Research On Recursive Filtering Method Of Stochastic Systems In Complex-Valued Domain

The filtering problem of stochastic systems has always been an important research content in automatic control theory.It is a method of estimating the internal state of a dynamic system based on the available measurement data.Therefore,it is necessary to research the filtering method of the system to understand and control this system.At present,most of the research results on the filtering problem of stochastic systems are based on the real-valued domain,which has received extensive attention in both theoretical research and engineering practice.As one of the key parts in the field of signal processing,complex-valued signals can understand and analyze the basic characteristics of the signal more comprehensively,and the corresponding stochastic system in the complex-valued domain can represent more complex actual phenomena.Therefore,the research of filtering methods of the complex-valued domain random system is of great significance.This paper mainly studies the recursive filtering of the complex-valued stochastic systems.Based on the existing basic stochastic system model,when the random variable is a complex-valued number,in order to simplify the structure of the filter,an expanded form is used to rewrite the system model.Considering the uncorrelated noise and one-step correlation of noise,the filtering problems with multiplicative noise and uncertain parameters in stochastic systems are studied separately,and appropriate filter gains are designed to meet the required performance requirements.Specifically,the main work of this paper is as follows:(1)This paper investigates the filtering problem of complex-valued domain stochastic systems with multiplicative noises.For complex-valued random variables,covariance and pseudocovariance are two important characteristics,which can reflect the second-order statistical characteristics of the variable.Therefore,considering that the random variables in the discrete system are all complex-valued,and considering the irrelevant characteristics of noises,a filter structure with unbiased characteristics is selected.The accurate value of the filter error covariance is obtained through the recursive method,and a suitable filter gain is designed to make the error covariance reach the minimum value.The effectiveness of the proposed scheme is verified by the example simulation.(2)Consider that there is multiplicative noises in the complex-valued stochastic system,and the process noise and measurement noise of the system have one-step correlation characteristics respectively.Based on the one-step prediction method,the one-step prediction filter error covariance and filter error covariance are obtained,respectively.The recursive method is used to design the required filter gain so that the filter error covariance reaches the minimum.Numerical simulation is carried out for this situation,and the images are analyzed to prove the effectiveness of the proposed filtering method.(3)Consider the case of uncertain complex-valued parameters in a stochastic system,and research its filtering problem.Unlike multiplicative noises,uncertain parameters will complicate system analysis,making it difficult to obtain an accurate value of the filter error covariance,so it is necessary to find alternative filter performance indicators.Consider that the noise of a stochastic system is non-circular,and its pseudo-covariance is not zero.The upper bound of the filter error covariance is obtained through mathematical calculations,and a filter gain is designed so that the upper bound reaches the minimum at every moment.A simulation example is presented to prove the correctness of this scheme.(4)In order to have more extensive research significance for the filtering problem of the complex-valued random system with uncertain parameters,in the design of the system model,this paper studies the situation that the process noise and measurement noise of the system have onestep correlation characteristics.Based on related theorems and knowledge of complex-valued numbers,the upper bound of the filter error covariance is obtained,and then the required filter gain is obtained through the recursive method to ensure and minimize the upper bound of the filter error covariance.Finally,a numerical simulation example in the complex-valued domain is given to prove the effectiveness of the proposed filter design.