Kernel Based Adaptive Filtering Algorithms And Its Applications On System Identification

Adaptive nonlinear filters are useful in the field of signal processing, extremely in the nonlinear and nonstationary environment for solving the complexity and non-convexity problems. In the past, nonlinear algorithms are computationally expensive because of the high dimensionality of block adaptation given by the huge number of involved input data. Nowadays, among nonlinear filtering methods, kernel based methods are the most popular nonlinear methods owning to the mathematical foundation and analytical power. Also, experimental results indicate the great success of kernel nonlinear models. Online kernel algorithms could save a lot of computation, which make them very indispensable for their flexibility in filtering design.Our work developed a class of on-line learning algorithms in reproducing kernel Hilbert spaces(RKHS). The reproducing kernel Hilbert space offers an elegant extension of obtaining nonlinear models of typical linear algorithms using the famed kernel trick with the expression of inner products. We presented kernel extensions for the well-known adaptive filtering method of least mean mixed-norm(LMMN) and studied their performance theoretically and validated in real applications.Kernel methods provide an efficient nonparametric model to produce adaptive nonlinear filtering(ANF) algorithms. However, in practical applications, standard squared error based kernel methods suffer from two main issues:(1) a constant step size is used, which degrades algorithm performance under non-stationary environment, and(2) additive noises are assumed to follow Gaussian distribution, while in practice the noises are in general non-Gaussian and follow other statistical distributions. Therefore, this paper proposes two novel kernel-based ANF algorithms to overcome the existing problems.Quantized kernel least-mean square(QKLMS) algorithm is an up-to-date adaptive nonlinear learning algorithm under Gaussian noise environment with strong control of the growing kernel. We propose a new algorithm called the quantized kernel least-mean mixed-norm(QKLMMN) for adaptive nonlinear learning corrupted with non-Gaussian statistical distribution noise. As an alternative of squared error, a combination of mixed-norm error is utilized in the algorithm. In addition, this paper also proposes a kernel normalized mixed-norm(KNMN) algorithm. Compared to standard squared error based kernel methods, KNMN extends linear mixed-norm adaptive filtering algorithms to Reproducing Kernel Hilbert Space(RKHS) and introduces a normalized step size as well as adaptive mixing parameter.The steady-state convergence analysis is provided for both algorithms. Experiments of nonlinear time series prediction and nonlinear system identification are also demonstrated. Simulation results verified the desirable performance and superiority of our proposed algorithms.