Abstract

Daughters $\left\{h_{ab}(t)=h[(t-b)/a]/\sqrt{\text{a}}\right\}$constructed by dilation aand shift bof a single mother wavelet h(t) form the basis of the wavelet transform (WT) coined by French geologists Morlet et al.in 1985. The physics behind the WT is a bank of matched filters¹of a constant resonator Qfor seismic imaging. The condition to be a mother wavelet is to be zero-dc square integrable. Let the Fourier transform (FT) be the inner product: $\text{FT}\left\{e\right\}=(s,e)={\displaystyle \int s}(t)e(t)dt$, with e(t) = exp(2πjft). Then a windowed FT becomes the Gabor GT{s} = (s,g), where g(t) = exp[−(t/τ)²]exp(2πjft), and Morlet's WT{s} = (s,g_ab), where ${}_{gab}(t)=g[(t-b)/a]/\sqrt{\text{a}}$. Applications are based either on an efficient, faithful, noise-immune wide band transient representation or on a sensitive, robust, truthful signal classsification. While the former optimizes the majority signals' energy, the latter works on the overlapping boundary and outliers. A superset of mother wave lets is introduced for adaptive WT using neural nets to explain the cocktail party hearing effect.

© 1992 Optical Society of America