On wideband deconvolution using wavelet transforms

A discussion on the expression proposed by Weiss et al. (see J. Acoust. Soc. Amer., vol.96, p.850-6 and p.857-66, 1994 and IEEE Signal Processing Mag., vol.11, p.13-32, 1994) for deconvolving the wideband density function is presented. We prove here that such an expression reduces to be proportional to the wideband correlation receiver output, or continuous wavelet transform of the received signal with respect to the transmitted one. Moreover, we show that the same result has been implicitly assumed by Weiss et al., when the deconvolution equation is derived. We stress the fact that the analyzed approach is just the orthogonal projection of the density function onto the image of the wavelet transform with respect to the transmitted signal. Consequently, the approach can be considered a good representation of the density function only under the prior knowledge that the density function belongs to such a subspace. The choice of the transmitted signal is thus crucial to this approach.