Chaotic signal denoising in a compressed sensing perspective

Nonlinear time series denoising is the premise for extracting useful information from an observable, for the applications in analyzing natural chaotic signals or achieving chaotic signal synchronizations. A good chaotic signal denoising algorithm processes not only a high signal-to-noise ratio (SNR), but also a good unpredictability of a signal. Starting from the compressed sensing perspective, in this work we provide a novel filtering algorithm for chaotic flows. The first step is to estimate the strength of the noise variance, which is not explicitly provided by any blind algorithm. Then the second step is to construct a deterministic projection matrix, whose columns are polynomials of different orders, which are sampled from the Maclaurin series. Since the noise variance is provided from the first step, then a sparsity level with regard to this signal can be fully constructed, and this sparsity value in conjunction with the orthogonal matching pursuit algorithm is used to recover the original signal. Our method can be regarded as an extension to the local curve fitting algorithm, where the extension lies in allowing the algorithm to choose a wider range of polynomial orders, not just those of low orders. In the analysis of our algorithm, the correlation coefficient of the proposed projection matrix is given, and the reason for shrinking the sparsity when the noise variance increases is also presented, which emphasizes that there is a larger probability of error column selection with larger noise variance. In the simulation, we compare the denoising performance of our algorithm with those of the wavelet shrinking algorithm and the local curve fitting algorithm. In terms of SNR improvement for the Lorenz signal, the proposed algorithm outperforms the local curve fitting method in an input SNR range from 0 dB to 20 dB. And this superiority also exists if the input SNR is larger than 9 dB when compared with the wavelet methods. A similar performance also exists concerning the Rossler chaotic system. The last simulation shows that the chaotic properties of the originals are largely recovered by using our algorithm, where the quantity for "chaotic degree" is described by using the proliferation exponent.