Then :

Calculate the integral

Fourier series and scalar product

Vectors (3D)

Functions

Orthogonality :Orthonormal base :

Decomposition :with

Scalar product :

2 vectors...1 number

Norm :

Orthonormal base :

Decomposition :

Scalar product :

Norm :

Fourier series and Differential Equations

Harmonic oscillator equation

2 independent solutions are

Any solution is the superposition

Essential ideas on Fourier Series

Periodic signal

Can be expressed as a series of

Representation of signals

Eric Aristidi

Laboratoire Lagrange - UMR 7293 UNS/CNRS/OCA

1. Fourier Series

2. Legendre polynomials

3. Spherical harmonics

4. Bessel functions

Nice - Porquerolles, 1 - 5 Juin 2015

Legendre Polynomials

Gravitational potential at

Introducing

Generating function of Legendre Polynomials

Taylor expansion atr=0:with

Degree :n

Recurrence relation :

Parity :n

Fourier-Legendre series

Scalar product :

Orthogonality :

Not orthonormal !

Coefficient determination :

Partial Fourier-Legendre seriesIndividual term

Signal :

Example of Fourier-Legendre reconstruction

Partial Fourier-Legendre seriesIndividual terms

Partial Fourier-Legendre seriesIndividual terms

Partial Fourier-Legendre seriesIndividual terms

Partial Fourier-Legendre seriesIndividual terms

For an even signal : all oddc

Error on the reconstruction :

Euclidean distance