Volterra Adaptive Filtering Algorithms And Their Applications

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.