A study of two-channel separable complex-valued wavelets and filter banks

In this paper, a general method for designing orthogonal and bi-orthogonal separable complex-valued wavelets and filter banks (SCVWFB) derived from existing real-valued wavelets and filter banks (RVWFB) is proposed. With this method, complex-valued wavelets and filter banks (CVWFB) are designed from existing RVWFBs regardless of wavelet family, length, or order of regularity, and without solving highly complex design equations. The resulting necessary and sufficient conditions derived in this work show that the only valid forms of SCVWFBs are complex rotations of RVWFBs and hybrid CVFWBs. Furthermore, the results show that none of the valid SCVWFB coefficients converge to wavelet coefficients. Thus, this work proves that no separable complex-valued wavelet (SCVW) coefficients exist.