Elliptic-cylindrical Wavelets: The Mathieu Wavelets

0.15 MB·English

by  M. M. S. Lira

Elliptic-Cylindrical Wavelets: The Mathieu Wavelets
M. M. S. Lira H. M. de Oliveira R. J. Cintra
∗ † †
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0 Abstract
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This note introduces a new family ofwaveletsanda multiresolutionanalysis,whichexploits
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a the relationship between analysing filters and Floquet’s solution of Mathieu differential equa-
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tions. Thetransferfunctionofboththe detailandthe smoothingfilterisrelatedtothesolution

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of a Mathieu equation of odd characteristic exponent. The number of notches of these filters
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canbe easilydesigned. Waveletsderivedby this methodhavepotentialapplicationinthe fields

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E of Optics and Electromagnetism.
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Keywords
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t Wavelets, Waveguides, Mathieu equation, Floquet’s Theorem
a
t
s
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1 Introduction

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v
5 In1868, theFrenchmathematicianE´.L´eonardMathieuintroducedafamilyofdifferentialequations
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2 nowadays termed Mathieu equations in his “memoir on vibrations of an elliptic membrane” [1].
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Mathieu’s equation is related to the wave equation for the elliptic cylinder. Mathieu is notably
0
. remembered for his discovery of sporadic simple groups [2]. This paper is particularly concerned
1
0 with the canonical form of the Mathieu Equation. For a R, q C, the Mathieu Equation is given
5 ∈ ∈
1 by
: d2y
v
+(a 2qcos(2ω))y = 0. (1)
i dω2 −
X
r The Mathieu equation is a linear second-order differential equation with periodic coefficients. This
a
equation was shownlater tobealsorelated toquantummechanicals; theparameters aandq denote
the energy level and an intensity, respectively. For q = 0 it reduces to the well-known harmonic
oscillator, a being the square of the frequency [3]. The solution of (1) is the elliptic-cylindrical
harmonic, known as Mathieu functions. In addition to being theoretically fascinating, Mathieu
functions are applicable to a wide variety of physical phenomena, e.g., diffraction, amplitude dis-
tortion, inverted pendulum, stability of a floating body, radio frequency quadrupole, and vibration
∗M. M. S.Lira is with thePower SystemsDigital Laboratory, Universidade Federal dePernambuco, Brazil.
†H. M. de Oliveira and R. J. Cintra are with the Signal Processing Group, Departamento de Estat´ıstica, Uni-
versidade Federal de Pernambuco, and were formely with the Communications Research Group, Departamento de
Eletroˆnica e Sistemas, Federal University of Pernambuco, Brazil. E-mail: {hmo,rjdsc}@de.ufpe.br
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in a medium with modulated density [4]. They have also long been applied on a broad scope of
waveguide problems involving elliptical geometry, including: (i) analysis for weak guiding for step
index elliptical core optical fibres [5], (ii) power transport of elliptical waveguides [6,7], (iii) evalu-
ating radiated waves of elliptical horn antennas [8], (iv) elliptical annular microstrip antennas with
arbitrary eccentricity [9], and (v) scattering by a coated strip [10].
The aim of this paper is to propose a new family of wavelets based on Mathieu differential
equations. Wavelets are a well-known tool for differential equation solving [11–13]. However, in
this work, we show another connection between wavelets and differential equations: the design of
new wavelets from the solution of a differential equation.
2 Mathieu Equations
In general, the solutions of (1) are not periodic. However, for a given q, periodic solutions exist for
infinitely many special values (eigenvalues) of a. For several physically relevant solutions y must
be periodic of period π or 2π. It is also convenient to distinguish even and odd periodic solutions,
which are termed Mathieu functions of first kind. One of four simpler types can be considered:
Periodic solution (π or 2π) symmetry (even or odd). For q = 0, the only periodic solution y
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corresponding to any characteristic value a = a (q) or a = b (q) has the following notation:
r r
Even periodic solution
ce (ω,q) = A cosmω for a = a (q), (2a)
r X r,m r
m
Odd periodic solution
se (ω,q) = A sinmω for a = b (q), (2b)
r X r,m r
m
wherethesumsaretakenovereven(respectivelyodd)valuesofmiftheperiodofy isπ (respectively
2π). Given r, wedenotehenceforthA byA , forshort. Ellipticcosine andelliptic sinefunctions
r,m m
arerepresentedbyceandse,respectively. Interestingrelationshipsarefoundwhenq 0,r = 0[14]:
→ 6
limce (ω,q) = cos(rω), limse (ω,q) = sin(rω). (3)
r r
q 0 q 0
→ →
One of the most powerful results of Mathieu’s functions is the Floquet’s Theorem [15]. It states
that periodic solutions of (1) for any pair (a,q) can be expressed in the form
y(ω) = F (ω) = ejνωP(ω) or
ν
(4)
y(ω) = F ( ω) = e jνωP( ω),
ν −
− −
whereν is aconstant dependingon a andq andP() is π-periodicin ω. Theconstant ν is called the
·
2
characteristic exponent. If ν is an integer, then F (ω) and F ( ω) are linear dependent solutions.
ν ν
−
Furthermore, y(ω+kπ) = ejνkπy(ω) or y(ω+kπ) = e jνkπy(ω), for the solution F (ω) or F ( ω),
− ν ν
−
respectively. We assume that the pair (a,q) is such that cosh(jνπ) < 1 so that the solution y(ω)
| |
is bounded on the real axis [16]. The general solution of Mathieu’s equation (q R, ν non-integer)
∈
has the form
y(ω)= c ejνωP(ω)+c e jνωP( ω), (5)
1 2 −
−
where c and c are arbitrary constants.
1 2
All bounded solutions —those of fractional as well as integral order— are described by an
infinite series of harmonic oscillations whose amplitudes decrease with increasing frequency. In the
wavelet framework we are basically concerned with even solutions of period 2π. In such cases there
exist recurrence relations among the coefficients [14]:
(a 1 q)A qA = 0,
1 3
− − − (6)
(a m2)A q(A +A )= 0, m 3, m odd.
m m 2 m+2
− − − ≥
In the sequel, wavelets are denoted by ψ(t) and scaling functions by φ(t), with corresponding
spectra Ψ(ω) and Φ(ω), respectively.
3 Mathieu Wavelets
Wavelet analysis has matured rapidly over the past years and has been proved to be invaluable
for scientists and engineers [17]. Wavelet transforms have lately gained extensive applications in
an amazing number of areas [18]. The equation φ(t) = √2 h φ(2t n), which is known as
Pn∈Z n −
the dilation or refinement equation, is the chief relation determining a Multiresolution Analysis
(MRA) [19].
3.1 Two Scale Relation of Scaling Function and Wavelet
Defining the spectrum of the smoothing filter h by H(ω) , 1 h e jωk, the central
{ k} √2 Pk∈Z k −
equations (in the frequency domain) of a Multiresolution analysis are [20]:
ω ω ω ω
Φ(ω) = H Φ and Ψ(ω) = G Φ , (7)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
2 2 2 2
where G(ω) , 1 g e jωk, is the transfer function of the detail filter.
√2Pk∈Z k −
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The orthogonality condition corresponds to [20]:
H(0) = 1 and H(π) = 0, (8a)
H(ω)2 + H(ω+π)2 = 1, (8b)
| | | |
H(ω)= e jωG (ω+π). (8c)
− ∗
−
3.2 Filters of a Mathieu MRA
Thesubtleliaison between Mathieu’s theory andwavelets was foundby observingthat the classical
relationship
ω ω
Ψ(ω) = e jω/2H π Φ (9)
− ∗(cid:16) (cid:17) (cid:16) (cid:17)
2 − 2
presents a remarkable similarity to a Floquet’s solution of a Mathieu’s equation, since H(ω) is a
periodic function.
As a first attempt, the relationship between the wavelet spectrum and the scaling function was
put in the form:
Ψ(ω) ω
= e jω/2H π . (10)
Φ ω − ∗(cid:16)2 − (cid:17)
(cid:0)2(cid:1)
Here, on the second member, neither ν is an integer nor H() has a periodπ. By an appropriate
·
scaling of this equation, we can rewrite it as
Ψ(4ω)
= e j2ωH (2ω π). (11)
− ∗
Φ(2ω) −
Defining a new function Y(ω) , Ψ(4ω)/Φ(2ω), we recognise that it has a nice interpretation in
the wavelet framework. First, we recall that Ψ(ω) = G ω Φ ω so that Ψ(2ω) = G(ω)Φ(ω).
(cid:0)2(cid:1) (cid:0)2(cid:1)
Therefore the function related to Mathieu’s equation is exactly Y(ω) = G(2ω). Introducing a new
variable z, which is defined according to 2z , 2ω π, it follows that Y z+ π = e j2zH (2z).
− − (cid:0) 2(cid:1) − ∗
The characteristic exponent can be adjusted to a particular value ν,
π
e j(ν 2)zY z+ = e jνzH (2z). (12)
− − (cid:16) (cid:17) − ∗
− 2
Defining now P( z) , H (2z) = c ejz2k, where c , 1 h , we figure out that the right-
− ∗ Pk∈Z 2k 2k √2 ∗k
side of the above equation represents a Floquet’s solution of some differential Mathieu equation.
The function P() is π-periodic verifying the initial condition P(0) = 1 h = 1, as expected.
· √2 Pk k
The filter coefficients are all assumed to be real. Therefore, there exist a set of parameters (a ,q )
G G
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such that the auxiliary function
π
y (z) , e j(ν 2)zY z+ (13)
ν − − ν(cid:16) (cid:17)
− 2
is a solution of the following Mathieu equation:
d2y
ν
+(a 2q cos(2z))y = 0, (14)
dz2 G − G ν
subject to y (0) = Y(π/2) = G(π) = 1 and cos(πν) y (π) =0, that is, y (π) = ( 1)ν.
ν ν ν
− − − − −
In order to investigate a suitable solution of (14), boundary conditions are established for pre-
determined a, q. It turnsoutthat when ν is zero or an integer, a belongs to the set of characteristic
values a (q). Furthermore, ν = r is associated with a (q). The even (2π-periodic) solution of such
r r
an equation is given by:
ce (z,q)
ν
y (z) = . (15)
ν
−ce (0,q)
ν
The Y (ω) function associated to y (z) and related to the detail filter of a “Mathieu MRA” is thus:
ν ν
Yν(ω) = Gν(2ω) = ej(ν−2)(ω−π2)ceν(cid:0)ω− π2,q(cid:1). (16)
ce (0,q)
ν
Finally, the transfer function of the detail filter of a Mathieu wavelet is
Gν(ω) = ej(ν−2)(ω−2π)ceν(cid:0)ω−2π,q(cid:1). (17)
ce (0,q)
ν
The characteristic exponent ν should be chosen so as to guarantee suitable initial conditions, i.e.,
G (0) = 0andG (π) = 1,whicharecompatiblewithwavelet filterrequirements. Therefore,ν must
ν ν
be odd. It is interesting to remark that the magnitude of the above transfer function corresponds
exactly to the modulus of a elliptic sine [16]:
ω
G (ω) = se , q /ce (0,q) . (18)
ν (cid:12) ν(cid:16) (cid:17) ν (cid:12)
| | (cid:12) 2 − (cid:12)
(cid:12) (cid:12)
The solution for the smoothing filter H() can be found out via QMF conditions [19], yielding:
·
Hν(ω) = e−jνω2 ceν(cid:0)ω2,q(cid:1). (19)
− ce (0,q)
ν
In this case, we find H (π) = 0 and
ν
ω
H (ω) = ce ,q /ce (0,q) . (20)
ν (cid:12) ν(cid:16) (cid:17) ν (cid:12)
| | (cid:12) 2 (cid:12)
(cid:12) (cid:12)
Given q, the even first-kindMathieu function with characteristic exponent ν is given by ce (ω,q) =
ν
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(a) (b)
Figure1: MagnitudeofthetransferfunctionforMathieumultiresolutionanalysisfilters: smoothing
filter H (ω) (solid line) and detail filter G (ω) (bold line) for a few Mathieu parameters. (a)
ν ν
| | | |
ν = 3, q = 3, a = 9.915506290452134; (b) ν = 5, q = 15, a = 31.957821252172874.
P∞l=0A2l+1cos(2l + 1)ω, in which ceν(0,q) = P∞l=0A2l+1. The G and H filter coefficients of a
Mathieu MRA can be expressed in terms of the values A2l+1 l Z of the Mathieu function as:
{ }∈
hν A /2 gν A /2
l = |2l−ν| and l = ( 1)l |2l+ν−2| . (21)
√2 − ceν(0,q) √2 − ceν(0,q)
It is straightforward to show that hν = hν , l > 0. The normalising conditions are
l l+ν ∀
−
√12P∞k=−∞hνk = −1 and P∞k=−∞(−1)khνk = 0.
4 Examples
Illustrative examples of filter transfer functions for a Mathieu MRA are shown in Fig. 1, for ν = 3
and 5, and a particular value of q (numerical solution obtained by 5-order Runge-Kutta method).
The value of a is adjusted to an eigenvalue in each case, leading to a periodic solution. Such
solutions present a number of ν zeroes in the interval ω < π. We observe lowpass behaviour
| |
(for the filter H) and highpass behaviour (for the filter G), as expected. Mathieu wavelets can be
derived from the lowpass reconstruction filter by the cascade algorithm. Infinite Impulse Response
filters (IIR) should be applied since Mathieu wavelet has no compact support. However a Finite
ImpulseResponse(FIR) approximation can begenerated by discardingnegligible filter coefficients,
say less than 10 10. In Fig. 2, emerging pattern that progressively looks like the wavelet shape is
−
shown for some couple of parameter a and q. Waveforms were derived using the Matlab wavelet
toolbox. Aswithmany wavelets thereisnoniceanalytical formulafordescribingMathieu wavelets.
5 Conclusions
Anewandwidefamilyofelliptic-cylindricalwaveletswasintroduced. Itwasshownthatthetransfer
functionsofthecorrespondingmultiresolutionfiltersarerelatedtoMathieuequationsolutions. The
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0.1 0.025 0.008
0.05 0.0125 0.004
0 0 0
-0.05 -0.0125 -0.004
-0.1 -0.025 -0.008
45 70 95 200 300 400 900 1300 1700
(a)
0.08 0.02 0.004
0.04 0.01 0.002
0 0 0
-0.04 -0.01 -0.002
-0.08 -0.02 -0.004
25 75 125 200 350 500 800 1400 2000
(b)
Figure 2: FIR-Based Approximation of Mathieu Wavelets as the number of iteration increases (2,
4, and 6 iterations, respectively). Filter coefficients holding h < 10 10 were thrown away (19
−
| |
retained coefficients per filter in both cases). (a) Mathieu Wavelet with ν = 3 and q = 3 and (b)
Mathieu Wavelet with ν = 5 and q = 15.
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magnitude of the detail and smoothing filters corresponds to first-kind Mathieu functions with odd
characteristic exponent. The number of zeroes of the highpass G(ω) and lowpass H(ω) filters
| | | |
within the interval ω < π can be appropriately designed by choosing the characteristic exponent.
| |
ThisseemstobethefirstconnectionfoundbetweenMathieuequationsandwavelettheory. Itopens
new perspectives on linking wavelets and solutions of other differential equations (e.g. Associated
Legendre functions).
Although there exist plenty of potential applications for Mathieu Wavelets, none are presented:
we just disseminate the major ideas, letting further research to be investigated. For instance, this
new family of wavelets could bean interesting tool for analysing optical fibres dueto its “elliptical”
symmetry. Theycouldaswellbebeneficialwhenexaminingmoleculardynamicsofchargedparticles
in electromagnetic traps such as Paul trap or the mirror trap for neutral particles [21,22].
Acknowledgements
The authors thank Dr. Harold V. McIntosh from Departamento de Aplicacio´n de Microcomputa-
doras, Universidad Aut´onoma de Puebla, Mexico for information about Mathieu’s functions. They
alsoexpresstheir indebtednesstoDr.R.M.CampellodeSouza(Federal University ofPernambuco,
Brazil) for stimulating comments.
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