Optimal wavelet expansion via sampled-data H∞ control theory

Wavelet expansion of an L² signal requires the L² inner product of the original signal and a scaling function. In digital signal processing, it is common to use sampled data of continuous-time signals instead of the inner product. This however causes a large reconstruction error, called "wavelet crime." We therefore design a causal system which produces an approximation of the inner product via sampled-data H^∞ control theory. We then make extensions to a multi-rate and a multi-wavelet case. By numerical examples, we show the effectiveness of the proposed method.