A new transform for solving the noisy complex exponentials approximation problem
This work provides a novel method for signal processing and inverse problems where complex exponentials are approximated from noisy data, but the results are preliminary and incremental.
The paper addresses the problem of estimating a complex measure composed of Dirac distributions from noisy complex moments. It proposes a new discrete transform that enables estimation of the unknown measure, with simulations validating the approach.
The problem of estimating a complex measure made up by a linear combination of Dirac distributions centered on points of the complex plane from a finite number of its complex moments affected by additive i.i.d. Gaussian noise is considered. A random measure is defined whose expectation approximates the unknown measure under suitable conditions. An estimator of the approximating measure is then proposed as well as a new discrete transform of the noisy moments that allows to compute an estimate of the unknown measure. A small simulation study is also performed to experimentally check the goodness of the approximations.