Design Of Nonlinear Adaptive Filtering Algorithm And Its Convergence Analysis

The desired signal can be separated from the mixture signal containing noise using low pass filter,high-pass filter,band-pass filter and band-stop filter.However,signal separation or noise cancellation cannot be achieved by traditional filter especially when the designed signal and noise are in the same frequency band.To address this issue,following the statistical change of environment,adaptive filter adjusts the system weight adaptively to perform filtering efficiently.On the construction of filtering model,adaptive filters can be categorized into the inner product-based model and state space-based model.Kernel adaptive filer is built in the inner product-based model,and Kalman filter and its nonlinear extensions are built in the state space-based model.Kernel adaptive filter transforms the original input space into the high-dimensional reproduce kernel Hilbert space(RKHS),so that the nonlinear issues can be solved linearly in the form of inner product in RKHS.However,the dimensionality of RKHS is generally high and even infinite,resulting in the curse of dimensionality when the inner product in RKHS is calculated.To address this issue,kernel trick can be used to calculate the inner product in RKHS by using kernel function effectively.In addition,kernel adaptive filter establishes a linear growth of network structure,which generates a large computational complexity.Therefore,sparsification and quantization methods have been proposed for curbing the network size.Compared with sparsification discarding redundant samples directly,quantization approach can utilize these samples for further updating system weight,and thus give higher filtering accuracy under the reduction of computational complexity.Unlike kernel adaptive filter,from the observable disturbed measurements,Kalman filter can achieve the optimal estimation of unknown state based on the state space model.The optimal estimation can therefore be regarded as a filtering process.In comparison with Wiener filer,as a kind of online algorithm,Kalman filter performs filtering iteratively.This paper focuses on the study of the kernel adaptive filter and Kalman filter,which are described as follows.(1)Quantization approach in kernel adaptive can solve the issue of the linearly increasing network structure induced by kernel method,generating the reduction of the computational complexity.When samples in the same quantization area have similarity in terms of Euclidean distance,the corresponding desired outputs can be different especially in the non-stationary environment.Therefore,weighted average approach is presented for evaluating the importance of these desired outputs.Moreover,in traditional quantization approach,the sample centers in the dictionary are unchanged.To further improve filtering accuracy,the sample center is updated adaptively by using the steepest descent in this paper.Therefore,weighted quantized kernel recursive least squares(WQKRLS)is proposed by applying the weighted average approach and dictionary update to quantized kernel recursive least squares(QKRLS).(2)Kernel adaptive filter transfers the nonlinear issues into the linear inner ones in high feature space to achieve higher filtering accuracy.However,traditional kernel adaptive filters mainly use a feedforward network,which means that the current output is decided only by the inputs,not by the previous outputs.Therefore,to further improve filtering accuracy,the feedback network is applied to KRLS in a linear form to generate kernel recursive least squares with multiple feedback(KRLS-MF).The introduction of the linear feedback form in KRLS-MF cannot increase computational complexity significantly,and can implement the improvement of filtering accuracy in comparison with KRLS.(3)On the stability of Kalman filter,the traditional convergence methods are mostly based on Lyapunov function so that the sufficient conditions for guaranteeing that the stability of state can be obtained.In comparison with measurements,unknown states are unobserved directly,resulting in that the traditional convergence method is difficult to perform.Based on the system associated with non-linear state and linear measurement functions,the convergence analysis based on innovation is proposed.Since the linear relationship exists in the estimation error of state and innovation,the stability of state estimation can be guaranteed by the convergence of innovation.