本书围绕非高斯噪声干扰，系统讲述鲁棒自适应信号处理，尤其是鲁棒自适应滤波算法的基本理论与方法，有效地反映了近年来该领域的新理论、新算法和新技术。内容包括 α 稳定分布模型、相关熵基本理论、线性自适应滤波基本原理、基于核函数的非线性自适应滤波基本原理、仿射投影类自适应滤波算法、基于最小二乘架构的自适应滤波算法、鲁棒核自适应滤波算法，以及它们在系统辨识、短期时间序列预测中的应用。本书提供了相关算法的伪代码及 MATLAB 程序示例。本书取材新颖、内容翔实、概念清楚，适合通信与电子信息类相关专业的高年级本科生、研究生、教师、研究人员及行业从业者阅读。

赵集，副教授，西南科技大学信息工程学院硕士研究生导师，一直从事电路与系统、信号与信息处理、自适应滤波等方面的教学与科研工作。

第 1 章 鲁棒自适应滤波概述···············································································.1
1.1 背景和意义 ··························································································.1
1.2 国内外研究现状和发展态势 ·····································································.2
1.2.1 α-SDM 研究及其应用 ····································································.2
1.2.2 自适应滤波原理及其典型应用·························································.3
1.2.3 自适应滤波算法的研究进展····························································.5
1.2.4 基于核方法的非线性自适应滤波算法················································.8
1.3 本书章节安排 ·······················································································.9
第 2 章 非高斯环境下的自适应滤波理论基础··························································10
2.1 α 稳定分布模型·····················································································10
2.1.1 α 稳定分布特征函数······································································10
2.1.2 α 稳定分布的重要性质···································································16
2.2 相关熵 ································································································18
2.2.1 相关熵的概念 ··············································································18
2.2.2 相关熵的性质 ··············································································19
2.2.3 广义相关熵的概念 ········································································23
2.2.4 广义相关熵的性质 ········································································24
2.3 常用的优化方法 ····················································································27
2.3.1 梯度法 ·······················································································27
2.3.2 牛顿递归法 ·················································································28
2.4 核自适应滤波算法 ·················································································29
2.4.1 Mercer 核函数··············································································29
2.4.2 重构核希尔伯特空间（RKHS）·······················································30
2.4.3 特征空间 ····················································································30
2.4.4 基于高斯核函数的自适应滤波算法···················································31
2.4.5 自适应滤波算法的性能指标····························································31
2.5 本章小结 ·····························································································34
第 3 章 仿射投影类自适应滤波算法······································································35
3.1 归一化最小均方（NLMS）算法································································35
3.1.1 LMS 算法原理 ·············································································35
3.1.2 NLMS 算法原理···········································································37
3.2 仿射投影（AP）算法 ·············································································38
3.2.1 AP 算法原理 ···············································································39
鲁棒自适应滤波原理、算法及应用 ·VI·
3.2.2 AP 算法的计算复杂度 ···································································40
3.2.3 快速 AP 算法——基于原始权重向量更新的快速近似方法 ·····················41
3.3 仿射投影符号算法（APSA） ···································································42
3.3.1 APSA 算法原理············································································42
3.3.2 APSA 算法均方稳定性分析·····························································44
3.4 仿射投影广义最大相关熵（APGMC）算法 ·················································47
3.4.1 广义最大相关熵准则·····································································47
3.4.2 APGMC 算法原理·········································································48
3.4.3 APGMC 算法计算复杂度分析··························································51
3.4.4 均方收敛稳定性分析·····································································51
3.4.5 其他鲁棒 AP 类算法······································································52
3.5 基于数据复用方法的 GMC 算法································································56
3.5.1 数据复用最大相关熵（DR-MCC）算法 ·············································57
3.5.2 数据复用广义最大相关熵（DR-GMC）算法·······································57
3.5.3 随机数据复用广义最大相关熵（RDR-GMC）算法·······························58
3.5.4 随机牛顿递归数据复用广义最大相关熵算法·······································61
3.6 面向稀疏系统辨识的仿射投影类算法 ·························································66
3.6.1 通用稀疏系统辨识 ········································································66
3.6.2 块/聚集型稀疏系统辨识 ·································································77
3.7 仿射投影类算法的实验仿真与分析 ····························································84
3.7.1 面向 APSA 算法的实验仿真与分析···················································84
3.7.2 面向 APGMC 算法的实验仿真与分析················································89
3.7.3 面向数据复用算法的实验仿真与分析················································92
3.7.4 面向块/聚集型稀疏系统辨识算法的实验仿真与分析·····························97
第 4 章 基于最小二乘架构的鲁棒自适应滤波算法·················································.103
4.1 递归最小二乘算法 ··············································································.103
4.1.1 最小二乘问题 ···········································································.103
4.1.2 递归最小二乘算法 ·····································································.105
4.1.3 递归最小 p 次幂算法··································································.107
4.2 不动点广义最大相关熵算法 ··································································.110
4.2.1 概要 ·······················································································.110
4.2.2 FP-GMC 算法原理 ·····································································.110
4.2.3 FP-GMC 算法收敛性分析 ····························································.112
4.2.4 FP-GMC 算法的在线形式 ····························································.115
4.2.5 自适应凸组合递归广义最大相关熵（AC-RGMC）算法······················.120
4.3 面向二阶 Volterra 滤波的 RGMC 算法······················································.124
4.3.1 二阶 Volterra 滤波器概述·····························································.124
4.3.2 面向 SOV 滤波器的基本 RGMC 算法 ·············································.125
目 录 ·VII·
4.3.3 具有可变遗忘因子的 RGMC 算法··················································.126
4.4 线性约束条件下的鲁棒递归自适应滤波算法 ·············································.129
4.4.1 递归约束广义最大相关熵（RCGMC）算法·····································.130
4.4.2 RCGMC 算法性能分析 ·······························································.134
4.4.3 RCGMC 算法的低计算复杂度方法 ················································.140
4.4.4 其他递归类约束算法··································································.144
4.5 RLS 型自适应滤波算法的实验仿真与分析················································.150
4.5.1 面向 RGMC 算法的实验仿真与分析···············································.150
4.5.2 面向 SOV 滤波器算法的实验仿真与分析 ········································.160
4.5.3 面向 RCGMC 算法的实验仿真与分析·············································.164
第 5 章 鲁棒核自适应滤波算法·········································································.171
5.1 核最小均方算法 ·················································································.172
5.2 核最小 p 次幂算法 ··············································································.174
5.2.1 核最小 p 次幂算法基本原理·························································.175
5.2.2 投影核最小 p 次幂算法·······························································.176
5.2.3 PKLMP 算法的收敛性分析 ··························································.180
5.2.4 PKLMP 算法的改进 ···································································.185
5.3 基于数据复用方法的归一化核最大相关熵算法··········································.188
5.3.1 核数据复用最大相关熵算法·························································.188
5.3.2 核数据复用广义最大相关熵算法···················································.191
5.4 带有反馈机制的核自适应滤波算法 ·························································.194
5.4.1 具有单时滞反馈结构的核最小均方（SF-KLMS）算法 ·······················.195
5.4.2 具有单时滞反馈结构的核广义最大相关熵（SF-KGMC）算法 ·············.201
5.4.3 具有多时滞反馈结构的非线性递归核自适应滤波算法························.208
5.4.4 基于随机傅里叶特征的 NR-KNLMS-MF 算法 ··································.215
5.5 基于递归方法的核自适应滤波算法 ·························································.219
5.5.1 核递归最小二乘算法··································································.219
5.5.2 核递归最大相关熵算法·······························································.221
5.5.3 具有加权输出信息的 KRMC 算法··················································.223
5.5.4 核递归广义最大相关熵算法·························································.225
5.5.5 具有投影加权输出信息的 KRGMC 算法 ·········································.228
5.6 核自适应滤波算法实验仿真与分析 ·························································.230
5.6.1 面向 PKLMP 算法的实验仿真与分析 ·············································.230
5.6.2 面向 KDNR-GMC 算法的实验仿真与分析·······································.241
5.6.3 面向反馈 KAF 算法的实验仿真与分析 ···········································.246
5.6.4 面向递归 KAF 算法的实验仿真与分析 ···········································.253
参考文献 ·······································································································.259