A new transform for solving the noisy complex exponentials approximation problem

Abstract

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 computing an estimate of the unknown measure. A small simulation study is also performed to experimentally check the goodness of the approximations.

Estimation of a new stochastic transform for solving the complex exponentials approximation problem: Computational aspects and applications

“…The following result, proved in [1], gives the relation between S n (z, σ ) and the unknown measure S(z).…”

“…In [2] it was proved that, when s = 0, in the limit for n → ∞ the condensed density is a distribution supported on the unit circle and it can be proved [1] that in the limit for σ → ∞ the generalized eigenvalues ξ j tend to concentrate on the unit circle and, in the limit for σ → 0, they concentrate around the true ξ j , j = 1, . .…”

“…. , 2p − 1, (1) or, equivalently [15], to find poles ξ j and corresponding residues r j = c j /ξ j of the rational function s(z) whose first 2p Taylor coefficients at z = 0 are s k , k = 0, 1, 2, . .…”

“…The problem is optimally conditioned when ξ j are equispaced on the unit circle. In this case in fact model (1) reduces to the Fourier model which is an orthogonal one. Clusters of ξ j are more difficult to estimate than well separated ones.…”

“…Complex exponentials with relatively small |c j | are more difficult to estimate than those with relatively large weight. Recently a new approach for solving the complex exponential approximation problem in a stochastic framework was proposed [1], which exploits the relation with generalized eigenvalue problems and with moments problems outlined above but without assuming to know p. It makes use of tools from the theory of logarithmic potential with external fields [23] and the theory of random polynomials [5,14] and provides an estimate of p and point and interval estimates of {c j , ξ j }.…”