信号的噪声抑制理论与技术研究
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摘要
信号的噪声抑制是信号处理领域一个基本的却又极富挑战的研究课题。由于信号的噪声抑制在自动检测、语音识别、无线通信、水声探测、生物医学工程、光纤通信等众多应用领域有着广泛的应用前景,其在最近的几年再次成为信号处理领域的研究热点。虽然在过去的几十年间,有关信号的噪声抑制的理论和算法得到的一些发展,包括信号的噪声抑制问题本身的可解性以及求解原理等方面的基本理论问题在一定程度上部分得到解决,并存在了一些在检测能力、内存需求、计算速度等方面性能各异的算法。但由于该问题的理论研究深度较大且算法实现难度较大,对信号的噪声抑制的研究仍未到非常成熟的阶段,许多理论问题和算法实现技术有待进一步探索。本论文主要做了以下几个方面的研究工作：
●在对信号进行稀疏建模的基础上,系统的研究了在此框架下的噪声抑制算法的基本原理。事实上信号的噪声抑制问题可以等效为欠定的线性系统方程的求解问题。该问题在过去看来是棘手的,但大量的实例表明,该问题通常存在稀疏解。论文说明了如何将观测信号进行稀疏模型化,完善了线性系统稀疏解存在的理论依据,并给出了在具体实例中的应用结果。论文指出：含有噪声的观测信号,可被视为目标信号和外加噪声的叠加,这等效为欠定的线性系统。唯一的先验知识是外加噪声的能量有限。如果我们可以解决一个代价函数,使得出的解含有最多不超过一定限度的非零解,即可有所得解与目标解的误差足够小(可以足够好的恢复目标信号)；线性系统方程稀疏解求解问题可概括为两个问题的求解,即稀疏解存在性和存在所需条件问题；以及在求出解后,如何验证其是否足够接近于全局最优解的问题。论文中针对这些问题,给出了稀疏性测度和唯一性测度的说明,列出了可用的寻踪算法,并评估了它们的性能。最后,揭示了问题的实质是由精确解的求解过渡到近似解的求解,在此情况下,相应的算法仍能保持良好的性能。基于前述,论文中提出将K一完全正交分解(K-complete orthogonal decomposition,K-COD)字典生成算法和正交匹配寻踪算法(orthogonal matching pursuit,OMP)相结合构成基于K-COD字典的稀疏表示(sparse representation,SR)噪声抑制算法,该算法的性能在实际的例子中得到验证,就我们所知,是迄今为止较好的解决方案。
·混沌信号在保密通信、光纤通信等诸多领域有广泛的应用,然而其易于受到加性高斯(Gaussian)白噪声的污染。由于混沌信号的频谱和Gaussian白噪声的频谱类似,对观测到的混沌信号进行噪声抑制的任务被认为较难实现,并且由于混沌信号的初值敏感性,其模态难于通过预先训练取得；此外混沌信号本身的稀疏性也不足,这就限制了采用基于K-COD字典的SR算法进行噪声抑制的可能性。在本论文中针对混沌信号不同于稀疏生成信号的特点,以及混沌信号固有的可相空间重构的特性,在基于前述解决方案的基础上,提出一种新的混沌信号噪声抑制算法—局部稀疏表示(local sparse representation,LSR)噪声抑制算法。LSR的基本原理是通过将SR局部应用于高维延迟坐标空间中信号的聚类簇上,来达到混沌信号噪声抑制的目的。通过将LSR与核主成分分析(kernel principal component analysis,KPCA)算法、局部独立分量分析(local independent component analysis,LICA)算法、延迟多信号抽取(delayed algorithm for multiple unknown signals extrac-tion,dAMUSE)算法的实证实验结果相对比,可以发现LSR能提供更好的噪声抑制性能。在此基础上,论文指出了LSR的优势的来源,并给出了相应的证明。由证明可得,LSR可视为局部正则化噪声抑制技术的一个分支。
●由前述LSR算法可知,在其实现过程中依赖于使用K-均值聚类(K-means clustering)算法和基于K-COD字典的SR,显然聚类算法和SR的性能影响着LSR算法的性能。对K-means聚类算法而言,其不但需要预先指定聚类的簇数,而且不能真正收敛到局部极值点。为了克服这些缺陷,本论文提出了密度聚类2.0(Density clustering 2.0,DENCLUE 2.0)算法,其原理是使用基于Gaussian核的核密度估计,且其中应用了优化爬山迭代更新过程,通过该迭代更新过程,DENCLUE 2.0算法本身可以自动确定簇数,自适应调整步长且不产生多余的计算开销,并精确收敛到局部极值点。对于提升SR性能的需求,本论文提出一种核模糊码本估计(kernel fuzzy codebook estimation,KFCE)算法用于从观测数据直接自动生成SR用字典。KFCE算法的原理在于将距离核技巧引入到模糊聚类技术中,.基于此产生SR用字典。我们将DENCLUE 2.0算法和KFCE算法引入到LSR中,构成了增强局部稀疏表示(enhanced LSR,ELSR)噪声抑制算法,并将其应用于实证数据中,由实证实验的结果可得,对比LSR算法,ELSR算法在性能上得到了进一步的提升。
●在综合考量基于K-COD字典的SR噪声抑制算法、LSR、ELSR算法的基础上,可以发现前述各算法所追求的是更为强大的噪声抑制性能；而处理海量的多媒体数据时,通常对性能的要求可略为降低,而对计算速度的要求更高。我们从前述的技术基础中得到启发,推导出一种基于最大后验估计(maximum a posteriori,MAP)的目标函数表达式；在回顾了现今的滤波算法的基础上,我们提出了一种修正无先导卡尔曼滤波(modified unscented Kalman filter,MUKF)算法,其原理是将迭代无先导卡尔曼滤波(iterated unscented Kalman filter,IUKF)的优势和均方根卡尔曼滤波(squared root unscented Kalman filter,SR-UKF)的优势相结合,同时优化了测量更新过程。我们将MUKF应用于目标函数表达式,构成了快速最大后验估计(Fast MAP,FMAP)噪声抑制算法。将FMAP算法应用于实证数据中,取得的实验结果表明,对比前述方法,该方法在效能上达到了设计的目标,其特别适合处理大规模的多媒体信号噪声抑制任务。
Noise suppression for signals is a fundamental and challenging research topic in signal processing field.Benefiting from the promising applications in automatic de-tection,speech recognition,wireless communication,sonar problem,biomedical en-gineering, fiber communication,and so on, oise suppression has become one of the hottest spots in signal processing field in recent years.After more than several decades, the theories and algorithms about noise suppression have great developments.Many effective algorithms have been presented,and their performances are different from de-tection capability, memory requirements,computational cost,etc.However,the noise suppression theory is profound and the related algorithms are difficult to implement. Until nowadays,the study on noise suppression is still far from mature.Many theory problems and noise suppression techniques are expected to continue to be discussed. The main contributions of this thesis are as follows:
●Based on sparse modeling technique of signals,we systematically study the basic principle of noise suppression algorithms.The noise suppression problem equals to underdetermined linear system equation solving problem.This problem appears to be difficult in the past,but many examples show that sparse solution of the problem is usually present.The thesis shows how to model the observed signal in a sparse way, demonstrates the theoretical basis for linear system sparse solution, and gives empirical experimental results.The thesis presents:the noisy observed signals can be regarded as the ideal signal and additive noise mixed,which is an underdetermined line system.The only priori knowledge is that noise energy is limited.If we can solve a cost function, the solution containing no more than a certain limit of non-zero solution,i.e.,the error between the estimated solution and the ideal solution is small enough (may good enough to restore the ideal signal).The sparse solution of linear system equations solving problem, can be summarized as two questions to solve:i).Sparse solution existence and necessary conditions for the existence,ⅱ).If a candidate solution is available,how to verify that the solution is the global minimizer.For these problems,this thesis defines the sparsity and uniqueness,lists the available pursuit algorithms,and evaluates their performances.Finally, essence of the problem is that when the exact so- lution solving problem transits to the approximate solution solving problem,the related algorithms still work well.Based on this,we combine the K-complete orthogonal decomposition(K-COD)algorithm with orthogonal matching pursuit (OMP)algorithm to form the sparse representation(SR) noise suppression algo-rithm based on K-COD.Implement this algorithm to the empirical instances,to our knowledge,it is the better solution by far.
●Chaotic signals are widely used in such as secure communication,optical fiber communication,and some other fields,however,they are vulnerable to be ad-ditive Gaussian white noise polluted.Since chaotic signal spectrum is similar to Gaussian white noise spectrum,the observed chaotic signal noise reduction task is considered difficult to achieve.Chaotic signals are initial sensitive;their patterns can not be learned by training.In addition,the sparsity of chaotic signals are insufficient,this limits the usage of the SR noise suppression algo-rithm based on K-COD.In this thesis,for that chaotic signal is different from the sparsely generated signal,as well as that chaotic signal can be reconstructed in phase space,we present a novel algorithm—local sparse representation (LSR) noise suppression algorithm.The LSR relies on applying SR locally to clusters of signals embedded in a high dimensional feature space of delayed coordinates. Compared with kernel principal component analysis(KPCA),local independent component analysis (LICA),and delayed algorithm for multiple unknown signals extraction (dAMUSE),LSR provides better noise suppression performance in the empirical experiments.We point out the advantages of the LSR, and gives cor-responding proof. From the proof, we conclude that LSR is a branch of the local regularization technique.
●LSR relies on K-means clustering algorithm and SR based on K-COD.Their performances affect the performance of LSR. In K-means,it needs to predefine the number of clusters,and can not converge to the local maximum. For overcoming these shortages,we adopt the Density clustering 2.0 (DENCLUE 2.0)algorithm in this thesis.DENCLUE 2.0 relies on kernel density estimation based on Gaussian kernel,and has an improved hill climbing procedure.It determines the number of clusters automatically,adjusts the step size automatically at no extra costs, and converges to the local maximum exactly. For enhancing the SR performance requirements,this thesis presents a kernel fuzzy codebook estimation (KFCE) algorithm to generate SR dictionary from the observed data directly. Its basic idea is to integrate the distance kernel trick with the fuzzy clustering algorithm to generate dictionary for SR. We use the DENCLUE 2.0 and SR based on KFCE to improve LSR, and this forms the enhanced LSR(ELSR)algorithm.From experimental results,ELSR has better performance than the LSR.
●After reviewing the SR noise suppression algorithm based on K-COD,LSR, and ELSR, we conclude that these algorithms pursuit better noise suppression per-formance.For processing the massive multimedia data, it usually decreases the performance level,and concentrates the computational speed.We enlighten from the previous demonstration,give an objective function based on maximum a pos-teriori (MAP)criterion. We propose a modified unscented Kalman filter (MUKF), whose principle is to integrate the advantage of square root unscented Kalman filter(SR-UKF) with that of the iterated unscented Kalman filter (IUKF),and utilize a modified measurement update procedure to achieve more accurate state estimation. We apply the MUKF to the MAP objective function, and then build the Fast MAP(FMAP)noise suppression algorithm. From experimental results, we find that FMAP match our design objectives,and it is suitable to be used in massive multimedia signals noise suppression.
●在对信号进行稀疏建模的基础上,系统的研究了在此框架下的噪声抑制算法的基本原理。事实上信号的噪声抑制问题可以等效为欠定的线性系统方程的求解问题。该问题在过去看来是棘手的,但大量的实例表明,该问题通常存在稀疏解。论文说明了如何将观测信号进行稀疏模型化,完善了线性系统稀疏解存在的理论依据,并给出了在具体实例中的应用结果。论文指出：含有噪声的观测信号,可被视为目标信号和外加噪声的叠加,这等效为欠定的线性系统。唯一的先验知识是外加噪声的能量有限。如果我们可以解决一个代价函数,使得出的解含有最多不超过一定限度的非零解,即可有所得解与目标解的误差足够小(可以足够好的恢复目标信号)；线性系统方程稀疏解求解问题可概括为两个问题的求解,即稀疏解存在性和存在所需条件问题；以及在求出解后,如何验证其是否足够接近于全局最优解的问题。论文中针对这些问题,给出了稀疏性测度和唯一性测度的说明,列出了可用的寻踪算法,并评估了它们的性能。最后,揭示了问题的实质是由精确解的求解过渡到近似解的求解,在此情况下,相应的算法仍能保持良好的性能。基于前述,论文中提出将K一完全正交分解(K-complete orthogonal decomposition,K-COD)字典生成算法和正交匹配寻踪算法(orthogonal matching pursuit,OMP)相结合构成基于K-COD字典的稀疏表示(sparse representation,SR)噪声抑制算法,该算法的性能在实际的例子中得到验证,就我们所知,是迄今为止较好的解决方案。
·混沌信号在保密通信、光纤通信等诸多领域有广泛的应用,然而其易于受到加性高斯(Gaussian)白噪声的污染。由于混沌信号的频谱和Gaussian白噪声的频谱类似,对观测到的混沌信号进行噪声抑制的任务被认为较难实现,并且由于混沌信号的初值敏感性,其模态难于通过预先训练取得；此外混沌信号本身的稀疏性也不足,这就限制了采用基于K-COD字典的SR算法进行噪声抑制的可能性。在本论文中针对混沌信号不同于稀疏生成信号的特点,以及混沌信号固有的可相空间重构的特性,在基于前述解决方案的基础上,提出一种新的混沌信号噪声抑制算法—局部稀疏表示(local sparse representation,LSR)噪声抑制算法。LSR的基本原理是通过将SR局部应用于高维延迟坐标空间中信号的聚类簇上,来达到混沌信号噪声抑制的目的。通过将LSR与核主成分分析(kernel principal component analysis,KPCA)算法、局部独立分量分析(local independent component analysis,LICA)算法、延迟多信号抽取(delayed algorithm for multiple unknown signals extrac-tion,dAMUSE)算法的实证实验结果相对比,可以发现LSR能提供更好的噪声抑制性能。在此基础上,论文指出了LSR的优势的来源,并给出了相应的证明。由证明可得,LSR可视为局部正则化噪声抑制技术的一个分支。
●由前述LSR算法可知,在其实现过程中依赖于使用K-均值聚类(K-means clustering)算法和基于K-COD字典的SR,显然聚类算法和SR的性能影响着LSR算法的性能。对K-means聚类算法而言,其不但需要预先指定聚类的簇数,而且不能真正收敛到局部极值点。为了克服这些缺陷,本论文提出了密度聚类2.0(Density clustering 2.0,DENCLUE 2.0)算法,其原理是使用基于Gaussian核的核密度估计,且其中应用了优化爬山迭代更新过程,通过该迭代更新过程,DENCLUE 2.0算法本身可以自动确定簇数,自适应调整步长且不产生多余的计算开销,并精确收敛到局部极值点。对于提升SR性能的需求,本论文提出一种核模糊码本估计(kernel fuzzy codebook estimation,KFCE)算法用于从观测数据直接自动生成SR用字典。KFCE算法的原理在于将距离核技巧引入到模糊聚类技术中,.基于此产生SR用字典。我们将DENCLUE 2.0算法和KFCE算法引入到LSR中,构成了增强局部稀疏表示(enhanced LSR,ELSR)噪声抑制算法,并将其应用于实证数据中,由实证实验的结果可得,对比LSR算法,ELSR算法在性能上得到了进一步的提升。
●在综合考量基于K-COD字典的SR噪声抑制算法、LSR、ELSR算法的基础上,可以发现前述各算法所追求的是更为强大的噪声抑制性能；而处理海量的多媒体数据时,通常对性能的要求可略为降低,而对计算速度的要求更高。我们从前述的技术基础中得到启发,推导出一种基于最大后验估计(maximum a posteriori,MAP)的目标函数表达式；在回顾了现今的滤波算法的基础上,我们提出了一种修正无先导卡尔曼滤波(modified unscented Kalman filter,MUKF)算法,其原理是将迭代无先导卡尔曼滤波(iterated unscented Kalman filter,IUKF)的优势和均方根卡尔曼滤波(squared root unscented Kalman filter,SR-UKF)的优势相结合,同时优化了测量更新过程。我们将MUKF应用于目标函数表达式,构成了快速最大后验估计(Fast MAP,FMAP)噪声抑制算法。将FMAP算法应用于实证数据中,取得的实验结果表明,对比前述方法,该方法在效能上达到了设计的目标,其特别适合处理大规模的多媒体信号噪声抑制任务。
Noise suppression for signals is a fundamental and challenging research topic in signal processing field.Benefiting from the promising applications in automatic de-tection,speech recognition,wireless communication,sonar problem,biomedical en-gineering, fiber communication,and so on, oise suppression has become one of the hottest spots in signal processing field in recent years.After more than several decades, the theories and algorithms about noise suppression have great developments.Many effective algorithms have been presented,and their performances are different from de-tection capability, memory requirements,computational cost,etc.However,the noise suppression theory is profound and the related algorithms are difficult to implement. Until nowadays,the study on noise suppression is still far from mature.Many theory problems and noise suppression techniques are expected to continue to be discussed. The main contributions of this thesis are as follows:
●Based on sparse modeling technique of signals,we systematically study the basic principle of noise suppression algorithms.The noise suppression problem equals to underdetermined linear system equation solving problem.This problem appears to be difficult in the past,but many examples show that sparse solution of the problem is usually present.The thesis shows how to model the observed signal in a sparse way, demonstrates the theoretical basis for linear system sparse solution, and gives empirical experimental results.The thesis presents:the noisy observed signals can be regarded as the ideal signal and additive noise mixed,which is an underdetermined line system.The only priori knowledge is that noise energy is limited.If we can solve a cost function, the solution containing no more than a certain limit of non-zero solution,i.e.,the error between the estimated solution and the ideal solution is small enough (may good enough to restore the ideal signal).The sparse solution of linear system equations solving problem, can be summarized as two questions to solve:i).Sparse solution existence and necessary conditions for the existence,ⅱ).If a candidate solution is available,how to verify that the solution is the global minimizer.For these problems,this thesis defines the sparsity and uniqueness,lists the available pursuit algorithms,and evaluates their performances.Finally, essence of the problem is that when the exact so- lution solving problem transits to the approximate solution solving problem,the related algorithms still work well.Based on this,we combine the K-complete orthogonal decomposition(K-COD)algorithm with orthogonal matching pursuit (OMP)algorithm to form the sparse representation(SR) noise suppression algo-rithm based on K-COD.Implement this algorithm to the empirical instances,to our knowledge,it is the better solution by far.
●Chaotic signals are widely used in such as secure communication,optical fiber communication,and some other fields,however,they are vulnerable to be ad-ditive Gaussian white noise polluted.Since chaotic signal spectrum is similar to Gaussian white noise spectrum,the observed chaotic signal noise reduction task is considered difficult to achieve.Chaotic signals are initial sensitive;their patterns can not be learned by training.In addition,the sparsity of chaotic signals are insufficient,this limits the usage of the SR noise suppression algo-rithm based on K-COD.In this thesis,for that chaotic signal is different from the sparsely generated signal,as well as that chaotic signal can be reconstructed in phase space,we present a novel algorithm—local sparse representation (LSR) noise suppression algorithm.The LSR relies on applying SR locally to clusters of signals embedded in a high dimensional feature space of delayed coordinates. Compared with kernel principal component analysis(KPCA),local independent component analysis (LICA),and delayed algorithm for multiple unknown signals extraction (dAMUSE),LSR provides better noise suppression performance in the empirical experiments.We point out the advantages of the LSR, and gives cor-responding proof. From the proof, we conclude that LSR is a branch of the local regularization technique.
●LSR relies on K-means clustering algorithm and SR based on K-COD.Their performances affect the performance of LSR. In K-means,it needs to predefine the number of clusters,and can not converge to the local maximum. For overcoming these shortages,we adopt the Density clustering 2.0 (DENCLUE 2.0)algorithm in this thesis.DENCLUE 2.0 relies on kernel density estimation based on Gaussian kernel,and has an improved hill climbing procedure.It determines the number of clusters automatically,adjusts the step size automatically at no extra costs, and converges to the local maximum exactly. For enhancing the SR performance requirements,this thesis presents a kernel fuzzy codebook estimation (KFCE) algorithm to generate SR dictionary from the observed data directly. Its basic idea is to integrate the distance kernel trick with the fuzzy clustering algorithm to generate dictionary for SR. We use the DENCLUE 2.0 and SR based on KFCE to improve LSR, and this forms the enhanced LSR(ELSR)algorithm.From experimental results,ELSR has better performance than the LSR.
●After reviewing the SR noise suppression algorithm based on K-COD,LSR, and ELSR, we conclude that these algorithms pursuit better noise suppression per-formance.For processing the massive multimedia data, it usually decreases the performance level,and concentrates the computational speed.We enlighten from the previous demonstration,give an objective function based on maximum a pos-teriori (MAP)criterion. We propose a modified unscented Kalman filter (MUKF), whose principle is to integrate the advantage of square root unscented Kalman filter(SR-UKF) with that of the iterated unscented Kalman filter (IUKF),and utilize a modified measurement update procedure to achieve more accurate state estimation. We apply the MUKF to the MAP objective function, and then build the Fast MAP(FMAP)noise suppression algorithm. From experimental results, we find that FMAP match our design objectives,and it is suitable to be used in massive multimedia signals noise suppression.
引文
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