摘要
非高斯信号处理是近年来发展起来的一个信号处理的新领域。传统的信号处理是基于高斯分布和二阶统计量的理论和技术,这是因为高斯模型比较简单,且在许多应用场合是适用的,在这种模型基础上设计的信号处理算法易于进行理论上的解析分析。
尽管高斯噪声假设能够很好的描述许多信号和噪声,然而,在实际应用中存在大量的非高斯信号和噪声,这些噪声的一个共同特点是它们的概率密度函数具有较厚的拖尾,并导致其时间波形上具有显著的脉冲特性。实际上,这种脉冲状噪声和较厚的统计拖尾正是分数低阶α稳定分布(FLOA)过程的显著特性。近年来,α稳定分布作为一种非高斯脉冲噪声的数学模型,已成为信号处理领域的热点研究课题。本文首先介绍了对称α稳定分布(SαS)中的α和γ参数的估计算法,随后着重对SαS分布噪声条件下的时间延迟估计进行了探讨,最后介绍了SαSG分布噪声条件下的自适应滤波算法,主要内容包括以下几点:
1.回顾了α稳定分布的研究背景,阐述了其研究现状、基本概念、基本特性、一般原理及应用前景。
2.介绍了对称α稳定分布的参数估计方法,并对基于样本分位数的参数估计方法和log SαS法进行了实验仿真,仿真结果显示两种算法均能给出较好的估计结果,能够满足后期的研究需要,而且log SαS法比基于样本分位数法计算量小,且具有闭合形式的计算公式,因此性能上更为优越。
3.在假设对称α稳定分布噪声相互独立的条件下,分析了基于分数低阶协方差(FLOC)的时间延迟估计算法中输入信号的分数低阶指数A、B依赖于α值的先验估计的缺点,并将其与反双曲正弦变换相结合推出了基于反双曲正弦的时间延迟估计算法,有效地避免了依赖于α值的先验估计选择A、B参数的缺点,易于对信号进行实时性的处理。仿真结果显示,该算法具有较高的估计精度,但同时也存在当对称α稳定分布噪声独立性不满足时性能显著退化的缺点。
4.在log SαS过程的基础上,利用log SαS过程的三阶矩定义代价函数,通过最小化代价函数给出新的时间延迟估计算法,算法中的α可由本文中介绍的参数估计方法得到。仿真结果显示,该算法具有较高的韧性,无论SαS噪声独立性是否满足都能给出较好的检测结果,很好地解决了SαS分布噪声不满足相互独立时性能显著退化的缺点。
5.当系统噪声为SαSG分布时,基于自适应混合范数(RMN)滤波算法,使用sigmoid函数对瞬时误差进行变换,可将误差信号转换成二阶矩过程,从而使对SαSG分布噪声的处理转变成对传统的二阶矩过程的处理。最优化标准化权值误差矢量的仿真实验验证了所给算法性能的提高。
最后对本文的工作做了总结,并对进一步研究工作进行了展望。
Non-Gaussian signal processing is a new signal processing field in recent years. The traditional signal processing is based on Gaussian distribution and second-order statistics in theory and technology. Because the Gaussian model is easier and reasonable in many instances, it often leads to analytically tractable solutions for signal processing problems.
Though in many instances the Gaussian assumption is reasonable, there are many non-Gaussian signals and noises in the actual applications, the remarkable characteristic of the sort of stochastic signals or noises is that the tails of the stable density are heavier and results in the remarkable impulsive nature in waveform. In nature, this impulsive and heavy tailed nature are remarkable characteristics of the fractional lower orderα-stable distribution(FLOA). Theα-stable distribution, which is one kind of mathematical model of non-Gaussian noises, is an important issue in signal processing field.
Firstly, this paper introduces the parameter estimation methods ofαandγin the symmetricα-stable distribution(SαS), then places extra emphasis on the algorithms of time delay estimation under the SαS distributed noise condition, in the end, introduces adaptive filter algorithms under the SαSG distributed noise condition, The main content includes the followings:
1. Review the research background ofα-stable distribution, expound research status, fundamental conceptions, features, principle and application prospects.
2. Introduce the parameter estimation methods of symmetricα-stable distribution. The computer simulations based on quantiles of samples method and log SαS method indicate that the two methods both can give better estimated results and the results can satisfy the needs of the study, and log SαS method has smaller computation and the closed-form formula for calculating, so it has a more superior performance.
3. Under the condition of symmetricα-stable distributed and spatial and temporal independent noises, the thesis analyzes the shortcoming of time delay estimation algorithm based on fractional lower order covariance (FLOC) which requests the previous estimation of characteristic exponentαin order to choose an appropriate value of fractional lower order exponents of input signals A and B, and combines time delay estimation algorithm based on FLOC and anti-hyperbolic sine transform to introduce time delay estimation algorithm based on anti-hyperbolic sine transform. The new algorithm overcomes the shortcoming of depending on the previous estimation of the characteristic exponentα. It is easy to real-time signal processing, and has got a high estimated precision through computer simulations. But it exits the shortcoming of significant degradation when the independence of the symmetricα-stable distributed noise can not be satisfied.
4. On the foundation of log SαS process, cost function is defined based on third-order moment of the log SαS process. A new time delay estimation method is introduced by minimizing the cost function. The parameterαcan be estimated by the new method introduced in the thesis. The experimental results indicate that robustness of this method is stronger and can give better test results regardless of SαS niose whether or not to meet the independence, and well resolves the problem of notably degeneration when the independence of SαS distributed noise can not be satisfied.
5. When the system noise satisfies SαSG distribution, the thesis adopts sigmoid function transform on instantaneous error based on RMN filter algorithm in order to transform the instantaneous error to two-order process. In this way the processing of SαSG distribution noise is converted into the processing of two-order process. The optimizing simulation results of normalizd weight error vector indicate that performance of this algorithm has been improved.
In the end, the work is summarized and the further research direction is pointed out.
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