Regularized Kernel Recursive Least Square Algoirthm

In most adaptive signal processing applications, system linearity is assumed and adaptive linear filters are thus used. The traditional class of supervised adaptive filters rely on error-correction learning for their adaptive capability. The kernel method is a powerful nonparametric modeling tool for pattern analysis and statistical signal processing. Through a nonlinear mapping, kernel methods transform the data into a set of points in a Reproducing Kernel Hilbert Space. KRLS achieves high accuracy and has fast convergence rate in stationary scenario. However the good performance is obtained at a cost of high computation complexity. Sparsification in kernel methods is know to related to less computational complexity and memory consumption 1 Linear And Nonlinear Adaptive Filter The term filter usually refers to a system that is designed to extract information about a prescribed quantity of interest from noisy data. An adaptive filter is a filter that self-adjusts its input output mapping according to an optimization algorithm like predictive coding [2] mostly driven by an error signal. Because of the complexity of the optimization algorithms, most adaptive filters are digital filters. With the available processing capabilities of current digital signal processors, adaptive filters have become much more popular and are now widely used in various fields such as sound wave based communication devices [23], face extraction [3, 26], camcorders and digital cameras, information retrieval [25] and medical monitoring equipments. 1.1 Linear Adaptive Filter In most adaptive signal processing applications, system linearity is assumed and adaptive linear filters are thus used. The traditional class of supervised adaptive filters rely on error-correction learning for their adaptive capability. Therefore, the error is the necessary element of a cost function, which is a criterion for optimum performance of the filter. The linear filter includes a set of adaptively adjustable parameters (also known as weights), which is marked as ω(n− 1), where n denotes discrete time. The input signal u(n) applied to the filter at time n, produces the actual response y(n) via y(n) = ω(n− 1)u(n) Then this actual response is compared with the corresponding desired response d(n) to produce the error signal e(n). The error signal, in turn, acts as a guide to adjust the weights ω(n− 1) by an incremental value denoted by △ω(n). On the next iteration, ω(n) becomes the latest value of the weights to be updated. The adaptive filtering process is repeated continuously until the filter reaches a stop condition, which normally is that the weights adjustment is small enough. An important issue in the adaptive design, no matter linear or nonlinear adaptive filter, is to ensure the learning curve is convergent with an increasing number of iterations. Under this condition, we define the system is in a steady-state. 1.2 Nonlinear Adaptive Filter Even though linear adaptive filtering can approximate non-linearity, the performance of adaptive linear filters is not satisfactory in applications where nonlinearities are significant. Hence more advanced nonlinear models are required.