Characterization of wavelet coefficients for ultrasonic signals

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Abstract
A wavelet transform has been widely used to investigate the characteristics of wave signals for a decade. However, only qualitative investigation of the spectrogram was made rather than a quantitative interpretation. On the other hand, an analytical closed-form representation of the wavelet transformed wave signal can be used as a basis function in estimating parameters using nonlinear least-squares optimization. We derived a quantitative closed-form equation directly from the analytical continuous wavelet transformation of a pulse with a Gaussian spectrum. A fundamental three-dimensional shape of a wavelet in the spectrogram was obtained, and the analytical form was compared quantitatively with numerical results.

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