Solvability of the Zero-Pinning Technique to Orthonormal Wavelet Design

A zero-pinning technique for orthonormal wavelet design proposed by Tay results in a system of linear equations. We first prove the existence and uniqueness to the solution of the linear system. For orthonormal wavelet filters, non-negativity is known to be a necessary condition. However, it is not sufficient. A tau-cycle condition is cited as one in verifying a wavelet filter being orthonormal. Finally, we show that the amplitude of ripples between two successive zeros of the parametric Bernstein polynomials decreases as the distance between these two zeros decreases.