|Place of Conferral||北京|
|Keyword||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|
|Other Abstract|| 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：|
; 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.
|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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