Cramér-Rao bounds for multiple poles and coefficients of quasi-polynomials in colored noise

In this paper, we provide analytical expressions of the Cramér-Rao bounds for the frequencies, damping factors, amplitudes, and phases of complex exponentials in colored noise. These expressions show the explicit dependence of the bounds of each distinct parameter with respect to the amplitudes and phases, leading to readily interpretable formulae, which are then simplified in an asymptotic context. The results are presented in the general framework of the Polynomial Amplitude Complex Exponentials (PACE) model, also referred to as the quasi-polynomial model in the literature, which accounts for systems involving multiple poles and represents a signal as a mixture of complex exponentials modulated by polynomials. This work looks further and generalizes the studies previously undertaken on the exponential and the quasi-polynomial models.