Cramér-Rao bound for estimates of frequencies and damping factors of real superimposed signals with multiple poles in noise

The problem of estimating frequencies and damping factors of real superimposed signals with multiple poles in white Gaussian noise is considered. Such signals are described by real quasipolynomials, i.e. by linear combinations of real damped sinusoids multiplied by power functions. In a particular case when poles are simple, a real quasipolynomial becomes a real damped sinusoid. An explicit expression of the Cramér-Rao bound (CRB) for the estimation of frequencies and damping factors of the signals is obtained. To derive the CRB, we use the expression for the Fisher information matrix (FIM) which we obtained in a previous paper for the model of complex quasipolynomials (i.e. complex exponentials multiplied by complex polynomials). We rewrite the model of real quasipolynomials as a model of complex quasipolynomials with constraints imposed on the parameter set. Then we make use of the formula presented by Gorman and Hero that allows us to obtain the CRB for the model with constraints from the FIM for the model without constraints. The results of numerical simulations are presented and discussed.