On the symmetry of orthogonal complex filter banks and wavelets

Recent wavelet research has primarily focused on real-valued wavelet bases. However, complex wavelet bases offer a number of potential advantages. For example, it has been recently suggested in literature 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 by-product, we propose a method to obtain a complex orthogonal wavelet basis having the symmetry property and approximately linear phase.