Wavelet interpolation: from orthonormal to the oversampled wavelet transform

The orthonormal wavelet transform is an efficient method of signal representation since there is no redundancy in its expression, but due to aliasing in the decimation stage it lacks the often desired property of shift invariance. On the other hand, the oversampled or nonorthogonal wavelet offers a finer resolution in translation; thus reducing the effect of shift of origin, it is more robust to changes in the initial phase of the signal. In some areas of signal processing, such as wideband correlation processing, sensitivity to time alignment necessitates the use of the nonorthogonal wavelet transform. The price paid for the advantage of robustness to shifting is the introduction of redundancy in the expression. In many applications, both of these properties are needed in different stages of signal processing. Thus there is a need to know the conditions under which the redundant and nonorthonormal wavelet transform coefficients can be derived from the orthonormal wavelet transform coefficients. The answer provides us with a convenient way to switch between these two forms: the orthonormal wavelet for efficient expression, and the nonorthogonal one whenever it is necessary for feature extraction. Alternatives to implementation of the translation axis oversampling by using filter banks are also suggested