Design of matched wavelets based on generalized Mexican-hat function

Based on a generalized Mexican-hat function proposed, the paper presents an algorithm for the design of a new class of continuous wavelets matched to arbitrary transient signals. While the generalized Mexican-hat wavelets are constructed from the standard Mexican-hat function by utilizing a polynomial series with the admissible condition imposed, the signal matching characteristics are achieved by minimizing the spectral differences between the reference signal and the generalized Mexican-hat wavelet via even and odd symmetry decomposition. Detail derivations are given to yield a set of linear simultaneous equations for determination of the appropriate polynomial coefficient values for the generalized Mexican-hat wavelet, thereby shaping its waveform to resemble the reference transient signal. The solution for the equations is shown to be given by a combination of analytical expressions and numerical approximation. Also included in the paper are computer simulations and practical examples to demonstrate the signal matching quality and the influence of various design parameters. Comparisons are made with the standard Mexican-hat wavelet and correlation methods to reveal the superior performance of the generalized and matched Mexican-hat wavelet, in terms of enhancement of signal features in the time-scale domain, and reliable detection of transient signals and their variants embedded in noise.