Nonlinear Filtering of Weak Chaotic Signals

Abstract

In recent years, the application of nonlinear filtering for processing chaotic signals has become relevant. A common factor in all nonlinear filtering algorithms is that they operate in an instantaneous fashion, that is, at each cycle, a one moment of time magnitude of the signal of interest is processed. This operation regime yields good performance metrics, in terms of mean squared error (MSE) when the signal-to-noise ratio (SNR) is greater than one and shows moderate degradation for SNR values no smaller than À3 dB. Many practical applications require detection for smaller SNR values (weak signals). This chapter presents the theoretical tools and developments that allow nonlinear filtering of weak chaotic signals, avoiding the degradation of the MSE when the SNR is rather small. The innovation introduced through this approach is that the nonlinear filtering becomes multimoment, that is, the influence of more than one moment of time magnitudes is involved in the processing. Some other approaches are also presented.

Ubiquitous Filtering for Nonlinear Problems

“…The reason behind this is the identification of Eq. (1) is an identification of the "inertial vector nonlinear system" which does not have an unique solution and can be formulated only for a previously defined class of nonlinear systems; the complexity of this task has been addressed elsewhere [8,9,18,20] and will not be considered in the following. As examples for F(•), which will be used in the following, there are the equations for the chaotic attractors corresponding to Rossler, Lorenz, and Chua types [8,9]…”

“…The solutions proposed hereafter might be encountered from the structural analysis of the quasi-optimum filtering algorithms for weak chaos in presence of AWGN [8,9] and are synthesized in the following for convenience.…”

“…The dynamic filtering approach, such as classic linear Kalman filtering, has been applied for many problems long ago [6] and recently as well [7]. However, in the following the dynamic filtering is proposed adopting a different (nonlinear) angle [8,9], namely, using signals from nonlinear chaotic attractors as a model for the desired signals arriving at the filtering structure. The modeling of real phenomena using chaos has been used for more than 50 years, and there is a wide range of scientific and practical applications, such as seismology [10][11][12], statistical theory of communication [13,14], control, geophysics, biomedical telemetry [15,16] under water signal processing [17], and many other areas related to applied physics as well [18].…”

“…Chaos modeling might be immensely useful because almost all quasi-optimum filtering algorithms (which formally are nonlinear but are essentially quasi-linear) show rather high precision in the sense of low NMSE (≤1%) and exhibit low computational complexity among other benefits. These properties were broadly discussed in [8,9], where theoretical proofs can be found. The material presented here contains some experimental applications, for rather different scenarios that apply and extend the ideas presented in [9], and so, both have to be considered together.…”

“…These properties were broadly discussed in [8,9], where theoretical proofs can be found. The material presented here contains some experimental applications, for rather different scenarios that apply and extend the ideas presented in [9], and so, both have to be considered together. In [9] it was impossible for the authors to include the present material, not even partially, for the lack of space.…”