摘要
线性滤波器理论基础成熟、易于分析与实现，已获得了较为广泛的应用。但随着科学研究的不断深入，人们开始更多地关注高速通信信道、卫星链路、回声对消等存在非线性干扰的场合，由于线性自适应滤波器自身存在的线性特性，限制了其探索非线性信号高阶冗余性和逼近非线性函数的能力，使滤波性能不尽如人意。因此，为了克服实际应用中线性滤波器的缺陷，提高系统性能，近年来非线性滤波理论已成为人们研究的热点。其中，Volterra滤波器是一种结构相对简单、性能良好的非线性滤波器，它综合了系统的线性结构和非线性结构，较适合构建各种系统的非线性模型，因此被广泛地应用于系统辨识、回波对消、信道均衡、图像处理、混沌预测等领域。
本文主要研究Volterra滤波器自适应算法，包括高斯噪声背景下的二阶Volterra变参数自适应滤波算法、高斯噪声背景下基于正交变换的Volterra自适应滤波算法和α稳定分布噪声背景下的Volterra自适应滤波算法。
首先，介绍Volterra滤波器的基本理论，为全文的研究工作打下基础。
其次，研究高斯噪声背景下的二阶Volterra变参数自适应滤波算法。一方面为了改善由于Volterra滤波器参数固定导致算法收敛性能和稳态性能差的情况，本文提出一种二阶Volterra变步长解相关NLMS算法；分别对Volterra滤波器线性部分和非线性部分输入信号进行解相关处理，并利用各阶采用不同变步长因子方法，改善算法性能。另一方面通过自适应调整输入信号数据块长度，提出一种二阶Volterra变数据块长LMS算法；利用当前时刻及该时刻以前更多输入信号和误差信号信息使收敛速度和稳态性能都得到较好的提高。
然后，研究高斯噪声背景下基于正交变换的Volterra自适应滤波算法。由于输入信号自身相关性以及Volterra滤波器各项间的耦合使Volterra滤波器自适应算法性能下降，本文提出两种基于不同正交变换的Volterra滤波器自适应算法。一方面利用格型滤波器对输入信号进行格型预处理得到相互正交的后向预测误差信号并将其作为滤波器的输入，从而大大降低一次项、平方项和交叉乘积项信号各项之间的耦合，改善算法性能。另一方面提出基于DCT的高阶Volterra全解耦VLMS滤波算法，利用实对称矩阵经DCT可变换成对角矩阵的特性，分别推导了Volterra滤波器偶次阶输出项和奇次阶输出项，它们均可表示成权系数向量与信号向量的内积，从而大大减少权系数个数，降低计算复杂度；与此同时采用全解耦结构调整权系数，有效地降低了非线性项耦合的影响，提高了算法性能。
最后，研究α稳定分布噪声背景下的Volterra自适应滤波算法。鉴于实际环境更接近于α稳定分布噪声，本文分别提出两种α稳定分布噪声背景下的自适应算法。首先提出一种二阶Volterra变记忆长度LMP算法，针对实际应用中非线性系统记忆长度未知致使Volterra自适应滤波器可能无法达到最优性能的问题，通过自适应调整记忆长度使其收敛到真实值，使算法在收敛速度、稳态性能和计算复杂度之间达到了较好的折中；然后通过对Volterra滤波器结构的改进，提出一种α稳定分布噪声背景下基于DCT的三阶Volterra滤波算法，相比传统的Volterra滤波器自适应算法，不仅降低了算法复杂度，而且提高了算法性能。
Linear filter has been used widely for its mature theoretical principle, easy analysis andrealization. However, in the occasion of existing nonlinear interference, such as high-speedcommunication channels, satellite links, echo cancellation and etc, the performance of a linear filteris not ideal, which is due to the linear adaptive filter’s linear nature that restricts the capability ofusing higher-order nonlinear signal redundancy and approximating nonlinear function. Therefore, tosolve this problem and improve the performance of system, nonlinear filtering theory is studieddetailedly. The Volterra filter is very suitable to construct various nonlinear models of systems,which has the structure of linear and nonlinear system, so it has been widely applied in fields ofsystem identification, echo cancellation, channel equalization, image processing, chaotic forecastingand so on.
Three aspects’contents are studied in the paper, including variable parameters adaptivefiltering algorithms for second order Volterra filter under Gaussian noise environment, adaptiveVolterra filtering algorithms based on orthogonal transformation under Gaussian noise environmentand adaptive Volterra filtering algorithms underα-stable distribution noise environment.
At first, the basic theory of Volterra filter is introduced, which is the foundation of the wholeresearch work.
Secondly, the variable parameters adaptive filtering algorithms for second order Volterra filterunder Gaussian noise environment are studied. On the one hand, in order to change the conditionthat the Volterra filter’s fixed parameters lead to the bad performance at convergence andsteady-state, a variable step size NLMS algorithm based on decorrelation for second-order Volterrafilter is proposed. In the algorithm, the decorrelation is used in linear and nonlinear input signals ofVolterra filter, respectively, and the different variable step factors are adopted to improve theperformance of the algorithm. On the other hand, by changing the input data block length at eachmoment, a variable data block length LMS algorithm for second-order Volterra filter is proposed,which uses the present moment and its previous moment abundant information of input signals anderror signals to increase the convergence performance and steady-state performance.
Thirdly, adaptive Volterra filtering algorithms based on orthogonal transformation underGaussian noise environment are studied. To overcome the problem that due to the correlation ofinput signal and the coupling among each order of Volterra filter, the performance of adaptaiveVolterra filter is decreased, two different adaptive Volterra filtering algorithms based on orthogonaltransformation are proposed. On the one hand, the input signals are pre-processed by a lattice filter to obtain mutually orthogonal backward prediction error signals which are used as the inputs of thefilter. Therefore the coupling of the linear terms, the quadratic terms and the cross-multiplicationterms is decreased, respectively, and the convergence performance of the algorithm is improved. Onthe other hand, using the character that a real symmetric matrix can be transformed into a diagonalmatrix by DCT, the Volterra filter’s even output terms and odd output terms are deduced as the innerproducts of weighting coefficient vectors and signal vectors, respectively. And then the number ofweighting coefficients is decreased and the computational complexity is reduced. Meanwhile, afully decoupled structure is used to adjust the weighting coefficients to effectively decrease theinfluence of nonlinear items’coupling and improve the performance of algorithm.
At last, adaptive Volterra filtering algorithms underα-stable distribution noise environmentare studied. Considering that the actual environment is more closed toα-stable distribution noiseenvironment, two adaptive algorithms underα-stable distribution noise environment are proposed.On the one hand, for the memory length of nonlinear system is unknown in practical applications,the performance of Volterra adaptive filter with a non-suitable memory length will not be optimal.Faced with this problem, a variable memory length LMP algorithm for second-order Volterra filteris proposed. By adaptive adjusting the memory length to its real value, the proposed algorithmachieves a better compromise among convergence rate, steady-state performance and computationcomplexity. On the other hand, by improving the structure of Volterra filter, an adaptive algorithmfor third-order Volterra filter based on DCT underα-stable distribution noise environment isproposed. Compared to traditional Volterra filtering algorithm, the new algorithm’s computationalcomplexity is reduced and its performance is improved.
引文
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