Research On Kernel Adaptive Filtering Algorithm With Feedback Mechanism

Filtering technology plays a significant role in the filed of signal processing. Since the signal is always contaminated by the disturbance or/and noise. It’s necessary to alleviate or remit the noise with the technology. There are four classical filers, i.e., low-pass filter, highpass filter, band-pass filter and band-stop filter. However, when the noise frequency coincides with signal frequency, the above filters are hard to weaken the noise. Adaptive filter has been as a modern statistical filtering technology, for to these fixed filters, its filtering frequency automatically adapts to changes of input signal. In the absence of any prior knowledge about the signal and noise, the adaptive filter can use the obtained parameters to automatically adjust the current parameters. Hence, it’s always received much attention since the adaptive filter is proposed. Nevertheless, with the increasement of nonlinearities of problems, the nonlinear adaptive filtering technologies are more and more important. Thus, many nonlinear adaptive filters with complex structures have been proposed but their performances are poor.Therefore, designing effective nonlinear adaptive filtering systems have gradually become a hot topic in the world. Fortunately, kernel method has a strong nonlinear processing capability,it can transform the difficult problems in a original space to the linear issues in a high dimensional space. The reason why kernel methods have been widely concerned is that, the inner product in the high dimensional space(or reproducing kernel Hilbert space, RKHS) can be efficiently evaluated by the kernel trick. In recent years, researchers innovatively combine kernel methods with the principle of adaptive filters to produce the kernel adaptive filters,KAFs. This becomes an important branch in the field of signal processing and control, due to the nonlinear processing power and the ability to make real-time processing.Nowadays, most KAFs belong to feed-forward type methods. But, this paper focuses on the study of the kernel adaptive filter with feedback mechanisms:(1) To investigate how the past information affects the algorithms, the previous instant output is fed into the systems. Then, the simplest but efficient cost function, i.e., instantaneouserror function, is considered, and the gradient descent method is applied to update connection weights. Finally, we obtain the Kernel least mean square with single feedback, SF-KLMS.Since SF-KLMS has many parameters and learning rates with larger fluctuations. A new KAF,with the same structure of SF-KLMS, is designed, which has fewer parameters and smoother learning rates. That KAF is variable learning rates kernel adaptive filter with single feedback,SF-VLRKAF. Furthermore, under the additive white gaussian noise, the energy conservation relation is adopted to analyze the stability of SF-VLRKAF. And, we obtain the effective mean square convergence conditions as well as the upper and lower bounds of the algorithm. It’s worthy noting that, the reason why SF-KLMS and SF-VLRKAF consider the previous output is that, based on the structure, the update forms of weights contain a momentum, which can improve the convergence speed and avoid the local minimum point. In addition, when the single feedback idea is expanded to random feedbacks, we can also get other KAF, i.e., kernel adaptive filter with random feedbacks, RFs-KAF. From the simulation results, these algorithms have their own advantages and disadvantages.(2) To improve the filtering performance, such as, convergence rate and testing mean square errors, different adaptive learning rates are designed in SF-KLMS, SF-VLRKAF and RFs-KAF. Because the learning rates of SF-VLRKAF have fewer parameters and smoother property, which becomes a role reason why we study the convergence of SF-VLRKAF.