Multiresolution Analysis Filters and Mathieu's Equation
Wavelets are denoted by and scaling functions by, with corresponding spectra and, respectively.
The equation, which is known as the dilation or refinement equation, is the chief relation determining a Multiresolution Analysis (MRA).
is the transfer function of the smoothing filter.
is the transfer function of the detail filter.
The transfer function of the "detail filter" of a Mathieu wavelet is
The transfer function of the "smoothing filter" of a Mathieu wavelet is
The characteristic exponent should be chosen so as to guarantee suitable initial conditions, i.e. and, which are compatible with wavelet filter requirements. Therefore, 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 zeroes in the interval .
The G and H filter coefficients of Mathieu MRA can be expressed in terms of the values of the Mathieu function as:
There exist recurrence relations among the coefficients:
for, m odd.
It is straightforward to show that, .
Normalising conditions are and .
Read more about this topic: Mathieu Wavelet
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