分数阶信号合成与滤波技术研究及应用
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摘要
本论文首先介绍带有重尾分布或长相关／局部相关性质的分数阶随机信号,然后介绍用于分析这些分数阶随机信号的分数阶信号处理技术。本文中的分数阶信号指具有重尾分布,长相关或局部相关特性的随机信号。科学和工程领域中很多信号都具有分数阶特性。典型的重尾分布信号包括水声信号,低频大气噪声,和许多人体生理信号等[1-3]。典型的长相关随机信号和局部相关随机信号包括金融数据,网络通信数据和人体生理信号等[4-61。这些特征,即重尾分布、长相关／长记忆和局部记忆特性的存在,导致很难准确估计这些复杂信号的统计特性并获取有价值信息。但是,这些分数阶特征不能被忽略,因为重尾分布的托尾厚度,长相关／长记忆,或局部记忆特性反映了自然现象及人体生理信号的重要本质特征。因此,一些分数阶信号处理技术逐渐被人们研究并应用于复杂分数阶随机信号的分析。分数阶信号处理技术的主要理论基础是分数阶微积分,alpha稳定分布和分数阶傅里叶变换。本文介绍的分数阶处理技术包括：可变阶次分数阶信号合成,可变阶次分数阶系统的物理实现,分布阶次滤波器以及最优分数阶阻尼器的研究。
随机信号合成是随机信号分析、处理的一种重要手段。经典“信号与系统”分析中,对高斯白噪声在时域进行整数阶滤波可以得到整数阶随机信号[7,8]。同样,对高斯白噪声或alpha稳定分布噪声在时域进行分数阶滤波可以得到分数阶随机信号[9,10]。通过对高斯白噪声,可变阶次分数阶高斯噪声和可变阶次分数阶布朗运动三者之间关系的研究,本文提出一种基于可变阶次分数阶运算的可变阶次分数阶高斯噪声合成方法。此外,本文应用分数阶电容Fractor实现了固定阶次分数阶微／积分器和依赖温度的可变阶次分数阶微／积分器,并且深入研究了可变阶次分数阶微／积分器的阶次与温度之间的对应关系。
本文讨论了几类分布阶次分数阶滤波器。对分布阶次分数阶滤波器的研究源于固定阶次分数阶滤波器具有整数阶滤波器所不具备的特殊性质,其中固定阶次分数阶滤波器是广义化的整数阶滤波器。分布阶次分数阶滤波器是固定阶次分数阶滤波器和整数阶的一种扩展形式。尤其,分布阶次分数阶滤波器是描述复杂性,复杂网络和多尺度特性的有效工具[11-15]。为了能够在控制、滤波和信号处理等领域中设计更加有效的滤波器,本文给出了分数阶滤波器的时域冲激响应及其数值离散化结果。
本文探讨了三种类型最优分数阶阻尼器的性能。根据系统实际输出与理想系统输出之间的偏差分析,本文对最优无时延固定阶次分数阶阻尼器,最优时延固定阶次分数阶阻尼器和最优分布阶次分数阶阻尼器分别进行了研究。应用频域搜索方法,本文根据平方误差积分准则(ISE)系统性能指标搜索最优时延分数阶阻尼器和最优分布阶次分数阶阻尼器。应用时域搜索方法,本文根据ISE,时间乘平方误差积分准则(ITSE)，绝对误差积分准则(IAE)和时间乘绝对误差积分准则(ITAE)搜索最优无时延固定阶次分数阶阻尼器,最优时延固定阶次分数阶阻尼器和最优分布阶次分数阶阻尼器。基于各种系统性能指标的最优整数阶阻尼器和三种最优分数阶阻尼器的搜索结果分别在文中给出。
分数阶信号处理技术在生物医学信号分析中的应用实例在本文的最后给出。这三个应用实例为如何将传统的整数阶信号处理方法归纳为分数阶信号处理方法,以及如何有效应用分数阶信号处理技术提供了启发作用。
In this dissertation, we will introduce some complex random signals which are characterized by the presence of heavy-tailed distribution or non-negligible dependence between distant observations, from the'fractional'point of view. Furthermore, the analysis techniques for these fractional signals are investigated using the'fractional thinking'. The term'fractional signals'in this dissertation refers to some random signals which manifest themselves by heavy-tailed distribution, long range dependence (LRD)/long memory, or local memory. Fractional processes are widely found in science, technology and engineering systems. Typical heavy-tailed distributed signals include underwater acoustic signals, low-frequency atmospheric noises, many types of man-made noises, and so on[1-3]. Typical LRD/long memory processes and local memory processes can be observed in financial data, communications networks data and biological data[4-6]. These properties, i.e., heavy-tailed distribution, LRD/long memory, and local memory always lead to certain trouble in correctly obtaining the statistical characteristics and extracting desired information from these fractional processes. These properties cannot be neglected in time series analysis, because the tail thickness of the distribution, LRD, or local memory properties of the time series are critical in characterizing the essence of the resulting natural or man-made phenomena of the signals. Therefore, some valuable fractional-order signal processing (FOSP) techniques were provided to analyze these fractional processes. FOSP techniques are based on the fractional calculus, FLOM and FrFT basic theories. This dissertation concentrates on the synthesis of variable-order fractional signals, the realization of variable-order fractional systems, distributed-order fractional filters, and optimal fractional-order damping systems.
Simulation of random signals is a valuable tool in random signal processing. Most random processes can be generated by performing time domain integer-order filtering on a white Gaussian process[7,8]. Similarly, the fractional random processes can be simulated by performing the time domain fractional-order filtering on a white Gaussian process or a whiteα-stable process[9，10]. In his dessertation, a synthesis method, which is based on variable-order fractional operators, for multifractional Gaussian noises (mGn) is proposed by studying the relationship of white Gaussian noise (wGn), mGn, and multifractional Brownian motion (mBm). Furthermore, a synthesis method for multifractionalα-stable processes, the generalization of mGn, is proposed in order to more accurately characterize the processes with local scaling characteristics and heavy-tailed distributions.
The discussion of the distributed-order fractiona filters is presented in the dessertation. The distributed-order fractionla filter is motivated by the advantages of the classical fractional-order filter which is a generalized case of the integer-order filters. It can be proved that both integer and fractional-order systems are special cases of distributed-order fractional systems. Particularly when the complexity, networks, nonhomogeneous, multi-scale and multispectral are considered, the distributed-order fractional filter becomes a more precise tool to describe the above phenomena[11-15]. Therefore, motivated by the applications of the distributed-order fractional filters to control, filtering and signal processing, the distributed-order fractional filters are derived step by step in the dessertation.
This dessertation also explores the potential benefits of three types of fractional-order damping systems using the performance criteria which are based on the step response error in frequency domain and time domain. Three types of fractional-order damping systems are: fractional-order damping without time delay; fractional-order damping with time delay, and distributedorder fractional damping. In frequency domain, the time-delayed fractional-order and distributed-order fractional damping systems were optimized by minimizing the integral of the squared step response error (ISE) performance criterion. In time domain, these three types of fractional-order damping systems were numerically optimized by finding the minimum of ISE, the integral of the time-weighted error squared (ITSE), the integral of absolute error (IAE), and the integral of time-weighted absolute error (ITAE) performance measures. The optimum coefficients and minimum performance indexes for ISE, ITSE, IAE and ITAE criteria of these three types of fractional-order damping systems are provided.
Three application examples of FOSP techniques in biomedical signals are presented in the end of this dessertation. These application examples provide the instructions on how to generalize the conventional signal processing methods to FOSP techniques, and how to obtain more valuable information by using FOSP techniques.
随机信号合成是随机信号分析、处理的一种重要手段。经典“信号与系统”分析中,对高斯白噪声在时域进行整数阶滤波可以得到整数阶随机信号[7,8]。同样,对高斯白噪声或alpha稳定分布噪声在时域进行分数阶滤波可以得到分数阶随机信号[9,10]。通过对高斯白噪声,可变阶次分数阶高斯噪声和可变阶次分数阶布朗运动三者之间关系的研究,本文提出一种基于可变阶次分数阶运算的可变阶次分数阶高斯噪声合成方法。此外,本文应用分数阶电容Fractor实现了固定阶次分数阶微／积分器和依赖温度的可变阶次分数阶微／积分器,并且深入研究了可变阶次分数阶微／积分器的阶次与温度之间的对应关系。
本文讨论了几类分布阶次分数阶滤波器。对分布阶次分数阶滤波器的研究源于固定阶次分数阶滤波器具有整数阶滤波器所不具备的特殊性质,其中固定阶次分数阶滤波器是广义化的整数阶滤波器。分布阶次分数阶滤波器是固定阶次分数阶滤波器和整数阶的一种扩展形式。尤其,分布阶次分数阶滤波器是描述复杂性,复杂网络和多尺度特性的有效工具[11-15]。为了能够在控制、滤波和信号处理等领域中设计更加有效的滤波器,本文给出了分数阶滤波器的时域冲激响应及其数值离散化结果。
本文探讨了三种类型最优分数阶阻尼器的性能。根据系统实际输出与理想系统输出之间的偏差分析,本文对最优无时延固定阶次分数阶阻尼器,最优时延固定阶次分数阶阻尼器和最优分布阶次分数阶阻尼器分别进行了研究。应用频域搜索方法,本文根据平方误差积分准则(ISE)系统性能指标搜索最优时延分数阶阻尼器和最优分布阶次分数阶阻尼器。应用时域搜索方法,本文根据ISE,时间乘平方误差积分准则(ITSE)，绝对误差积分准则(IAE)和时间乘绝对误差积分准则(ITAE)搜索最优无时延固定阶次分数阶阻尼器,最优时延固定阶次分数阶阻尼器和最优分布阶次分数阶阻尼器。基于各种系统性能指标的最优整数阶阻尼器和三种最优分数阶阻尼器的搜索结果分别在文中给出。
分数阶信号处理技术在生物医学信号分析中的应用实例在本文的最后给出。这三个应用实例为如何将传统的整数阶信号处理方法归纳为分数阶信号处理方法,以及如何有效应用分数阶信号处理技术提供了启发作用。
In this dissertation, we will introduce some complex random signals which are characterized by the presence of heavy-tailed distribution or non-negligible dependence between distant observations, from the'fractional'point of view. Furthermore, the analysis techniques for these fractional signals are investigated using the'fractional thinking'. The term'fractional signals'in this dissertation refers to some random signals which manifest themselves by heavy-tailed distribution, long range dependence (LRD)/long memory, or local memory. Fractional processes are widely found in science, technology and engineering systems. Typical heavy-tailed distributed signals include underwater acoustic signals, low-frequency atmospheric noises, many types of man-made noises, and so on[1-3]. Typical LRD/long memory processes and local memory processes can be observed in financial data, communications networks data and biological data[4-6]. These properties, i.e., heavy-tailed distribution, LRD/long memory, and local memory always lead to certain trouble in correctly obtaining the statistical characteristics and extracting desired information from these fractional processes. These properties cannot be neglected in time series analysis, because the tail thickness of the distribution, LRD, or local memory properties of the time series are critical in characterizing the essence of the resulting natural or man-made phenomena of the signals. Therefore, some valuable fractional-order signal processing (FOSP) techniques were provided to analyze these fractional processes. FOSP techniques are based on the fractional calculus, FLOM and FrFT basic theories. This dissertation concentrates on the synthesis of variable-order fractional signals, the realization of variable-order fractional systems, distributed-order fractional filters, and optimal fractional-order damping systems.
Simulation of random signals is a valuable tool in random signal processing. Most random processes can be generated by performing time domain integer-order filtering on a white Gaussian process[7,8]. Similarly, the fractional random processes can be simulated by performing the time domain fractional-order filtering on a white Gaussian process or a whiteα-stable process[9，10]. In his dessertation, a synthesis method, which is based on variable-order fractional operators, for multifractional Gaussian noises (mGn) is proposed by studying the relationship of white Gaussian noise (wGn), mGn, and multifractional Brownian motion (mBm). Furthermore, a synthesis method for multifractionalα-stable processes, the generalization of mGn, is proposed in order to more accurately characterize the processes with local scaling characteristics and heavy-tailed distributions.
The discussion of the distributed-order fractiona filters is presented in the dessertation. The distributed-order fractionla filter is motivated by the advantages of the classical fractional-order filter which is a generalized case of the integer-order filters. It can be proved that both integer and fractional-order systems are special cases of distributed-order fractional systems. Particularly when the complexity, networks, nonhomogeneous, multi-scale and multispectral are considered, the distributed-order fractional filter becomes a more precise tool to describe the above phenomena[11-15]. Therefore, motivated by the applications of the distributed-order fractional filters to control, filtering and signal processing, the distributed-order fractional filters are derived step by step in the dessertation.
This dessertation also explores the potential benefits of three types of fractional-order damping systems using the performance criteria which are based on the step response error in frequency domain and time domain. Three types of fractional-order damping systems are: fractional-order damping without time delay; fractional-order damping with time delay, and distributedorder fractional damping. In frequency domain, the time-delayed fractional-order and distributed-order fractional damping systems were optimized by minimizing the integral of the squared step response error (ISE) performance criterion. In time domain, these three types of fractional-order damping systems were numerically optimized by finding the minimum of ISE, the integral of the time-weighted error squared (ITSE), the integral of absolute error (IAE), and the integral of time-weighted absolute error (ITAE) performance measures. The optimum coefficients and minimum performance indexes for ISE, ITSE, IAE and ITAE criteria of these three types of fractional-order damping systems are provided.
Three application examples of FOSP techniques in biomedical signals are presented in the end of this dessertation. These application examples provide the instructions on how to generalize the conventional signal processing methods to FOSP techniques, and how to obtain more valuable information by using FOSP techniques.
引文
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