Even-length, 3-balanced

There are 12 different minimal-length pairs of scaling filters. The best solution, obtained using Gröbner bases, is shown in this figure, and is given by the formulas in the MATLAB program symbal3e.m. The numerical solutions are tabulated in the file symbal3e.float.

The scaling filters h0 and h1 were found by converting the nonlinear design equations (eqs) into a lexical Gröbner basis (gb.lp), and factorizing that into six disjoint Gröbner bases (gb.lp.fact.1, gb.lp.fact.2, gb.lp.fact.3, gb.lp.fact.4, gb.lp.fact.5, gb.lp.fact.6). Notice that the factored form of the lexical Gröbner basis is much more convenient than the unfactored form and has much shorter coefficients. All 12 minimal-length pairs of scaling filters can be found by solving these 6 Gröbner bases. The smoothest solution, shown in the figure, is given by one of the solutions of gb.lp.fact.3.

To obtain the wavelet filters h2 and h3, the nonlinear design equations (eqs.B) for the total system (scaling and wavelet filters) were appended to the Gröbner basis corresponding to the best scaling functions (gb.lp.fact.3). For this set of equations, the lexical Gröbner basis (gb.B.lp) was obtained, which yields both the scaling filters h0, h1 and the wavelet filters h2, h3 tabulated in the MATLAB program symbal3e.m. Note that in the design equations, the filters h0 and h1 are normalized so that the their DC gain is 1.

Image of scaling/wavelet functions/filters.


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