Mathieu wavelet

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"

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" \frac {d^2 y} {dw^2} +(a-2q \cos 2w )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. 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: analysis for weak guiding for step index elliptical core optical fibres power transport of elliptical wave guides evaluating radiated waves of elliptical horn antennas elliptical annular microstrip antennas with arbitrary eccentricity \nu ) scattering by a coated strip. For q \ne 0 , the only periodic solutions y corresponding to any characteristic value a=a_r(q) or a=b_r(q) have the following notations: ce and se are abbreviations for cosine-elliptic and sine-elliptic, respectively. Even periodic solution: ce_r(\omega,q)= \sum_m A_{r,m} \cos {m \omega}\text{ for }a = a_r(q) Odd periodic solution: se_r(\omega,q)= \sum_m A_{r,m} \sin {m \omega}\text{ for }a = b_r(q) where the sums are taken over even (respectively odd) values of m if the period of y is \pi (respectively 2\pi ). Given r, we denote henceforth A_{r,m} by A_m , for short. Interesting relationships are found when q \to 0 , r \ne 0 : \lim_{q \to 0} ce_r(\omega,q)= \cos {r \omega} \lim_{q \to 0} se_r(\omega,q)= \sin {r \omega} Figure 1 shows two illustrative waveform of elliptic cosines, whose shape strongly depends on the parameters \nu and q. The equation \phi(t)= \sqrt {2} \sum_{n \in Z} h_n \phi(2t-n) , which is known as the dilation or refinement equation, is the chief relation determining a Multiresolution Analysis (MRA). H(\omega)= \frac {1} {\sqrt 2} \sum_{k \in Z} h_k e^{j \omega k} is the transfer function of the smoothing filter. G(\omega)= \frac {1} {\sqrt 2} \sum_{k \in Z} g_k e^{j \omega k} is the transfer function of the detail filter. The transfer function of the "detail filter" of a Mathieu wavelet is G_{\nu}(\omega)=e^{j(\nu-2)[ \frac {\omega - \pi} {2}]}. \frac {ce_{\nu} ( \frac {\omega-\pi} {2},q)} . The transfer function of the "smoothing filter" of a Mathieu wavelet is H_{\nu}(\omega)=-e^{j\nu [ \frac {\omega} {2}]}. \frac {ce_{\nu}( \frac {\omega} {2},q)} . The characteristic exponent \nu should be chosen so as to guarantee suitable initial conditions, i.e. G_{\nu}(0)=0 and G_{\nu}(\pi)=1 , which are compatible with wavelet filter requirements. Therefore, \nu must be odd. 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 zeroes in the interval 0 \le |\omega | \le \pi . The G and H filter coefficients of Mathieu MRA can be expressed in terms of the values \{ A_{2 l +1} \}_{l \in Z} of the Mathieu function as: \frac {h_l} {\sqrt{2}}=- \frac {A_ /2} {ce_{\nu}(0,q)} \frac {g_l} {\sqrt{2}}=(-1)^l \frac {A_ /2} {ce_{\nu}(0,q)} There exist recurrence relations among the coefficients: (a-1-q)A_1-qA_3 =0 (a-m^2)A_m-q(A_{m-2}+A_{m+2} =0 for m \ge 3 , m odd. It is straightforward to show that h_{-l}=h_{|l|-1} , \forall l>0 . Normalising conditions are \sum_{k=- \infty}^{k=+ \infty} {h_k =-1} and \sum_{k=- \infty}^{k=+ \infty} {(-1)^k h_k =0} .

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