Abstract
The notion of phase tracking has found its applications in signal processing, navigation system, radar and communication systems. The nonlinear phase tracking filtering is attributed to nonlinear noisy observables that combines dynamics and observables. The precised phase angle estimation is always a potential task in practical problems. With this motivation, in this paper, we propose a new estimation method, which is a relatively very scarce, but exploits an appealing framework because of its concerning algorithmic convenience stemming from the bilinearizability. The novelty of the method is the Carleman linearization-based filtering. The Carleman linearization-based phase tracking filtering is contrasted with the two-benchmark filters, the EKF and a second-order Gaussian filtering. This paper develops a phase tracking filtering algorithm in the Carleman framework for the case, where the process noise is the OU process and measurement system noise is the Wiener process. Notably, this paper develops mathematics of stochastic phase tracking problem in the Carleman framework.
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Appendix
Appendix
Consider the Carleman linearized Itô bilinear SDE,
We wish to calculate the expectation of the term \(E(d\xi_{t} d\xi_{t}^{T} ).\) The term signifies the diffusion coefficient term in the variance evolution of the filtering. Thus,
and
After applying the Itô differential rule, we have
where \(\overbrace {{\xi_{t} \xi_{t}^{T} }}^{\Lambda } = E(\xi_{t} \xi_{t}^{T} \left| {t_{0} } \right.,\xi_{{t_{0} }} ) = V_{t} + \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\xi }_{t} \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\xi }_{t}^{T} .\) Recall the Riccati equation in the linear filtering. The generalization of the Riccati equation is natural because of the bilinearity. The ‘generalized’ Riccati equation combines the Riccati equation and the corrections to the diffusion coefficient, i.e. the terms associated with the coefficient of the term \(dt\) of the right-hand side of the above equation. Thus, the variance evolution of the Carleman linearized SDE is a generalized Riccati equation, i.e.
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Medewar, P.G., Sharma, S.N. & Patel, H.G. Carleman framework-based filtering for a nonlinear phase tracking problem. Int. J. Dynam. Control 11, 194–204 (2023). https://doi.org/10.1007/s40435-022-00937-1
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DOI: https://doi.org/10.1007/s40435-022-00937-1