Optimal discrete filtering of samples of a continuous random process against the background of correlated Markov noise

DOI: 10.34759/trd-2022-126-16

Аuthors

Detkov A. N.

State Institute of Aviation Systems, 7, Victorenko str., Moscow, 125319, Russia

e-mail: detkov@gosniias.ru

Abstract

In modern onboard information-measuring systems (IMS) of UAV, algorithms for optimal or quasi-optimal processing of random processes are practically implemented using digital signal processing tools. Therefore, these algorithms must be synthesized in a discrete form in the form of recurrent relations that are convenient for such an implementation. However, in most cases, the mathematical models of the processes evaluated and observed in the IMS have a continuous form of recording, which is due to the physical nature of the phenomena occurring with the signals. In this regard, the problem of synthesizing optimal algorithms for discrete filtering of counts of continuous random processes arises. At the same time, the modern element base makes it possible to use a high sampling rate in on-board IMS, which leads to autocorrelation of measurement readings. Therefore, in the statistical synthesis of filters, it is necessary to take into account the correlation of measurement noise, which significantly affects the generated filtering estimates. In this paper, proposes a new discrete Kalman filter (KF) based on a statistically equivalent discrete representation of continuous state and observation vector models to solve the problem of optimal linear filtering of samples of a continuous vector Markov random process, taking into account the known statistical characteristics of additive vector Markov correlated noise. The problem of filtering a state-space linear model with colored measurement noise and precise noise covariance matrices is converted to a problem of estimating a state-space linear model with white Gaussian measurement noise and precise noise covariance matrices using the measurement difference method and without state vector augmentation. The structure of analog-to-digital conversion and discrete KF is presented. The main feature of this algorithm is the accumulation (integration) of continuous observation on the time intervals between adjacent samples of the estimated state vector. To illustrate the potential characteristics of the accuracy and noise immunity of the synthesized algorithm, a simple example of filtering a continuous Gaussian Markov random process against the background of Markov random noise is considered. From the graphs presented, one can quantify how much analog and discrete algorithms for optimal filtering of a continuous random process depend on the ratio of the width of the spectral density of the measurement noise to the width of the spectral density of the random process and the signal-to-noise ratio. The presented discrete filtering algorithm allows us to quantify the deterioration of the potential characteristics of accuracy and noise immunity in the case of optimal estimation of continuous random processes due to the coloration of measurement and sampling noise. The use of the method of difference measurements does not increase the computational costs in the discrete KF, since the dimension of the state vector remains unchanged, as in the case of filtering continuous random processes against the background of white Gaussian noise. At the same time, the analog part of the analog-to-digital converter is significantly simplified in comparison with similar algorithms, since a one-cycle delay is implemented in a discrete KF after the ADC. It should also be noted that there is no operation of analog differentiation inherent in classical algorithms for filtering continuous random processes in continuous time using the method of difference measurements.

Keywords:

continuous random process, optimal discrete filtering, correlated noise, difference measurement method

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