Mathieu wavelet

<h2 id="elliptic-cylinder-wavelets">Elliptic-cylinder wavelets</h2>
This is a wide family of wavelet system that provides a <a href="/facts/Multiresolution_analysis/yQtKJuNx">multiresolution analysis</a>. The magnitude of the detail and smoothing filters corresponds to first-kind <a href="/facts/Mathieu_function/zmsyWw0K">Mathieu functions</a> 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 <a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> possess potential application in the fields of <a href="/facts/Optics/p0vq4S1T">optics</a> and <a href="/facts/Electromagnetism/oAkmALHT">electromagnetism</a> due to its symmetry.

<h2 id="mathieu-differential-equations">Mathieu differential equations</h2>
Mathieu's equation is related to the wave equation for the elliptic cylinder. In 1868, the French mathematician <a href="/facts/%C3%89mile_L%C3%A9onard_Mathieu/yDRel2S1">Émile Léonard Mathieu</a> introduced a family of differential equations nowadays termed <a href="/facts/Mathieu_equation/zmsyWw0K">Mathieu equations</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>
Given 
 
 
 
 a
 ∈
 
 R
 
 ,
 q
 ∈
 
 C
 
 
 
 {\displaystyle a\in \mathbb {R} ,q\in \mathbb {C} }
 
, the Mathieu equation is given by

d
                
                  2
                
              
              y
            
            
              d
              
                w
                
                  2
                
              
            
          
        
        +
        (
        a
        −
        2
        q
        cos
        ⁡
        2
        w
        )
        y
        =
        0.
      
    
    {\displaystyle {\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.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>
The solution of the Mathieu equation is the elliptic-cylinder harmonic, known as <a href="/facts/Mathieu_functions/zmsyWw0K">Mathieu functions</a>. They have long been applied on a broad scope of wave-guide problems involving elliptical geometry, including:

<ol><li>analysis for weak guiding for step index elliptical core <a href="/facts/Optical_fibre/5ruuIFbj">optical fibres</a></li>
<li>power transport of elliptical <a href="/facts/Wave_guide/tJS9Y0GH">wave guides</a></li>
<li>evaluating radiated waves of elliptical <a href="/facts/Horn_antenna/kVroJAmN">horn antennas</a></li>
<li>elliptical annular <a href="/facts/Microstrip_antenna/scmm9bBq">microstrip antennas</a> with arbitrary eccentricity 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
)</li>
<li>scattering by a coated strip.</li></ol>
<h2 id="mathieu-functions-cosine-elliptic-and-sine-elliptic-functions">Mathieu functions: cosine-elliptic and sine-elliptic functions</h2>
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 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 or 
 
 
 
 2
 π
 
 
 {\displaystyle 2\pi }
 
. It is convenient to distinguish even and odd periodic solutions, which are termed <a href="/facts/Mathieu_function/zmsyWw0K">Mathieu functions</a> of first kind.
One of four simpler types can be considered: Periodic solution (
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 or 
 
 
 
 2
 π
 
 
 {\displaystyle 2\pi }
 
) symmetry (even or odd).
For 
 
 
 
 q
 ≠
 0
 
 
 {\displaystyle q\neq 0}
 
, the only periodic solutions y corresponding to any characteristic value 
 
 
 
 a
 =
 
 a
 
 r
 
 
 (
 q
 )
 
 
 {\displaystyle a=a_{r}(q)}
 
 or 
 
 
 
 a
 =
 
 b
 
 r
 
 
 (
 q
 )
 
 
 {\displaystyle a=b_{r}(q)}
 
 have the following notations:
ce and se are abbreviations for cosine-elliptic and sine-elliptic, respectively. 

<ul><li>Even periodic solution: 
 
 
 
 c
 
 e
 
 r
 
 
 (
 ω
 ,
 q
 )
 =
 
 ∑
 
 m
 
 
 
 A
 
 r
 ,
 m
 
 
 cos
 ⁡
 
 m
 ω
 
 
  for 
 
 a
 =
 
 a
 
 r
 
 
 (
 q
 )
 
 
 {\displaystyle ce_{r}(\omega ,q)=\sum _{m}A_{r,m}\cos {m\omega }{\text{ for }}a=a_{r}(q)}
 
</li>
<li>Odd periodic solution: 
 
 
 
 s
 
 e
 
 r
 
 
 (
 ω
 ,
 q
 )
 =
 
 ∑
 
 m
 
 
 
 A
 
 r
 ,
 m
 
 
 sin
 ⁡
 
 m
 ω
 
 
  for 
 
 a
 =
 
 b
 
 r
 
 
 (
 q
 )
 
 
 {\displaystyle se_{r}(\omega ,q)=\sum _{m}A_{r,m}\sin {m\omega }{\text{ for }}a=b_{r}(q)}
 
</li></ul>
where the sums are taken over even (respectively odd) values of m if the period of y is 
 
 
 
 π
 
 
 {\displaystyle \pi }
 
 (respectively 
 
 
 
 2
 π
 
 
 {\displaystyle 2\pi }
 
).
Given r, we denote henceforth 
 
 
 
 
 A
 
 r
 ,
 m
 
 
 
 
 {\displaystyle A_{r,m}}
 
 by 
 
 
 
 
 A
 
 m
 
 
 
 
 {\displaystyle A_{m}}
 
, for short.
Interesting relationships are found when 
 
 
 
 q
 →
 0
 
 
 {\displaystyle q\to 0}
 
, 
 
 
 
 r
 ≠
 0
 
 
 {\displaystyle r\neq 0}
 
:

 
 
 
 
 lim
 
 q
 →
 0
 
 
 c
 
 e
 
 r
 
 
 (
 ω
 ,
 q
 )
 =
 cos
 ⁡
 
 r
 ω
 
 
 
 {\displaystyle \lim _{q\to 0}ce_{r}(\omega ,q)=\cos {r\omega }}

lim
          
            q
            →
            0
          
        
        s
        
          e
          
            r
          
        
        (
        ω
        ,
        q
        )
        =
        sin
        ⁡
        
          r
          ω
        
      
    
    {\displaystyle \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 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
 and q.

<h2 id="multiresolution-analysis-filters-and-mathieus-equation">Multiresolution analysis filters and Mathieu's equation</h2>
<a href="/facts/Wavelets/3XcX4uIP">Wavelets</a> are denoted by 
 
 
 
 ψ
 (
 t
 )
 
 
 {\displaystyle \psi (t)}
 
 and <a href="/facts/Wavelet/3XcX4uIP">scaling functions</a> by 
 
 
 
 ϕ
 (
 t
 )
 
 
 {\displaystyle \phi (t)}
 
, with corresponding spectra 
 
 
 
 Ψ
 (
 ω
 )
 
 
 {\displaystyle \Psi (\omega )}
 
 and 
 
 
 
 Φ
 (
 ω
 )
 
 
 {\displaystyle \Phi (\omega )}
 
, respectively.
The equation 
 
 
 
 ϕ
 (
 t
 )
 =
 
 
 2
 
 
 
 ∑
 
 n
 ∈
 Z
 
 
 
 h
 
 n
 
 
 ϕ
 (
 2
 t
 −
 n
 )
 
 
 {\displaystyle \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 <a href="/facts/Multiresolution_Analysis/yQtKJuNx">Multiresolution Analysis</a> (MRA).

 
 
 
 H
 (
 ω
 )
 =
 
 
 1
 
 2
 
 
 
 
 ∑
 
 k
 ∈
 Z
 
 
 
 h
 
 k
 
 
 
 e
 
 j
 ω
 k
 
 
 
 
 {\displaystyle 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
 (
 ω
 )
 =
 
 
 1
 
 2
 
 
 
 
 ∑
 
 k
 ∈
 Z
 
 
 
 g
 
 k
 
 
 
 e
 
 j
 ω
 k
 
 
 
 
 {\displaystyle 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
          
            ν
          
        
        (
        ω
        )
        =
        
          e
          
            j
            (
            ν
            −
            2
            )
            [
            
              
                
                  ω
                  −
                  π
                
                2
              
            
            ]
          
        
        .
        
          
            
              c
              
                e
                
                  ν
                
              
              (
              
                
                  
                    ω
                    −
                    π
                  
                  2
                
              
              ,
              q
              )
            
            
              c
              
                e
                
                  ν
                
              
              (
              0
              ,
              q
              )
            
          
        
        .
      
    
    {\displaystyle G_{\nu }(\omega )=e^{j(\nu -2)[{\frac {\omega -\pi }{2}}]}.{\frac {ce_{\nu }({\frac {\omega -\pi }{2}},q)}{ce_{\nu }(0,q)}}.}

The transfer function of the "smoothing filter" of a Mathieu wavelet is

H
          
            ν
          
        
        (
        ω
        )
        =
        −
        
          e
          
            j
            ν
            [
            
              
                ω
                2
              
            
            ]
          
        
        .
        
          
            
              c
              
                e
                
                  ν
                
              
              (
              
                
                  ω
                  2
                
              
              ,
              q
              )
            
            
              c
              
                e
                
                  ν
                
              
              (
              0
              ,
              q
              )
            
          
        
        .
      
    
    {\displaystyle H_{\nu }(\omega )=-e^{j\nu [{\frac {\omega }{2}}]}.{\frac {ce_{\nu }({\frac {\omega }{2}},q)}{ce_{\nu }(0,q)}}.}

The characteristic exponent 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
 should be chosen so as to guarantee suitable initial conditions, i.e. 
 
 
 
 
 G
 
 ν
 
 
 (
 0
 )
 =
 0
 
 
 {\displaystyle G_{\nu }(0)=0}
 
 and 
 
 
 
 
 G
 
 ν
 
 
 (
 π
 )
 =
 1
 
 
 {\displaystyle G_{\nu }(\pi )=1}
 
, which are compatible with wavelet filter requirements. Therefore, 
 
 
 
 ν
 
 
 {\displaystyle \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 
 
 
 
 ν
 
 
 {\displaystyle \nu }
 
 zeroes in the interval 
 
 
 
 0
 ≤
 
 |
 
 ω
 
 |
 
 ≤
 π
 
 
 {\displaystyle 0\leq |\omega |\leq \pi }
 
.

The G and H filter coefficients of Mathieu MRA can be expressed in terms of the values 
 
 
 
 {
 
 A
 
 2
 l
 +
 1
 
 
 
 }
 
 l
 ∈
 Z
 
 
 
 
 {\displaystyle \{A_{2l+1}\}_{l\in Z}}
 
 of the Mathieu function as:

h
              
                l
              
            
            
              2
            
          
        
        =
        −
        
          
            
              
                A
                
                  
                    |
                  
                  2
                  l
                  +
                  1
                  
                    |
                  
                
              
              
                /
              
              2
            
            
              c
              
                e
                
                  ν
                
              
              (
              0
              ,
              q
              )
            
          
        
      
    
    {\displaystyle {\frac {h_{l}}{\sqrt {2}}}=-{\frac {A_{|2l+1|}/2}{ce_{\nu }(0,q)}}}

g
              
                l
              
            
            
              2
            
          
        
        =
        (
        −
        1
        
          )
          
            l
          
        
        
          
            
              
                A
                
                  
                    |
                  
                  2
                  l
                  −
                  3
                  
                    |
                  
                
              
              
                /
              
              2
            
            
              c
              
                e
                
                  ν
                
              
              (
              0
              ,
              q
              )
            
          
        
      
    
    {\displaystyle {\frac {g_{l}}{\sqrt {2}}}=(-1)^{l}{\frac {A_{|2l-3|}/2}{ce_{\nu }(0,q)}}}

There exist recurrence relations among the coefficients:

(
        a
        −
        1
        −
        q
        )
        
          A
          
            1
          
        
        −
        q
        
          A
          
            3
          
        
        =
        0
      
    
    {\displaystyle (a-1-q)A_{1}-qA_{3}=0}

(
        a
        −
        
          m
          
            2
          
        
        )
        
          A
          
            m
          
        
        −
        q
        (
        
          A
          
            m
            −
            2
          
        
        +
        
          A
          
            m
            +
            2
          
        
        =
        0
      
    
    {\displaystyle (a-m^{2})A_{m}-q(A_{m-2}+A_{m+2}=0}

for 
 
 
 
 m
 ≥
 3
 
 
 {\displaystyle m\geq 3}
 
, m odd.
It is straightforward to show that 
 
 
 
 
 h
 
 −
 l
 
 
 =
 
 h
 
 
 |
 
 l
 
 |
 
 −
 1
 
 
 
 
 {\displaystyle h_{-l}=h_{|l|-1}}
 
, 
 
 
 
 ∀
 l
 >
 0
 
 
 {\displaystyle \forall l>0}
 
.
Normalising conditions are 
 
 
 
 
 ∑
 
 k
 =
 −
 ∞
 
 
 k
 =
 +
 ∞
 
 
 
 
 h
 
 k
 
 
 =
 −
 1
 
 
 
 {\displaystyle \sum _{k=-\infty }^{k=+\infty }{h_{k}=-1}}
 
 and 
 
 
 
 
 ∑
 
 k
 =
 −
 ∞
 
 
 k
 =
 +
 ∞
 
 
 
 (
 −
 1
 
 )
 
 k
 
 
 
 h
 
 k
 
 
 =
 0
 
 
 
 {\displaystyle \sum _{k=-\infty }^{k=+\infty }{(-1)^{k}h_{k}=0}}
 
.

<h2 id="waveform-of-mathieu-wavelets">Waveform of Mathieu wavelets</h2>
Mathieu wavelets can be derived from the lowpass reconstruction filter by the <a href="/facts/Cascade_algorithm/LNLMHHc0">cascade algorithm</a>. Infinite Impulse Response filters (<a href="/facts/IIR_filter/0KLkxylJ">IIR filter</a>) should be use since Mathieu wavelet has no <a href="/facts/Compact_support/BTuMGbsb">compact support</a>. 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.

<h2 id="references">References</h2>

<ol>
<li id="fn:1">L. Ruby, “Applications of the Mathieu Equation,” Am. J. Phys., vol. 64, pp. 39–44, Jan. 1996 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">M.M.S. Lira, H.M. de Oiveira, R.J.S. Cintra. Elliptic-Cylindrical Wavelets: The Mathieu Wavelets,IEEE Signal Processing Letters, vol.11, n.1, January, pp. 52–55, 2004. <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">É. Mathieu, Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique, J. Math. Pures Appl., vol.13, 1868, pp. 137–203. <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">N.W. McLachlan, Theory and Application of Mathieu Functions, New York: Dover, 1964. <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Mathieu wavelet open-in-new

Mathieu wavelet