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基于算子的信号分解相关理论与应用研究
郭宝奎
学位类型工学博士
导师彭思龙
2017-05-22
学位授予单位中国科学院大学
学位授予地点北京
关键词信号分解与表示 调幅调频信号 基于算子的信号分解 微分方程 信号解调 复值微分算子 多算子分解 线性等式约束的最优化问题
其他摘要       自然现象或人工系统(比如雷达,生物工程,语音处理等)中产生的信号多为复杂的多成分信号。而将多成分信号分解为简单的基本信号之和一直是科学研究的中的热点。相对于多成分信号而言,基本信号可称为单成分信号。近年来,有许多信号分解的方法被提出,比如基于数据驱动的经验模态分解,基于时频表示的重赋值与同步压缩,以及基于算子的零空间追踪算法等。总的来说,信号的分解可以分为传统的基于基投影的方法(比如短时傅里叶变换,小波变换,基于稀疏表示的匹配追踪等)与基于数据驱动的信号自适应分解(比如经验模态分解,同步压缩,基于算子的信号自适应分解等)。基于算子的信号自适应分解方法将单成分信号定义为在某种算子的零空间的函数,而将信号分解建模为从一个多成分信号中,提取一个单成分子信号,使得这个子信号在某种算子的零空间中。一般来说,算子是由微分算子及单成分信号的一些参数(比如瞬时幅值,瞬时频率等)构成。本文针对基于算子的信号分解及零空间追踪算法展开了详细的研究,内容涉及算子的设计,求解模型的建立,算法的收敛性证明,单成分信号的解调,以及实际信号(比如音频信号)的应用等。具体的,本文主要研究内容及创新点有以下4点:
       1. 提出了基于复值微分算子的信号自适应分解方法。在基于算子的信号分解方法最初的研究中,信号及算子都是实值的,算子只能近似零化信号。通过Hilbert变换,实信号变换到复信号,其优点是可以得到原信号的幅值及相位信息。进一步,本文提出了能够完全零化复调幅调频(AM-FM)信号的复值微分算子。在研究以往算子方法的求解问题时,指出其参数更新方式中存在的问题,提出新的参数更新方式,并证明了新的算法是二阶收敛的。简言之,在复值微分算子方面本文主要贡献是:
(1)将实信号的分解转移到复信号,使得分解时可以利用信号的更多信息;
(2)提出能够完全零化复调幅调频信号的复值微分算子;
(3)在求解模型中提出了新的参数更新方式并证明了算法的收敛性。
2. 提出了基于多个算子的信号分解方法。基于算子的信号分解模型最初是用一个算子将信号逐次分解,每次分解中对剩余信号的约束是最小化二范数,如果剩余信号存在非高斯噪声的子成分,那么这个约束是不合理的。另外,由于原分解模型倾向于将幅值相对较大的信号提取出来,因此当待分解的信号是非平稳非线性的时,会造成提取出的信号出现混叠,即提取出的信号是幅值大的子信号片段拼接而成。对此,本文提出了基于多算子的分解模型,其优点是:
(1)该模型建立在将信号的每个子成分都对应一个算子,能够避免成分混叠问题;
(2)剩余信号不再由二范数约束,使得模型最优解和信号理想分解一致;
(3)在一定程度上,多算子模型是对单算子模型的推广。
3. 提出了非参数迭代优化算法。本文研究了在算子分解方法中的最优化问题及用普通交替优化时信号变化规律,指出在以往求解模型中由规整参数带来的分解结果敏感性问题,提出了求解带约束优化问题的新的迭代算法。新的方法中参数是自适应更新的,因此本文称之为非参数迭代优化算法。在该算法的研究中,本文主要贡献为:
(1)分析了算法中参数新的更新方式,指出在迭代中信号及参数是按照约束矩阵的奇异值大小逐次收敛的。证明了这个算法以约1/2的指数收敛速率收敛;
(2)该算法可以用在算子分解方法中,能够精确提取算子的零空间;
(3)解决了约束矩阵有扰动问题:根据迭代规律,提出算法停止条件,使得算法能够提前停止,避免因约束矩阵不精确而使算法收敛到无意义的解。
4. 提出了表示AM-FM信号的精确微分方程模型,并应用到信号解调及分解中。基于算子的信号分解中,算子的设计起到至关重要的作用,以往提出的算子都只能近似零化实AM-FM信号。由于信号最基本的表示就是多个AMFM信号的和,因此有必要找到一个能完全零化(而不是近似零化)AM-FM信号的算子。本文提出了一类新的二阶微分方程,其系数由信号的瞬时频率与瞬时幅值构成,由此建立了微分方程与AM-FM信号的关系,导出了可以完全零化AM-FM信号的算子。基于该微分方程,本文做了两个主要工作:
(1)将AM-FM信号的解调建模为反解微分方程的参数,并提出解调算法;
(2)将该微分方程导出的算子与3中所提到的非参数优化算法结合,得到新的
信号分解算法。
; Most signals stemming from nature or man-made systems (e.g. radar, biological engineering, speech processing etc.) are complex signals with many components. Thus decomposing the multi-component signals into a sum of several basic signals is always a hot issue in scientific research. Comparing with multicomponent signals, the basic signals are referred to mono-component signals. In recent years, many separation methods have been proposed, e.g. data-driven based empirical mode decomposing, time-frequency representation based reassignment and synchrosqueezing transform, operator-based null space pursuit algorithm etc. In general, signal separation method can be classified into two groups: traditional methods which are based on base projection (e.g. shorttime Fourier transform, wavelet transform, sparse representation based matching pursuit) and data-driven method which decomposing a signal adaptively (e.g. empirical mode decomposing, synchrosqueezing transform, operator-based signal
separation). The operator-based signal separation method defines the monocomponent signals as functions which are in the null space of some operators, and models the signal separation as extracting a mono-component signal from a multi-component signal such that the extracted signal is in the null space of some operators. Generally, the operators are characterized by some parameters of the mono-component signals (e.g. instantaneous amplitude, instantaneous frequency). This dissertation focus on the operator-based signal separation, involving the design of operators, the solving model establishment, the convergence problem of associated algorithms, the demodulation of mono-component signals and the real-life signal application (e.g. speech signal). Specifically, the research contents and innovation points are including the following 4 aspects:
1. We proposed a complex-valued differential operator based adaptive signal separation method. In the original study, signals and operators are all real-valued, and the operators can just approximately annihilate the signals. Through Hilbert transform, the real signal can be transformed to its analytic signal, the benefit of which is that the information of amplitude and phase of original signals can be used. Furthermore, we proposed a complex-valued differential operator which can absolutely annihilate the amplitude-modulation and frequency-modulation signals (AM-FM). Besides, after analyzing the problem in solving algorithm of original methods, we proposed a novel parameters updating method, and proved
the convergence of the new algorithm. In a word, in the aspect of complex-valued differential operator, we make 3 contributions in This dissertation as follows:
(1) proposing a new idea that transforming the real signal into complex signal in separation problem, and the benefits are that one can use more information about signals;
(2) proposing a novel complex-valued differential operator which can absolutely annihilate the complex AM-FM signals;
(3) proposing a new method for parameters updating in the algorithm of solving separation problem and the new algorithm is proved to be convergent.
2. We proposed a multi-operator based signal separation method. In the original operator based signal separation method, signals are separated sequently, i.e. the input signal is separated into two subcomponent, one is in the null space of some operators and the other is refered as the residual signal, and recursively separating the residual signals, the original signal will be decomposed into several
components. In each separation, the constraint on the residual signals are minimizing the 2-norm of signals. This is unreasonable since some components which are not Gaussian noise may exist in the residual signal. What’s more, because the original separation model tents to extract components with larger amplitude, the extracted signal may be mixed when the input signal is nonstationary and nonlinear, that is the extracted signalss may be a mixture of fragments with larger amplitude. Thus This dissertation proposed the multi-operator based signal separation model, the advantages of which includes:
(1) this model is established in the assumption that each component of input signal is corresponding to an operator, i.e. the constraints on each component are that they are in the null space of associated operators, which avoid the mode mixing problem;
(2) the residual signal is not constraint by 2-norm which makes the optimal solution of separation model is equal to the ideal separation of a multi-component signal;
(3) the multi-operator based method is a generalization of original single operator based method in some sense.
3. We proposed a nonparametric iteration method for optimization problem with linear equality constraints. After studying the optimization problem in operator based signal separation issue and the change rules of signals in original separation algorithm which based on alternative optimization, we proposed a novel iteration method for optimization problem. In the new method, the parameter is updated adaptively, thus it is called nonparametric iteration optimization algorithm. The contributions of This dissertation on the proposed optimization algorithm are as follows:
(1) analyzing the parameter updating rules, and deriving results that the signals in the algorithm converge according to the singular values of the constraint matrix, and proving that this algorithm converges with exponential rate which is close to 1/2;
(2) with this algorithm, the operator-based signal separation method can extract an exact solution in the null space of associated operators;
(3) solving the inaccuracy problem of the constraint matrix: we proposed a stopping criterion according to the iteration rules, with which the algorithm can stop earlier before it converge to a meaningless solution.
4. We proposed a novel AM-FM signal representation by differential equation model and applied it in signal demodulation and separation. In the operator based signal separation method, the design of operator is critical since the operator determines which signals can be extracted from the input signals. Since the basic signal representation is a sum of several AM-FM signals, it is necessary to find operators which can absolutely, not approximately, annihilate the real AM-FM signals. This dissertation proposed a series of new second order differential equations whose coefficients compose of instantaneous frequency and instantaneous amplitude. The differential equations derive new operators which can absolutely annihilate the AM-FM signals. Based on the proposed differential equations, two contributions are made as follows:
(1) modeling the AM-FM signal demodulation problem as estimating the coefficients of proposed differential equations and proposing the demodulation algorithm;
(2) deriving a new signal separation method by combining the proposed differential equations and the nonparametric optimization method in 3.
文献类型学位论文
条目标识符http://ir.ia.ac.cn/handle/173211/14728
专题毕业生_博士学位论文
作者单位中国科学院自动化研究所
推荐引用方式
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郭宝奎. 基于算子的信号分解相关理论与应用研究[D]. 北京. 中国科学院大学,2017.
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