Necessary conditions on the lifting scheme for existence of wavelets in L2(R)

Over the years many different libraries of biorthogonal wavelets have been designed. In addition, the lifting scheme has been developed which unifies the design of biorthogonal wavelets using a simple approach. The lifting scheme allows the custom design of wavelets while maintaining the biorthogonal constraint on the filter bank. A drawback of the lifting scheme is that it does not guarantee that the wavelet associated with the iterated filter bank exists in L₂(R). In practice this raises the possibility that the filter bank may produce numerical artifacts. We present some necessary conditions on the lifting scheme for the resulting wavelets to exist in L₂(R). Using these conditions, we show how valid wavelet bases can be easily generated from randomly perturbed biorthogonal filter banks