The Mathieu equation is a linear second-order differential equation with periodic coefficients. The French mathematician, E. Léonard Mathieu, first introduced this family of differential equations, nowadays termed Mathieu equations, in his “Memoir on vibrations of an elliptic membrane” in 1868. "Mathieu functions are applicable to a wide variety of physical phenomena, e.g., diffraction, amplitude distortion, inverted pendulum, stability of a floating body, radio frequency quadrupole, and vibration in a medium with modulated density"[1]
This is a wide family of wavelet system that provides a multiresolution analysis. The magnitude of the detail and smoothing filters corresponds to first-kind Mathieu functions with odd characteristic exponent. The number of notches of these filters can be easily designed by choosing the characteristic exponent. Elliptic-cylinder wavelets derived by this method [2] possess potential application in the fields of optics and electromagnetism due to its symmetry.
Mathieu's equation is related to the wave equation for the elliptic cylinder. In 1868, the French mathematician Émile Léonard Mathieu introduced a family of differential equations nowadays termed Mathieu equations.[3]
Given
a\inR,q\inC
d2y | |
dw2 |
+(a-2q\cos2w)y=0.
The Mathieu equation is a linear second-order differential equation with periodic coefficients. For q = 0, it reduces to the well-known harmonic oscillator, a being the square of the frequency.[4]
The solution of the Mathieu equation is the elliptic-cylinder harmonic, known as Mathieu functions. They have long been applied on a broad scope of wave-guide problems involving elliptical geometry, including:
\nu
In general, the solutions of Mathieu equation 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
\pi
2\pi
One of four simpler types can be considered: Periodic solution (
\pi
2\pi
For
q\ne0
a=ar(q)
a=br(q)
ce and se are abbreviations for cosine-elliptic and sine-elliptic, respectively.
where the sums are taken over even (respectively odd) values of m if the period of y is
\pi
2\pi
Given r, we denote henceforth
Ar,m
Am
Interesting relationships are found when
q\to0
r\ne0
Figure 1 shows two illustrative waveform of elliptic cosines, whose shape strongly depends on the parameters
\nu
Wavelets are denoted by
\psi(t)
\phi(t)
\Psi(\omega)
\Phi(\omega)
The equation
\phi(t)=\sqrt{2}\sumnhn\phi(2t-n)
H(\omega)=
1 | |
\sqrt2 |
\sumkhkej
G(\omega)=
1 | |
\sqrt2 |
\sumkgkej
The transfer function of the "detail filter" of a Mathieu wavelet is
G\nu
| ||||||
(\omega)=e |
.
| ||||||||||
{ce\nu(0,q) |
The transfer function of the "smoothing filter" of a Mathieu wavelet is
H\nu
| ||||||
(\omega)=-e |
.
| ||||||||||
{ce\nu(0,q) |
The characteristic exponent
\nu
G\nu(0)=0
G\nu(\pi)=1
\nu
The magnitude of the transfer function corresponds exactly to the modulus of an elliptic-sine:
Examples of filter transfer function for a Mathieu MRA are shown in the figure 2. The value of a is adjusted to an eigenvalue in each case, leading to a periodic solution. Such solutions present a number of
\nu
0\le|\omega|\le\pi
The G and H filter coefficients of Mathieu MRA can be expressed in terms of the values
\{A2\}l
hl | |
\sqrt{2 |
gl | |
\sqrt{2 |
There exist recurrence relations among the coefficients:
(a-1-q)A1-qA3=0
2)A | |
(a-m | |
m-q(A |
m-2+Am+2=0
for
m\ge3
It is straightforward to show that
h-l=h|l|-1
\foralll>0
Normalising conditions are
k=+infty | |
\sum | |
k=-infty |
{hk=-1}
k=+infty | |
\sum | |
k=-infty |
{(-1)khk=0}
Mathieu wavelets can be derived from the lowpass reconstruction filter by the cascade algorithm. Infinite Impulse Response filters (IIR filter) should be use since Mathieu wavelet has no compact support. Figure 3 shows emerging pattern that progressively looks like the wavelet's shape. Depending on the parameters a and q some waveforms (e.g. fig. 3b) can present a somewhat unusual shape.