Periodic Wavelet Transforms and Periodicity Detection

The theory of periodic wavelet transforms presented here was originally developed to deal with the problem of epileptic seizure prediction. A central theorem in the theory is the characterization of wavelets having time and scale periodic wavelet transforms. In fact, we prove that such wavelets are precisely generalized Haar wavelets plus a logarithmic term. It became apparent that the aforementioned theorem could not only be quantified to analyze seizure prediction, but could also provide a technique to address a large class of periodicity detection problems. An essential step in this quantification is the geometric and linear algebra construction of a generalized Haar wavelet associated with a given periodicity. This gives rise to an algorithm for periodicity detection based on the periodicity of wavelet transforms defined by generalized Haar wavelets and implemented by wavelet averaging methods. The algorithm detects periodicities embedded in significant noise. The algorithm depends on a discretized version W p ψf(n, m) of the continuous wavelet transform. The version defined provides a fast algorithm with which to compute W p ψf(n, m) from W p ψf(n−1, m) or W p ψf(n, m − 1). This has led to the theory of non–dyadic wavelet frames in l2(Z) developed by the second-named author, and which will appear elsewhere.