Orthogonal complex filter banks and wavelets: some properties and design

Previous wavelet research has primarily focused on real-valued wavelet bases. However, complex wavelet bases offer a number of potential advantageous properties. For example, it has been suggested that the complex Daubechies wavelet can be made symmetric. However, these papers always imply that if the complex basis has a symmetry property, then it must exhibit linear phase as well. In this paper, we prove that a linear-phase complex orthogonal wavelet does not exist. We study the implications of symmetry and linear phase for both complex and real-valued orthogonal wavelet bases. As a byproduct, we propose a method to obtain a complex orthogonal wavelet basis having the symmetry property and approximately linear phase. The numerical analysis of the phase response of various complex and real Daubechies wavelets is given. Both real and complex-symmetric orthogonal wavelet can only have symmetric amplitude spectra. It is often desired to have asymmetric amplitude spectra for processing general complex signals. Therefore, we propose a method to design general complex orthogonal perfect reconstruct filter banks (PRFBs) by a parameterization scheme. Design examples are given. It is shown that the amplitude spectra of the general complex conjugate quadrature filters (CQFs) can be asymmetric with respect the zero frequency. This method can be used to choose optimal complex orthogonal wavelet basis for processing complex signals such as in radar and sonar