【作者】 刘强；

【导师】 王世元；

【作者基本信息】 西南大学 ， 电子与通信工程（专业学位）， 2022， 硕士

【摘要】 作为信号处理领域的重要组成部分,自适应滤波器(AF,adaptive filter)已经广泛应用于系统辨别等实际环境中。通过自适应调节滤波系数,AF能够实现对未知系统的辨别和跟踪。在AF算法中,成本函数是数据系统与学习模型间的相似性度量。在高斯噪声环境中,使用二阶相似性度量的AF因其能够提供良好的滤波性能而被广泛使用。然而,在非高斯噪声环境中,使用二阶相似性度量的AF的滤波性能下降。为对抗非高斯噪声环境中离群值的干扰,信息理论学习(ITL,information theoretic learning)从概率角度出发,有效地解决非高斯噪声环境中滤波性能下降的问题。在非线性系统中,线性AF的滤波性能会显著下降。为解决上述问题,将核方法引入到AF中,从而推导出核自适应滤波器(KAF,kernel adaptive filter)。KAF能够有效地解决非线性问题。然而在算法学习的过程中,其线性增长的网络尺寸会带来较高的计算和存储成本。为控制KAF的网络尺寸,稀疏化方法被提了出来。稀疏化方法包括样本稀疏化和结构稀疏化。样本稀疏化方法是通过矢量量化和新奇准则等方法来抑制网络尺寸的增长。结构稀疏化是通过高维特征空间的映射函数,将原始数据映射到固定维的空间,从而控制算法的网络尺寸。然而,以上两种稀疏化方法都是以牺牲一定的滤波性能为代价来降低滤波器的计算复杂度。针对传统稀疏化KAF在滤波精度下降的问题,基于最近中心估计(NICE,Nearest-Instance-Centroid-Estimation)的KAF组被提了出来。该KAF组可以看作是单核多滤波器,它能够针对不同的输入信号,选择最合适的子滤波器,以此来提高传统稀疏化KAF的滤波精度。并且与传统KAF相比,KAF组可以在更低的时间成本下达到与传统KAF相近的滤波精度。本文的研究围绕以上KAF展开,具体如下。(1)为提高传统滤波器对抗非高斯噪声的能力,本文设计出了一种广义对数核损失函数。同时,为了降低KAF的网络尺寸,结合稀疏Nystr(?)m方法,最终推导出一种新的鲁棒Nystr(?)m核递归最小广义对数核损失算法。该算法不仅在非高斯噪声环境中表现出较高的鲁棒性,与传统KAF相比,还具备较低的计算复杂度。(2)为提高Nystr(?)m稀疏方法近似数据集核矩阵的精度,在采样阶段使用概率密度秩量化(PRQ,probability density rank-based quantization)采样方法,有效地避开了随机采样中可能会抽取到的异常值,从而提高了Nystr(?)m稀疏方法的近似精度。(3)稀疏Nystr(?)m法通过对输入数据进行固定维映射,进而控制算法的网络尺寸。在固定维映射的过程中,由于Gram矩阵中充斥的大量数值近似为零且分布无规律的元素,滤波器在计算输出的过程会进行很多无效的运算。本文利用NICE的特性来优化Nystr(?)m法中Gram矩阵,避免了Gram矩阵中的无效值参与到输入数据的固定维映射中,从而降低算法的计算复杂度。优化后的KAF算法可以看作是将原本结构庞大的KAF分割成多个并行的子滤波器,滤波器的输出是多个子滤波器输出的叠加和。进一步,根据高概率密度函数的置信概率可推出,只有离当前输入最近的子滤波器才会影响到KAF的输出,其他子滤波器的输出几乎为零,可直接忽略。将优化后的KAF称为KAF组。因此,在迭代学习中,KAF组算法的各子滤波器交替工作,进一步提高了算法的学习效率。由于每次迭代学习的输出都是由最优的子滤波器计算所得,故相比于传统稀疏Nystr(?)m滤波器算法,该方法可以获取更高的滤波精度。更多还原

【Abstract】 As an important part of signal processing field,adaptive filter(AF)has been widely used in system recognition and other practical environments.AF can realize the recognizing and tracking of unknown system by adjusting the filter coefficient adaptively.In AF algorithms,cost function is the similarity measure between data system and learning model.In Gaussian noise environment,AF using second-order similarity measure is widely used because of its good filtering performance.However,in the non-Gaussian noise environment,the filtering performance of AF using the second-order similarity measure decreases.In order to resist the interference of outliers in non-Gaussian noise environment,Information theoretic learning(ITL)effectively solves the problem of filtering performance degradation in non-Gaussian noise environment from the perspective of probability.In nonlinear system,the performance of linear AF declines significantly.Therefore,kernel method is introduced into AF and then kernel adaptive filter(KAF)is derived.KAF can effectively deal with nonlinear problems.However,in the process of algorithm learning,its linear growth of network size will lead to higher calculation and storage costs.In order to control the network size of KAF,sparsity methods including sample sparsity and structure sparsity are proposed.The sample sparsity method is to restrain the growth the network size through vector quantization and novel criteria.Structural sparsity is to map the original data to the fixed dimensional space through the mapping function of the high dimensional feature space,so as to control the network size of the algorithm.However,the above two sparsity methods reduce the complexity of the filter at the cost of sacrificing certain filtering performance.Aiming at the problem that the filtering accuracy of traditional sparse KAF decreases,the KAF group based on Nearest-Instance-Centroid-Estimation(NICE)is proposed.This KAF group can be regarded as single-core multi-filter,which can select the most appropriate sub-filter for different input signals,so as to improve the filtering accuracy of traditional sparse KAF.Compared with traditional KAF,the KAF group can achieve similar filtering accuracy with lower time cost.This thesis focuses on the above KAF,as follows.(1)In order to improve the ability of traditional filter against non-Gaussian noise,a generalized logarithmic kernel loss function is designed in this thesis.At the same time,in order to reduce the network size of KAF,a new robust Nystr(?)m kernel recursive minimum generalized logarithmic kernel loss algorithm is finally derived combining with sparse Nystr(?)m method.This algorithm not only shows better robustness in non-Gaussian noise environment,but also has lower computational complexity compared with traditional KAF.(2)In order to improve the accuracy of Nystr(?)m sparse method,PRQ sampling method is used in the sampling phase.PRQ sampling method can effectively avoid the outliers that may be extracted in random sampling,thus improving the approximate accuracy of sparse Nystr(?)m method.(3)The sparse Nystr(?)m method controls the network size of the algorithm by mapping input data to fixed dimensions.In the process of fixed dimension mapping,because of the large number of elements with values approximate to zero and irregular distribution are filled in Gram matrix,the filter will perform many invalid operations in the process of calculating the output.in this thesis,the Gram matrix in Nystr(?)m method is optimized by using the property of NICE,and the invalid values in Gram matrix are avoided from participating in the fixed dimension mapping of input data,thus reducing the computational complexity of the algorithm.The optimized KAF algorithm can be regarded as dividing the originally huge structure of KAF into multiple parallel sub-filters,and the output of the filter is the superposition sum of the outputs of multiple sub-filters.Further,according to the confidence probability of the high probability density function,it can be deduced that only the subfilter closest to the current input will affect the output of KAF,while the output of other subfilters is almost zero and can be directly ignored.The optimized KAF is called the KAF group.Therefore,in iterative learning,each subfilter of KAF group algorithm works alternately,which further improves the learning efficiency of the algorithm.Since the output of each iteration learning is calculated by the optimal sub-filter,this method can obtain higher filtering accuracy compared with the traditional sparse Nystr(?)m filter.更多还原