#### On Hilbert transform and related integral transforms of wavelets

###### Autor: NIKHIL KHANNA .

###### 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.