## Section: New Results

### Approximation and parametrization of wavelets

Participants : Martine Olivi, Bernard Hanzon, Ralf Peeters [ Univ. Maastricht ] , Jean-Paul Marmorat, Vikentiy Mikheev, Jérémie Giraud-Telme.

An application of our rational approximation methods to orthogonal wavelets has been investigated.
The problem is to implement wavelets in analog circuits in view of medical signal processing applications. A dedicated method has been developed [75] based on an L^{2} -approximation of the wavelet by the impulse response of a stable causal low order filter. However, this method fails to find an accurate and sufficiently small order approximation in some difficult cases (Daubechies db7 and db3). The idea was to use the software RARL2 to perform a model reduction on an accurate high order (100-200) approximation. However, an admissibility condition for wavelets is that the integral of a wavelet equals zero, which means that it has one vanishing moment. The low order approximation is still required to have an integral zero, otherwise undesired bias will show up when the wavelet is used in an application. We thus had to adapt a version of the RARL2 software to address this constraint. Since we are dealing with a quadratic optimization problem under a linear constraint, this can be solved analytically. We could thus reformulate the problem of L^{2} -approximation subject to this constraint into an optimization problem over the class of lossless systems. This could be handled by the software with only minor changes and we were able to perform an accurate approximation of order 8 for db7 (Figure 6 ).
This way to address a linear or a convex constraint could be used for other purposes, for example to impose passivity.

In close connection, we investigate the possibility to parametrize wavelets with (more) vanishing moments using interpolation theory at the boundary.
A very useful and concise description of the class of filter banks leading to orthogonal wavelets is by means of its associated *lossless* polyphase filter [89] .
A vanishing moment condition can be expressed as a boundary interpolation condition for the lossless polyphase filter. We thus exploited our previous works on the parametrization
of lossless matrix functions with interpolation conditions. We got explicit parametrizations of 2×2 polyphase matrices of arbitrary order n with (up to) 3 vanishing moments built in, in terms of angular derivative (positive) parameters. However, the conditions were cleverly handled in an unusual recursive fashion that we still do not completely understand. These results have been presented at the ANR-AHPI meeting.