On Hilbert transform and related integral transforms of wavelets

Coautor(es): Dr. Shiv Kumar Kaushik
During the past two decades, wavelet theory has entrenched itself as one of the most efficacious mathematical tools for a scopic extent of signal processing applications, such as data and image compression, transient detection, noise reduction, texture analysis, pattern recognition, and singularity detection. It is well-known that Hilbert transform of a wavelet is again a wavelet. Hilbert transform of wavelets are orthogonal to their translates, form a basis for L 2 (R) and define a multiresolution analysis (MRA). The fundamental reasons for the seamless integration of Hilbert transform into the multiresolution framework of wavelets are its scale and translation invariances and its energy-preserving (unitary) nature. In this talk, we study wavelets obtained by applying Hilbert transform and other related integral transforms. Various results are given to approximate the functions in L 2 (R) and sufficient conditions have been obtained for higher vanishing moments of such wavelets. Finally, convolution and cross-correlation theorems are given to study convolved and cross-correlated signals using such wavelets.