由于自适应滤波器的组合需要同时独立地并行运行两个或多个自适应滤波器，所以计算复杂度高。同时，在大步长的自适应滤波器和小步长的自适应滤波器之间的收敛区域，自适应滤波器的组合呈现出慢的收敛/跟踪速度。为了解决这些问题，本文提出了组合步长(CSS)的新概念，开展了CSS设计方法研究，提出了一系列CSS自适应滤波算法，形成了CSS自适应滤波理论与方法。因为提出的CSS自适应滤波器每时每刻只需要一个滤波器运行，所以它比传统的自适应滤波器的组合具有更低的计算复杂度。同时，由于CSS扮演着变步长的角色，所以提出的CSS自适应滤波器比传统的自适应滤波器的组合具有更好的收敛/跟踪行为。提出的CSS方案为变步长的设计提供了一个全新的设计方案和理念。本文主要工作有：
1、为了降低传统的最小均方(LMS)自适应滤波器的组合算法的计算复杂度和改善它的收敛/跟踪速度，提出了CSS-LMS自适应滤波算法。提出的CSS-LMS算法使用一个组合因子来自适应地组合两个不同大小的步长，大步长执行快的收敛/跟踪速度和小步长提供小的稳态误差。组合因子定义为sigmoid激活函数的输出。然后，使用随机梯度下降法最小化系统输出误差的二范数间接地更新组合因子。为了快速地获得大步长的快收敛/跟踪速度和小步长的小稳态误差，对sigmoid激活函数作了放大、平移和截断的改进。基于提出的CSS新概念，从代价函数角度出发，提出了CSS归一化LMS(CSS-NLMS)算法和CSS成比例NLMS(CSS-PNLMS)/CSS改进的成比例NLMS(CSS-IPNLMS)算法。提出的CSS-NLMS算法和CSS-PNLMS/CSS-IPNLMS算法比对应的NLMS自适应滤波器的组合算法和PNLMS/IPNLMS自适应滤波器的组合算法具有更低的计算复杂度。仿真已经证明提出的CSS-LMS类自适应滤波算法比传统的LMS类自适应滤波器的组合算法获得更好的收敛/跟踪速度。
2、提出了CSS仿射投影(AP)类自适应滤波算法，包括CSS-AP算法、CSS成比例AP/CSS改进的成比例AP算法、CSS仿射投影符号(APS)算法和CSS成比例APS/CSS改进的成比例APS算法。提出的CSS-AP类自适应滤波算法除了计算复杂度远远低于传统的AP类自适应滤波器的组合算法外，其计算复杂度与原型AP类算法相当，尤其当投影阶数很大的时候，提出的CSS-AP类算法额外增加的计算复杂度非常轻微。仿真研究表明提出的CSS-AP类算法比传统的AP类自适应滤波器的组合算法具有更快的收敛/跟踪速度。甚至，提出的CSS-APS算法比变步长APS算法呈现出更好的性能。
3、为了避免了广义的最大相关熵准则(GMCC)算法的高计算开销的指数运算和产生更小的稳态误差，用广义的箕舌线函数作代价函数，提出了最大箕舌线准则(MVC)算法。为了进一步产生更好的性能，也提出了CSS-MVC算法。为了改善MVC算法在相关输入信号下的收敛性能，结合数据重用方法，提出了仿射投影箕舌线(APV)算法，并分析了它的计算复杂度、稳定性、稳态EMSE和引入快速滤波技术降低其计算复杂度。接着，为了保障小的稳态误差的同时产生快的收敛速度，提出CSS-APV算法。针对通信传输信道的稀疏性，提出了CSS成比例APV/CSS改进的成比例APV算法，用于估计稀疏的传输信道的脉冲响应。这里，所有组合因子的更新都是基于箕舌线函数的最大化。仿真结果显示提出的算法获得了好的性能。
4、为了解决变步长(VSS)步长标量(SSS)归一化子带自适应滤波器(NSAF)算法由恒定参数引起的在快收敛速度和小稳态误差之间的权衡问题，以及改善它在系统发生突变时的跟踪性能，提出了CSS变参的SSS NSAF(CSS-VPSSS-NSAF)算法，并分析了它的稳定性。使用随机梯度法最小化子带误差向量的一范数或系统输出误差的一范数来间接地更新提出的CSS-VPSSS-NSAF算法的组合因子。仿真研究表明提出的CSS-VPSSS-NSAF算法无需使用重置算法仍然比使用重置算法的VSS-SSS-NSAF算法获得更好的跟踪性能。利用箕舌线函数的饱和特性，提出CSS变参的基于箕舌线的NSAF(CSS-VPV-NSAF)算法，然后分析了其稳定性。仿真结果展示提出的CSS-VPV-NSAF算法比VSS-SSAF算法和CSS-VPSSS-NSAF算法具有更好的性能。
5、针对偏差补偿(BC)归一化最大相关熵准则(NMCC)算法在快的收敛速度和小的稳态误差之间的权衡问题，应用箕舌线代价函数，提出了一个CSS BC归一化MVC(CSS-BC-NMVC)算法，然后分析了它的稳定性。为了改善BC-NSAF算法在非高斯脉冲干扰下的鲁棒性，利用箕舌线代价函数的饱和特性可以有效地抑制非高斯脉冲干扰的特点，提出了CSS BC基于箕舌线的NSAF(CSS-BC-V-NSAF)算法，并分析了它的稳定性。仿真证明提出的CSS-BC-NMVC算法具有良好的鲁棒性和比BC-NMCC算法产生更快的收敛速度和更小的稳态误差；提出的CSS-BC-V-NSAF算法不但解决了BC-NSAF算法在快的收敛速度和小的稳态误差之间的折衷问题，而且克服了BC-NSAF算法在非高斯脉冲干扰下丧失鲁棒性的缺点。

Combinations of adaptive filters have high computational burdens due to the two or several adaptive filters running at the same time. Meanwhile, they also have poor convergence or tracking behavior at the intersection of the large step size filter and the small step size filter. To overcome these drawbacks, this paper proposes a new concept of combined-step-size (CSS), studies the design method of CSS, proposes a series of CSS adaptive filtering algorithms, and forms the theory and method of CSS adaptive filtering. The proposed CSS scheme has lower computational complexity than the conventional combination scheme of adaptive filters because it only requires one filter run at every moment. Because the proposed CSS plays the role of variable step size, the convergence or tracking behavior of the proposed CSS adaptive filter is much better than that of the traditional combination of adaptive filters. The proposed CSS scheme provides a new design scheme and idea for variable step size design. The major contributions of this dissertation are as follows:
1、In order to reduce the computational complexity of combination of least mean square (LMS) adaptive filters and improve its convergence or tracking behavior, a CSS-LMS adaptive filter is developed. The proposed CSS-LMS adaptive filter uses a combining factor to adaptively combine two different step sizes of one LMS adaptive filter, where the large step size affords a fast convergence or tracking rate and the small one offers a small steady-state error. The combining factor is defined as the output of a sigmoidal activation function and it updates indirectly by using the stochastic gradient descent method to minimize the L2-norm of the system output error. In order to quickly obtain the fast convergence or tracking rate of the large step size and the small steady-state error of the small one, the sigmoid activation function is modified by using enlarged, shifted and truncated methods. Based on the proposed new concept of CSS, the CSS normalized LMS (CSS-NLMS) adaptive filter and the CSS proportionate NLMS (CSS-PNLMS)/CSS improved proportionate NLMS (CSS-IPNLMS) adaptive filter are also proposed for reducing the computational complexities of combination of the corresponding adaptive filters and improving their convergence or tracking behavior. Simulations have demonstrated that the proposed CSS-LMS family adaptive filters obtain the superior convergence or tracking performance than the combinations of LMS family adaptive filters.
2、Based on the proposed new concept of CSS, a CSS affine projection (AP) family adaptive filtering algorithms are developed. The proposed CSS-AP family algorithms include the CSS-AP algorithm, CSS proportionate AP (CSS-PAP)/CSS improved proportionate AP (CSS-IPAP) algorithm, CSS affine projection sign (CSS-APS) algorithm, and CSS proportionate APS (CSS-PAPS)/CSS improved proportionate APS (CSS-IPAPS) algorithm. Except that the computational complexities of the proposed CSS-AP family algorithms are much lower than those of the traditional combination algorithms of AP family adaptive filters, their computational complexities are comparable to those of the original AP family algorithms. Especially when the projection order is very large, the additional computational complexities of the proposed CSS-AP family algorithms are very slight. Simulation results show that the proposed CSS-AP family algorithms have faster convergence or tracking rates than the traditional combination algorithms of AP family adaptive filters. Furthermore, the proposed CSS-APS algorithm presents superior performance than the variable step size APS algorithm.
3、In order to avoid the high computational burden of the exponential term of the generalized maximum correntropy criterion (GMCC) algorithm and produce smaller steady-state error, a maximum Versoria criterion (MVC) algorithm is proposed by maximizing the generalized Versoria function. Then, its CSS variant is also developed for producing better filter performance. To accelerate the convergence rate of the MVC algorithm for the correlated input signals, an affine projection Versoria (APV) algorithm is derived by maximizing the summation of Versoria-cost-reusing with a constraint on the square of the L2-norm of the filter weight vector difference. Its computational complexity, stability, and steady-state excess mean-square error analyses are carried out and a fast recursive filtering technique is introduced to reduce its complexity. A CSS-APV algorithm is then proposed for solving the tradeoff problem of fast convergence rate and small steady-state error of the APV algorithm. For the sparsity of communication transmission channel, a CSS proportionate APV/CSS improved proportionate APV algorithm is proposed to estimate the impulse response of sparse transmission channels. In this section, all combining factors of CSSs are updated indirectly by using the stochastic gradient method to maximize the Versoria cost function. Simulation results show that the proposed Versoria family algorithms achieve good performance in terms of the convergence or tracking rate and the steady-state error.
4、To address the tradeoff between the fast convergence rate and small steady-state error of the variable step-size (VSS) normalized subband adaptive filter (NSAF) with a fixed parametric step-size scaler (SSS) and improve its tracking performance in abrupt change scenarios, a combined-step-size (CSS) NSAF with a variable-parametric SSS (VPSSS) is proposed and its stability is analyzed. The combining factor of the CSS-VPSSS-NSAF algorithm is updated indirectly by using the stochastic gradient method to minimize the L1-norm of the subband error vector or system output error. Simulation results show that the proposed CSS-VPSSS-NSAF algorithm without the reset algorithm still achieves better tracking performance than the VSS-SSS-NSAF algorithm with the reset algorithm. Utilizing the saturation property of the Versoria cost function, a CSS variable-parametric Versoria-based NSAF (VPV-NSAF) algorithm is proposed and its stability is analyzed. Simulations show that the proposed CSS-VPV-NSAF algorithm performs much better than the VSS sign subband adaptive filter algorithm and CSS-VPSSS-NSAF algorithm.
5、The bias-compensated (BC) normalized maximum correntropy criterion (NMCC) algorithm encounters the conflicting requirement of fast convergence rate and small steady-state error. In order to overcome this drawback, a CSS BC normalized MVC (CSS-BC-NMVC) algorithm is proposed, which is obtained by combining the CSS scheme and the Versoria cost function. Its stability is then analyzed. For improving the robustness of the BC-NSAF algorithm in the presence of non-Gaussian impulsive interferences, by utilizing the Versoria cost function with the saturation property which can suppress the non-Gaussian impulsive interferences effectively, a CSS BC V-NSAF (CSS-BC-V-NSAF) algorithm is presented and its stability is analyzed. Simulation results have verified that the proposed CSS-BC-NMVC algorithm has good robustness and produces faster convergence rate and smaller steady-state error than the BC-NMCC algorithm. Also, the proposed CSS-BC-V-NSAF algorithm not only solves the tradeoff problem between fast convergence rate and small steady-state error of the BC-NSAF algorithm, but also overcomes the disadvantage that the BC-NSAF algorithm loses robustness in the presence of non-Gaussian impulsive interferences.

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