Multiscale deconvolution of sensor array signals via sum-of-cumulants

This correspondence presents a solution to a multiscale deconvolution problem using higher order spectra where the data to be deconvolved consist of noise-corrupted sensor array measurements. We assume that the data are generated as a convolution of an unknown wavelet with reflectivity sequences that are linearly time-scaled versions of an unknown reference reflectivity sequence. This type of data occurs in many signal processing applications, including sonar and seismic processing. Our approach relies on exploiting the redundancy in the measurements due to time scaling and does not require knowledge of the wavelet or the reflectivity sequences. We formulate and solve the deconvolution problem as a quadratic minimization subject to a quadratic constraint in the sum-of-cumulants (SOC) domain. The formulation using the SOC approach reduces the effect of additive Gaussian noise on the accuracy of the results when compared with the standard time-domain formulation. We demonstrate this improvement using a simulation example.