A new class of orthonormal symmetric wavelet bases using a complex allpass filter

This paper considers the design of the WSS paraunitary filterbanks composed of a single complex allpass filter and gives a new class of real-valued orthonormal symmetric wavelet bases. First, the conditions that the complex allpass filter has to satisfy are derived from the symmetry and orthonormality conditions of wavelets, and its transfer function is given to satisfy these conditions. Second, the paraunitary filter banks are designed by using the derived transfer function from the viewpoints of the regularity and frequency selectivity. A new method for designing the proposed paraunitary filterbanks with a given degrees of flatness is presented. The proposed method is based on the formulation of a generalized eigenvalue problem by using the Remez exchange algorithm. Therefore, the filter coefficients can be easily obtained by solving the eigenvalue problem, and the optimal solution is attained through a few iterations. Furthermore, both the maximally flat and minimax solutions are also included in the proposed method as two specific cases. The maximally flat filters have a closed-form solution without any iteration. Finally, some design examples are presented to demonstrate the effectiveness of the proposed method.