Orthonormal bases of bandlimited wavelets with frequency overlap

Motivated by their utility in coding of digital communication signals, numerous examples of orthonormal wavelet sets that are generated from a bandlimited mother wavelet have been constructed. In most of these examples, orthogonality of the replicates at different scales is achieved by ensuring that the frequency domain supports at every scale are disjoint. Given the mother wavelet for such a set, this paper describes how it can be extended to yield a new bandlimited mother wavelet which generates an orthonormal set and whose dilated replicates overlap in the frequency domain. The utility of these wavelets to improve robustness of wavelet-coded communication signals with respect to demodulation by an unintended receiver is also discussed. The possibility of developing a new class of symbols by unitary warping of the frequency axis is also considered, but is shown not to be viable.