Introduction
This book addresses robust signal processing problems in complex radar systems (CRSs) and describes their features. Both traditional problems of synthesis and analysis of the main digital signal processing operations and new problems of the robust signal processing in noise, in particular, the generalized approach to signal processing in noise under coherent filtering, are presented in this book. Problems of adaptation and control of functioning processes in CRSs are also new problems both by the problem statement and by the methods to solve these problems. Successes in automation allow us to make definite generalizations that will promote further development.
When designing CRSs from a modern perspective, it is important to support an interrelation between the individual robust signal processing algorithms and designing the multipronged attack on an all-round algorithm of CRS functioning in parallel with a choice of robust signal processing algorithms. In this regard, this book focuses on the problems related to system design. In this book, the complexity and difficulty in realization of robust signal processing algorithms are defined by specific systems approaches, allowing us to identify realistically the requirements of CRSs.
Construction of CRSs for information and control purposes is a multistage, long-term process. One of the important stages of system construction is design. That said, the actual problem lies in essentially increasing the quality of design while simultaneously reducing system construction time. To solve this problem, extensive use of science-based methods of CRS construction, taking into account features and functioning conditions, plays an important role.
An important problem is the definition of signal parameters. The process of defining signal parameters is related to theoretical and experimental investigation methods of stochastic processes. Experimental investigations of stochastic processes are applied in the following cases:
Under analysis of signal transformations by linear and nonlinear systems when we do not have a priori information with respect to statistic of the input stochastic process and the physical source generating the input stochastic process.
There is a need to check the correctness of theoretical approaches applied to the investigation of CRSs.
Mathematical description of physical processes in CRSs is cumbersome and impractical.
Analysis of statistical regularities for the design of optimal and quasi-optimal measuring systems to define CRS signal parameters based on the theory of statistical estimations is presented in this book. Much attention is also paid to the analysis of systematic and stochastic errors of signal parameter definition as a function of observation time and noise power. Procedures of experimental investigations of CRS stochastic processes are much more difficult when compared to methods of experimental investigations of deterministic processes. This can be explained due to the following reasons:
To describe the stochastic process completely, extensive evidence of various signal parameters is required.
It is impossible, in practice, to carry out a measurement of individual signal parameters in accordance with their definition.
Detailed investigations have been carried out for various methods to measure the main statistical characteristics of stochastic processes and their estimates such as mean, variance, correlation function, power spectral density, probability density function, probability distribution function, and so on. Analysis of measuring procedures and errors of these procedures under robust signal processing as well as block diagrams of programs for digital measuring systems are presented in this book. Structures of optimal metering systems for definition of the mean, variance, and parameters of the correlation functions are designed, and the drifts and estimate variances of these characteristics are also determined. The procedure of measurement of the mathematical expectation and variance of nonstationary stochastic processes under robust signal processing is discussed in detail. The general mathematical relations for the drifts and estimate variances of the main stochastic process characteristics are presented in an appropriate form for analytical calculations in the cases of the Gaussian and Rayleigh distributions of stochastic processes.
The following principal problems of mathematical statistics are frequently encountered under investigation of stochastic processes in robust signal processing based on the generalized approach to signal processing in noise in CRSs:
Definition of unknown probability distribution functions and probability density functions
Definition of unknown parameters of probability distribution function and probability density function
Test of statistical hypotheses
It must be noted that the last of these problems is a rare case compared to the first two problems.
The law of large numbers lies at the heart of experimental methods to define characteristics of stochastic processes under robust signal processing used in CRSs. According to this law, the probability of an event can be changed by the corresponding event appearance frequency, and the mathematical expectation can be changed by an average. In practice, given that a large number of tests are conducted, we can conclude that the probability of an event and its characteristics obtained in this manner are close to the true values. However, there are situations when we need to use only a limited number of tests. For these cases, we apply the same mathematical formulas as under the large number of tests. As a result, an additional problem, related to defining the estimate of characteristics obtained based on testing and comparing this estimate with the potentially achieved accuracy of measuring devices, arises.
Definition of extreme accuracy to estimate stochastic process parameters for our purposes is based on methods and procedures of statistical decision-making theory. This theory is developed to construct and design optimal measuring devices for deterministic and quasideterministic signals in noise and to analyze signal parameters and their estimations [1–5]. Investigations devoted to the methods of experimental study of stochastic processes and accurate definitions of estimations of statistical parameters are also widely covered in the literature [6–21].
In this book, we attempt to analyze the measuring methods of the main parameters of stochastic processes using the unified methodological approach, taking into consideration the theory of statistical estimates. The stationary ergodic stochastic processes and analog procedures to measure their parameters, in general, are investigated owing to the highest measuring accuracy. Specificity of digital measuring technique lies in analog-to-digital transform of signals under robust signal processing in CRSs and in the use of radar computer subsystems [22].
This book provides a definition of and investigates potential accuracy measurement of the probability distribution and probability density functions, correlation and covariance functions, mathematical expectation, variance, spikes of energy spectrum as a function of observation time, correlation interval of considered stochastic processes, and signal-to-noise ratio. In doing so, the bias, variance, and correlation function of estimations widely used in the theory of statistical estimates and mathematical statistics are employed to measure the accuracy of stochastic processes. To obtain more simple and illustrative examples, we use the approximated solutions acceptable in practice.
This book summarizes the investigations carried out by the author over the last 30 years. It consists of three parts. Part I discusses the main design principles of the modern robust digital signal processing algorithms used in CRSs. Special attention is paid to the generalized approach of signal processing in noise. Part II covers the main principles of computer system design for modern robust digital signal processing algorithms. Some examples of designing actual CRSs are discussed in this part. Part III deals with experimental measurements of the main statistical parameters of stochastic processes and definitions of their estimations. Important estimates of statistical parameters of stochastic processes, such as mathematical expectation, variance, correlation function, probability density function, probability distribution function, and other frequency–time parameters, are defined by experimental investigations.
The book consists of 15 chapters. Chapter 1 discusses the principles of systems approach to design CRSs. It focuses on the design methodology of and main requirements of CRSs. Problems in the system design of complex automated radar systems are covered. Representation of signal processing subsystems as an object of design is also investigated.
Chapter 2 deals with signal processing by the digital generalized detector in CRSs based on the digital signal processing procedures. The main principles of analog-to-digital conversion of signals are discussed. A comparative analysis of the generalized approach to signal processing in noise and matched filtering for digital signal processing procedures is done and the key results are analyzed. In addition, comparison between the digital matched filtering and digital generalized detector constructed for coherent pulse signal is presented. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 3 presents robust digital interperiod signal processing algorithms. It investigates robust digital signal processing algorithms for selection of moving targets. Digital generalized detectors of target return signals, for cases when the noise is with known and unknown statistical parameters, are discussed. Comparative analysis with robust digital generalized detectors is carried out. Digital measuring systems of signal parameters are investigated. Complex interperiod signal processing algorithms, including robust digital generalized signal processing algorithms are also investigated and a comparison is made. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 4 explores problems of robust signal detection algorithms based on the generalized approach to signal processing in noise and trajectory tracking of targets on digital measurements carried out by measuring subsystems. The principal stages and secondary operations of robust signal processing are discussed. Trajectory detection and tracking of targets through surveillance of CRS data are investigated and discussed. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 5 provides a definition of filtering algorithms based on the generalized approach to signal processing in noise for robust signal processing and extrapolation of target trajectory parameters using the measured data obtained by CRSs. In this chapter, the initial premises to determine the estimations and errors of trajectory parameters are defined. Investigation of the filtered input stochastic process at the front end of the generalized receiver is carried out. A statistical approach to define random unknown signal parameters by filtering technique is discussed. Algorithms of linear filtering and extrapolation under fixed sample size are defined and investigated. Recurrent filtering algorithms for the definition of trajectory parameters of the nondistorted polynomial trajectory are presented and discussed. Adaptive filtering, which allows us to define the trajectory parameters of maneuvering targets, is investigated in detail. Logical block diagrams of all-round algorithms for secondary signal processing are also discussed. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 6 deals with the principles of design and construction of control algorithms by CRS functioning in a dynamic regime. Organization principles and structural block diagrams of CRS control circuits are discussed. Principles of direct control of parameters and, in particular, device parameters, are formulated. Procedures of radar scanning control under new target searching are constructed and discussed. The principles of resource management under target tracking are defined. Distribution of energy resources under overlapping of target searching and tracking operations is described. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 7 deals with the principles for designing all-round algorithms of CRS computer sub-systems. The methods of assignment of all-round algorithms for computer subsystems are defined. An estimation of work content for the realization of all-round algorithms in computer subsystems is carried out. Principles of parallelization of computational processes by computer subsystems are discussed. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 8 focuses on the designing principles of CRS computer subsystems employing robust signal processing algorithms based on the generalized approach to signal processing in noise. The structure and technical requirements and parameters of computer subsystems are defined. The technical requirements concerning an effective processing speed are validated, taking into consideration the memory size and memory structure of computer subsystems. The technical characteristics of central computer subsystem microprocessors are defined. The structure and elements of computer subsystems are also discussed. The requirements and structure of central high-performance computer systems are proposed. Programmable microprocessors for robust signal preprocessing are designed. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 9 looks into an example to design CRS digital signal processing subsystems. The main statements and initial conditions are defined. The structure of designing computer subsystems is developed and validated. The microprocessor structure for robust coherent and incoherent signal preprocessing is also developed and validated. The requirements for microprocessor structure and characteristics under secondary robust signal processing are formulated. An example of a computer subsystem for robust signal processing of target return signals is discussed. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 10 is devoted to the analysis of digital signal processing subsystem samples. One of the variants of digital signal processing subsystems is proposed and analyzed. Analysis of radar measuring devices of the kinds. “n – 1 – 1,” “n – n – 1,” and “n – m – 1” is carried out, and the results are discussed. A comparative analysis of the proposed and considered versions of the target tracking subsystem design is carried out, and the results are discussed. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 11 focuses on the theory and experimental measurement of statistical estimation of stochastic process parameters. Basic definitions are presented, point estimations are defined, and their main features are discussed. Effective estimations, the loss function and average risk, and Bayesian estimations for various loss functions are also defined and discussed. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 12 explores the mathematical expectation estimation of stochastic processes defined by experimental tests. The conditional functional of probability density function with estimates at the mathematical expectation is defined by experimental study. The maximal mathematical expectation likelihood estimate and the Bayesian estimate of the quadrature loss function mathematical expectation are defined experimentally. Applied approaches to estimate the mathematical expectation by tests are discussed. The mathematical expectation estimate under analysis of parameters of stochastic processes at discrete time instants is defined experimentally. The estimate of the mathematical expectation under amplitude of stochastic processes is defined experimentally. The optimal estimate of the Gaussian stochastic process mathematical expectation varying as a function in time is defined experimentally. The optimal estimate of the mathematical expectation varying as a function in time under averaging in time the stochastic process is defined by experiment at robust signal processing. The estimate of stochastic process mathematical expectation is defined experimentally by interactive approaches. The estimate of stochastic process mathematical expectation with unknown time is defined by a test. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 13 deals with the estimate of stochastic process variance, which is defined by experimental investigation. The optimal estimate of Gaussian stochastic process variance is defined by a test. The estimate of stochastic process variance at averaging in time is defined experimentally. Errors caused by difference of transducer performance from the quadratic function and limitations in instantaneous values are defined experimentally. The estimate of stochastic process variance varying as a function of time is defined experimentally. Measurements of stochastic process variance under stimulus of the noise are carried out by tests. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 14 deals with the estimate of the probability distribution function and probability density function of stochastic process defined by experimental investigation. The main features of estimates of the probability distribution function and probability density function of stochastic process are defined by test. The major characteristics of the probability distribution function and probability density function estimate of stochastic process as well as the estimation variance of the Gaussian and Rayleigh probability distribution functions are also defined experimentally. Estimation of the probability density function of stochastic process based on estimations of expansion coefficients is defined by a test. The designing principles of measuring devices of the probability distribution function and probability density function of stochastic process are discussed and confirmed by experimental investigations. Theoretical study is strengthened by computer modeling results that are discussed.
Chapter 15 focuses on frequency–time parameter estimations of stochastic process defined by experimental investigation. Estimation of the correlation function of stochastic process is defined by experimental investigation. Estimation of the correlation function of stochastic process based on series expansion is defined by experiment. Estimation of the optimal correlation function parameter of Gaussian stochastic process is defined experimentally. Methods of correlation function definition based on other principles of estimations are defined experimentally. Estimations of the power spectral density and of spike parameters of stochastic process are defined experimentally. The average quadratic frequency estimation of power spectral density of stochastic process and measurements of the correlation functions and power spectral densities of stochastic process by digital signal processing approaches are presented experimentally. Theoretical study is strengthened by computer modeling results that are discussed.
This book presents the different principles of optimization in the structure of computer subsystems in the design and construction of CRSs, taking into account all peculiarities of robust signal processing of target return signals and control algorithms, by which we are able to define the parameters of the target and target trajectory. Additionally, it describes numerous procedures and methods to measure the principal statistical parameters of target and target trajectory and to define their estimates under signal and image processing.
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