自适应滤波器的稳态性能研究
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摘要
自适应滤波器的性能分析通常检测其瞬时行为和稳态行为,前者提供了自适应滤波器的稳定性和收敛速率信息,后者提供了滤波器达到稳态后的均方误差信息。一般,稳态性能通常可以通过瞬时性能的极限得到,但是这也遇到了比较大的困难,主要表现在三个方面:(1)针对误差信号具有非线性的自适应滤波器,瞬时性能分析比较困难;(2)瞬时性能分析要求一些比较严格的简化假设;(3)针对不同自适应算法,瞬时分析一般需要各自进行分析。为此,针对稳态性能分析,很有必要发展一种独立于瞬时分析的统一理论以适应大量不同的自适应算法。
本文以自适应滤波器迭代过程中的能量守恒关系为出发点,利用复数形式和实数形式的泰勒级数展开,深入研究了具有非线性估计误差信号的自适应滤波器的稳态均方误差和跟踪性能。主要从几个方面着手:(1)基于分离原理,推导了估计误差信号具有非线性的自适应滤波算法的稳态性能通用解析表达式,包括:稳态均方误差表达式、稳态跟踪均方误差表达式、最优步长表达式和最小稳态均方误差表达式,并给出了这些通用稳态表达式成立时迭代步长应该满足的条件。然后举例说明了这些通用解析表达式适用于常用的LMS算法、LMF算法和LMMN算法等,仿真实验验证了理论结果。(2)当输入信号为高斯白噪声时,不依赖分离原理条件,给出了具有非线性估计误差信号的自适应滤波算法的稳态性能通用解析表达式,并给出了这些通用解析表达式成立时迭代步长应该满足的条件。相同条件下,同基于分离原理得到的稳态性能结果相比较,两者相差非常微小,并且在迭代步长充分小条件下,这些解析表达式同基于分离原理得到的解析表达式是完成一样的。(3)研究了在高斯噪声和均匀分布噪声环境下LMP算法(不同参数p)和LMMN算法的稳态性能通用解析表达式,仿真表明理论结果同仿真结果相吻合。(4)比较了LMP算法和LMS算法在高斯白噪声模型和均匀分布噪声模型下的跟踪性能。说明在高斯噪声环境下,LMS算法的跟踪性能优于LMP算法(参数p大于2);而在均匀分布噪声环境下,LMP算法的跟踪性能优于LMS算法。
常用的调制信号如QPSK、8PSK、4QAM、8QAM、16QAM、64QAM、256QAM等的发射信号星座图具有对称特性。论文充分利用这些对称特性,在没有利用循环假设条件下,推导了Bussgang盲均衡算法的稳态超量均方误差的表达式,并证明其与A.Goupil和J.Palicot在文献[127]中得到的理论结果是一致性的。
并行软判决恒模算法(CCMA+SDD)同常用的恒模算法(CMA)相比,其在QAM信道中的均衡能力得到了很大提高,具有收敛速度快,收敛后的均方误差小等特点。但是,由于CCMA+SDD算法具有双并行FIR滤波器结构,而且其估计误差信号具有非线性,因此此算法的性能分析一直被解决。本文针对CCMA+SDD算法的并行双迭代结构,提出一种等效的具有单FIR结构的软判决恒模算法:然后应用前面得到的Bussgang盲均衡算法的稳态性能通用解析表达式,给出了CCMA+SDD算法在实值数据和复值数据下的的稳态解析表达式,仿真实验验证了理论结果。
The performance of an adaptive filter is generally measured in terms of its transient behavior and its steady-state behavior. The former provides information about the stability and the convergence rate of an adaptive filter, whereas the latter provides information about the mean square error (MSE) of the filter once it reaches steady state. In general, the steady state performance can be obtained as the limiting case of a transient analysis, but it encounters some difficulties. Firstly, transient analyses tend to be laborious, especially for adaptive filters with nonlinear estimation error signals. Secondly, transient analyses tend to require some simplifying assumptions, which at times can be restrictive. Finally, it is common in the literature to perform transient analyses of different adaptive filters separately by studying each nonlinear update form individually. These points motivate the development of a unified approach in the thesis to the steady-state performance for a large class of adaptive filters that bypass several of the difficulties encountered in obtaining steady-state results as the limiting case of a transient analysis.
In the thesis, based on the real form and complex form Taylor series expansions and the energy conservation relation during two successive iteration update, the steady-state MSE and tracking performance analyses for the adaptive filters with nonlinear estimation error signals are studied. Firstly, based on separation principle, some unified closed analytical expressions for the steady-state MSE, the tracking MSE, the optimal step-size and the minimum MSE, respectively, are derived for adaptive filters with nonlinear estimation error signals, and the restrictive conditions for these expressions are also given. The steady-state performance analyses for some common algorithms, such as least mean square (LMS) algorithm, least mean forth (LMF) algorithm and least mean mixed norm (LMMN) algorithm are special cases for these general expressions. In addition, the extensive computational simulations for adaptive filters validate these theoretical resutls. Secondly, rather using separation principle, the steady-state performance analyses can be obtained while the regressors are white white Gaussian noise, and the restrictive conditions for these expressions are also given. Comparing with the results obtained by separation principle, it can be found that only small distinction exists. In addition, in view of small step-size enough, their expressions are same. Thirdly, the steady-state performance analyses for least mean p-order (LMP) algorithm with different parameter p and LMMN algorithm are presented in both Gaussian noise environments and uniformly distributed noise environments. The experimental and theoretical results are matched reasonable well. Finally, comparison of the tracking performance between the LMP algorithm and LMS algorithm are implemented both in Gaussian noise environments and in uniformly distributed noise environments. Simulations show the superiority of the LMS algorithm over LMP with parameter p>2 for tracking nonstationary systems in Gaussian noise environments, and the superiority of the LMP algorithm over LMS in uniformly distributed noise environments.
Most of modulation signals, such as QPSK, 8PSK, 4QAM, 8QAM, 16QAM, 64QAM, 256QAM, and so on, all have symmetrical constellations. In the thesis, based on the constellation symmetric characteristics of the modulated signals, some expressions for the steady-state EMSE of the Bussgang algorithms (BA) are derived without appealing to the circularity assumption, and are proved that they are same as the results derived by A. Goupil and J. Palicot in [127].
The concurrent constant modulus algorithm and decision-directed scheme (CCMA+SDD) blind equalization achieves a considerable improvement in equalization performance over the constant modulus algorithm (CMA) for high-order quadrature amplitude modulation (QAM) channels, such as faster convergence and lower steady-state MSE. However, the actual steady-state performance has largely been left undone because of its concurrent adaptive filter structure and the complexity to analyze the time evolution of the weight update estimation error that arises from the nonlinear estimation error. In the thesis, an equivalent blind equalizer described by a hybrid cost function of CMA and SDD scheme is proposed. Then, based on the previous steady-state EMSE expressions for BA, some closed analytical expressions for EMSE for CCMA+SDD algorithm are given, for real-valued cases and for complex-valued cases, respectively. Extensive computational simulations for CCMA+SDD algorithm validate these theoretical results.
本文以自适应滤波器迭代过程中的能量守恒关系为出发点,利用复数形式和实数形式的泰勒级数展开,深入研究了具有非线性估计误差信号的自适应滤波器的稳态均方误差和跟踪性能。主要从几个方面着手:(1)基于分离原理,推导了估计误差信号具有非线性的自适应滤波算法的稳态性能通用解析表达式,包括:稳态均方误差表达式、稳态跟踪均方误差表达式、最优步长表达式和最小稳态均方误差表达式,并给出了这些通用稳态表达式成立时迭代步长应该满足的条件。然后举例说明了这些通用解析表达式适用于常用的LMS算法、LMF算法和LMMN算法等,仿真实验验证了理论结果。(2)当输入信号为高斯白噪声时,不依赖分离原理条件,给出了具有非线性估计误差信号的自适应滤波算法的稳态性能通用解析表达式,并给出了这些通用解析表达式成立时迭代步长应该满足的条件。相同条件下,同基于分离原理得到的稳态性能结果相比较,两者相差非常微小,并且在迭代步长充分小条件下,这些解析表达式同基于分离原理得到的解析表达式是完成一样的。(3)研究了在高斯噪声和均匀分布噪声环境下LMP算法(不同参数p)和LMMN算法的稳态性能通用解析表达式,仿真表明理论结果同仿真结果相吻合。(4)比较了LMP算法和LMS算法在高斯白噪声模型和均匀分布噪声模型下的跟踪性能。说明在高斯噪声环境下,LMS算法的跟踪性能优于LMP算法(参数p大于2);而在均匀分布噪声环境下,LMP算法的跟踪性能优于LMS算法。
常用的调制信号如QPSK、8PSK、4QAM、8QAM、16QAM、64QAM、256QAM等的发射信号星座图具有对称特性。论文充分利用这些对称特性,在没有利用循环假设条件下,推导了Bussgang盲均衡算法的稳态超量均方误差的表达式,并证明其与A.Goupil和J.Palicot在文献[127]中得到的理论结果是一致性的。
并行软判决恒模算法(CCMA+SDD)同常用的恒模算法(CMA)相比,其在QAM信道中的均衡能力得到了很大提高,具有收敛速度快,收敛后的均方误差小等特点。但是,由于CCMA+SDD算法具有双并行FIR滤波器结构,而且其估计误差信号具有非线性,因此此算法的性能分析一直被解决。本文针对CCMA+SDD算法的并行双迭代结构,提出一种等效的具有单FIR结构的软判决恒模算法:然后应用前面得到的Bussgang盲均衡算法的稳态性能通用解析表达式,给出了CCMA+SDD算法在实值数据和复值数据下的的稳态解析表达式,仿真实验验证了理论结果。
The performance of an adaptive filter is generally measured in terms of its transient behavior and its steady-state behavior. The former provides information about the stability and the convergence rate of an adaptive filter, whereas the latter provides information about the mean square error (MSE) of the filter once it reaches steady state. In general, the steady state performance can be obtained as the limiting case of a transient analysis, but it encounters some difficulties. Firstly, transient analyses tend to be laborious, especially for adaptive filters with nonlinear estimation error signals. Secondly, transient analyses tend to require some simplifying assumptions, which at times can be restrictive. Finally, it is common in the literature to perform transient analyses of different adaptive filters separately by studying each nonlinear update form individually. These points motivate the development of a unified approach in the thesis to the steady-state performance for a large class of adaptive filters that bypass several of the difficulties encountered in obtaining steady-state results as the limiting case of a transient analysis.
In the thesis, based on the real form and complex form Taylor series expansions and the energy conservation relation during two successive iteration update, the steady-state MSE and tracking performance analyses for the adaptive filters with nonlinear estimation error signals are studied. Firstly, based on separation principle, some unified closed analytical expressions for the steady-state MSE, the tracking MSE, the optimal step-size and the minimum MSE, respectively, are derived for adaptive filters with nonlinear estimation error signals, and the restrictive conditions for these expressions are also given. The steady-state performance analyses for some common algorithms, such as least mean square (LMS) algorithm, least mean forth (LMF) algorithm and least mean mixed norm (LMMN) algorithm are special cases for these general expressions. In addition, the extensive computational simulations for adaptive filters validate these theoretical resutls. Secondly, rather using separation principle, the steady-state performance analyses can be obtained while the regressors are white white Gaussian noise, and the restrictive conditions for these expressions are also given. Comparing with the results obtained by separation principle, it can be found that only small distinction exists. In addition, in view of small step-size enough, their expressions are same. Thirdly, the steady-state performance analyses for least mean p-order (LMP) algorithm with different parameter p and LMMN algorithm are presented in both Gaussian noise environments and uniformly distributed noise environments. The experimental and theoretical results are matched reasonable well. Finally, comparison of the tracking performance between the LMP algorithm and LMS algorithm are implemented both in Gaussian noise environments and in uniformly distributed noise environments. Simulations show the superiority of the LMS algorithm over LMP with parameter p>2 for tracking nonstationary systems in Gaussian noise environments, and the superiority of the LMP algorithm over LMS in uniformly distributed noise environments.
Most of modulation signals, such as QPSK, 8PSK, 4QAM, 8QAM, 16QAM, 64QAM, 256QAM, and so on, all have symmetrical constellations. In the thesis, based on the constellation symmetric characteristics of the modulated signals, some expressions for the steady-state EMSE of the Bussgang algorithms (BA) are derived without appealing to the circularity assumption, and are proved that they are same as the results derived by A. Goupil and J. Palicot in [127].
The concurrent constant modulus algorithm and decision-directed scheme (CCMA+SDD) blind equalization achieves a considerable improvement in equalization performance over the constant modulus algorithm (CMA) for high-order quadrature amplitude modulation (QAM) channels, such as faster convergence and lower steady-state MSE. However, the actual steady-state performance has largely been left undone because of its concurrent adaptive filter structure and the complexity to analyze the time evolution of the weight update estimation error that arises from the nonlinear estimation error. In the thesis, an equivalent blind equalizer described by a hybrid cost function of CMA and SDD scheme is proposed. Then, based on the previous steady-state EMSE expressions for BA, some closed analytical expressions for EMSE for CCMA+SDD algorithm are given, for real-valued cases and for complex-valued cases, respectively. Extensive computational simulations for CCMA+SDD algorithm validate these theoretical results.
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