Odd-length, 2-balanced
There are 4 different minimal length pairs of scaling filters. Obtained using Gröbner bases, they are tabulated in the MATLAB program symbal2o.m. To obtain the 4 solutions, use the programs with the following syntax:
[h0,h1] = symbal2o([0,0]) [h0,h1] = symbal2o([0,1]) [h0,h1] = symbal2o([1,0]) [h0,h1] = symbal2o([1,1])The best solution, given by [h0,h1]=symbal2o([0,0]) is shown in this figure. The numerical solutions are tabulated in the file symbal2o.float.
The scaling filters h0 and h1 were found by converting the nonlinear design equations into a lexical Gröbner basis (gb.lp), and factorizing it to remove repeated roots. The new lexical Gröbner basis (gb.lp.fact) is `square free'. All 4 minimal length pairs of scaling filters can be found by solving this Gröbner basis.
We also provide for this example the Maple program for automatically generating the equations (setup), the Singular program for obtaining the Gröbner bases (sfile), and the Maple program for solving the the Gröbner basis (result). Executing these programs in sequence will regenerate the filter coefficients. The design equations normalize the filters so that their DC gain is 1.
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