混沌信号的压缩感知去噪

Chaotic signal denoising in a compressed sensing perspective

对非线性时间序列进行噪声抑制是从中提取有效信息的前提.混沌信号的去噪算法不仅要使滤波后的信号具有较高的信噪比,也要具有较好的不确定性.从压缩感知的角度出发,提出了一种新的噪声抑制方法.该方法包括估计噪声方差,以及依据动态的稀疏度将观测值往确定的过完备字典上投影.仿真实验表明,该方法比常用的小波阈值法和局部曲线拟合法具有更高的输出信噪比,而原始信号的混沌特性也能得到较大程度的恢复.

Nonlinear time series denoising is the premise for extracting useful information from an observable, for the applica- tions 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 R6ssler 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.