Optimal estimation of amplitude and phase modulated signals (Q2724973)

The present paper deals with an on-line estimation of phases and amptitudes of a non-stationary discrete model \(y_t=C(\theta_t)a_t+Dw_t\), where \(C(\theta_t)=(cos\theta_t(1),\dots ,cos\theta_t(K))\), \(w_t\) i.i.d. \(\sim {N(0,1)}, a_t=(a_t(1), \dots, a_t(K))^T = Aa_{t-}+Bv_t,v_t,w_t\) are mutually independet noises with \(v_t\) i.i.d. \(\sim {N(0,I)},a_0 \sim {N(\widehat {a}_0,P_0)}\), \(\theta_t\) is a Markov process with \( \theta_0 \sim {U([0,\pi ]^K}\) and a transition density. An efficient sequential Monte Carlo approach is developed to perform the sequential Bayesian estimation of parameters \( \theta_t,a_t\). The main point is that the estimation of Bayesian density \( p(\theta_t,a_t|y_1, \dots, y_t)\) can be reduced to estimate the marginal Bayesian density \(p(\theta_t|y_1,\dots, y_t)\). NEWLINENEWLINENEWLINETo obtain the estimation of the sequential posterior density, the particle filtering, initiated by \textit{N. J. Gotden, D. J. Salmond} and \textit{A. F. M. Smith} [Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings-F, 140, 107-113 (1993)], is adopted and the propagation of the particles over time, that is, from the estimation of \( \theta_0, \dots, \theta_{t-1}\) to attain an estimation of \( \theta_t\), is given by using a combination technique of the sequential importance sampling, the Bootstrap resampling, the extended Kalman filtering and the work of approaches. It then provides pratically an efficient way of obtaining the on-line estimation of phases and amplitudes. NEWLINENEWLINENEWLINEThe idea of this paper is something related to \textit{A. Doucet, S. J. Godsill} and \textit{C. Andrieu} [On sequential Monte Carlo sampling methods for Bayesian filtering. Statist. Comput. 10, 197-208 (2000)].