Orthogonal complex filter banks and wavelets: Some properties and design

Recent wavelet research has primarily focused on real-valued wavelet bases. However, complex wavelet bases offer a number of potential advantageous pr operties, For example, it has been recently suggested that the complex Daub echies wavelet can be made symmetric. However, these papers always imply th at if the complex basis has a symmetry property, then it must exhibit linea r phase as well. In this paper, we prove that a linear-phase complex orthog onal wavelet does not exist. We study the implications of symmetry and line ar phase for both complex and real-valued orthogonal wavelet bases. As a by product, we propose a method to obtain a complex orthogonal wavelet basis h aving the symmetry property and approximately linear phase. The numerical a nalysis of the phase response of various complex and real Daubechies wavele ts is given, Both real and complex symmetric orthogonal wavelet can only ha ve symmetric amplitude spectra, It is often desired to have asymmetric ampl itude spectra for processing general complex signals. Therefore, we propose a method to design general complex orthogonal perfect reconstruct filter b anks (PRFB's) by a parameterization scheme. Design examples are given. It i s shown that the amplitude spectra of the general complex conjugate quadrat ure filters (CQF's) 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.