The polynomial system of equations

The coefficients of polynomials $H_0,\ldots H_{M-1}$ from the cascade in [4] are polynomials in terms of the cosines and sines of the angles. The flatness equations for the polynomials $H_0,\ldots H_{M-1}$ are linear combinations of the coefficients of the polynomials $H_0,\ldots H_{M-1}$.This shows the principle leading to the use of techniques for solving polynomial systems in this signal processing context.

Actually the straightforward application of existing algorithms to the polynomial system obtained by writing the flatness equations in terms of the cascade parameters cannot design filters with support larger than 6$\times$6 and 2 degrees of flatness. So to design filter banks with higher regularity we propose a change of variables, splitting the system into two smaller subsystems, which make it possible to design filters with support 16$\times$16 and 5 degrees of flatness.

The cascade form we use in the sequel, borrowed from [4], and the formulation of the issue, maximizing the flatness of filters of given size, turns it into a polynomial system.