Differential Operator And Its Application In Signal Processing

Since the1960s, with the development of computer and electronic information technology, digital signal processing has become an emerging discipline, rapidly. Meanwhile, it attracted lots of scholars to research. So far, most of the signals models in the signal processing were represented by a series of simple singles. Then, the approach of signal separation has been the fundamental problem in many researches.In2008, Peng and Hwang proposed an adaptive signal separation algorithm--the Null Space Pursuit algorithm based on a differential operator, which makes use of an adaptive operator to decompose a signal into a sum of simple signals, and these simple signals belong to the null space of the differential operator. This method attracts wide attentions because of its adaptability and characteristics of fully data-driven. However, it has a good effect just for FM signal; in order to expanded the scope of signal that could be decomposed, Xiyuan Hu improved the algorithm. After that, someone improved the order of the differential operator to third and fourth.This paper can be divided into two parts. The first part is the Null Space Pursuit based on an arbitrary even-order differential operator. In this part, we firstly propose an arbitrary even-order differential operator and use it to the Null Space Pursuit algorithm; then, we estimate the value of the parameters by solving an optimization problem, thereby we obtain the concrete steps of the algorithm; finally, we conduct the experiments simulation by Matlab procedure and do some analysis and discussion.The second part is the application of the Runge-Kutta algorithm in Null Space Pursuit. First, we introduced the Runge-Kutta method and the classical Runge-Kutta formula. Then, we give the process of solving differential equations of higher order using the Runge-Kutta formula, and integrate it into the Null Space Pursuit algorithm. Finally, we take some examples to show the feasibility of the algorithm.