CramÉr–Rao Bounds for Multiple Poles and Coefficients of Quasi-Polynomials in Colored Noise

Abstract

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 quasipolynomial 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 quasipolynomial models.

Performance of ESPRIT for Estimating Mixtures of Complex Exponentials Modulated by Polynomials

“…They match the theoretical variance, which confirms the validity of our perturbation analysis for this SNR. The variation rate of the variances is similar to that of the Cramér Rao bounds [12]: it is broken at ∆f = 1 N = 5 10 −3 , which corresponds to the resolution limit of Fourier analysis. At this limit point, the relative variance of the amplitude estimate is still lower than -60 dB, which shows the good resolution of the ESPRIT algorithm.…”

“…Therefore we focus here on the dependency on the frequency gap between two components (section V-A1), the damping factor (section V-A2), the spectral flatness of the noise (section V-A3), and the order of a pole (section V-A4). For these simulations, the same synthetic signals as those introduced in [12] are used. In the figures below, the solid lines represent the theoretical variance of the frequency estimators or that of the damping factor estimators, which are equivalent according to (17).…”

“…An analysis of the Cramér-Rao bounds for the frequencies and damping factors of complex quasipolynomials in white noise was proposed in [10]. In [12], we derive analytic expressions of the Cramér-Rao bounds for the frequencies, damping factors, amplitudes and phases of quasipolynomials in colored noise, and these expressions are simplified in an asymptotic context. In particular, it is shown that the CRB for the parameters associated to a multiple pole present an exponential increase with the order of the pole, which suggests that the practical estimation of the PACE model is only possible if the exponentials are modulated by polynomials of low order.…”

Stability and super-resolution of generalized spike recovery

“…In Section 6, Theorem 6.1 we specialize the above results to the system (2), and subsequently discuss their relation to existing works in the literature, in particular [6,16,17,18,23,30,40,41,42,43,47,48,54].…”

“…In [40] the authors only prove the CRB estimates for K = 1, 2 and N ≫ 1, for the system (2). On the other hand, the authors of [6] consider the more general system (3) (called PACE model), and derive asymptotic estimates for N ≫ 1. These results are qualitatively similar to our Theorem 2.2 and Theorem 2.1.…”