Diss / 10

where γrel is the relative threshold. The root-mean-square value of potential error under delay measurement of the unmodulated in frequency bell-shaped pulse, which can be given by

where τs is the duration of scanning bell-shaped pulse, at the level 0.46 has the following form:

The root-mean-square value of potential error under measurement of the Doppler frequency of the coherent unmodulated in frequency and bell-shaped pulse with duration τs at the level 0.46 takes the following form:

The root-mean-square value of potential error under measurement of the angular coordinates is determined by

where leff is the effective length of antenna aperture normalized with respect to the wavelength and is defined as

where

d is the length of antenna aperture

χ is the constant

Analogous relations take place for more complex models of signal processing techniques.

Thus, the SNR is a generalized parameter that can be used as QoS (the criterion of effectiveness) in the design of the complex radar system and radar signal processing methods.

1.3  PROBLEMS OF SYSTEM DESIGN FOR AUTOMATED COMPLEX RADAR SYSTEMS

The CRS contains a great number of interdependent elements and blocks and belongs to the complex system class. As mentioned earlier, the first step in the design of the system is a definition of functional purposes in the higher order system. In Section 1.2, we have introduced the complex radar target designation and control firing systems. Henceforth, we are limited by consideration of system design problems of the CRS carrying out operations of searching, detection, and tracking a set of targets within the limits of radar coverage and providing information with the required QoS on the reassigned line.

The main task of the system design is the choice and justification of a structural block diagram of the system. For this task, a designing process is based on existing experience of construction

Energy parameters of the complex radar system are the following:

•The power of scanning pulse Pscan

•The duration of pulse τs

•The transmitting antenna gain Gt

•The receiving antenna gain Gr

•The effective area of the receiving antenna Seff

Choice of these parameters is accomplished in accordance with the end use of the CRS, the level of development of the corresponding element base, the technology of production, the technology and procedures of adjustment, and taking into account the allowable charges on production manufacturing and operating costs. As a rule, under designing the system, a definition of scanning pulse parameters, ways of scanning pulse generation, and emission is basic, and results of definition are initial data to design the receiving path and target return signal processing algorithms.

In designing the receiving path and the target return signal processing algorithms, the energy parameters of the CRS are considered as the fixed and external parameters. Thus, attention is focused on solution of problems in accordance with which there is a need to define the target return signal processing algorithms ensuring a maximal effect that can be characterized by probability performance and accuracy of definition of the target return signal parameters required by the user. Ultimately, the solution of this problem is reduced to a definition of the target return signal processing algorithms and a choice of computer subsystems for signal processing tasks at all stages, from preamplification and signal conditioning to data preparation and radar information output to the user. To solve these problems, the target return signal processing algorithms well developed in the statistical signal processing area are used, which are invariant to methods of transmission and receiving the signals. Consequently, the receiving path and the target return signal processing algorithms can be developed individually from other blocks and components of the CRSs. Henceforward, a totality of signal processing algorithms and receivers and/or detectors, which are needed to design, is called radar signal ­processing system.

Considering the radar signal processing system as an autonomous subsystem of the complex radar system, we can define and solve the design problems of the system as applied to its classes and groups, not only the individual CRS, based on functional purposes in high-order systems. The control system of the CRS is, per se, the autonomous system, but by the nature of solving problems, it is the system of higher order in comparison with the considered CRS. Naturally, designing of the control system can be considered as the solo problem within the limits of requirement specification presented for the CRS as a whole.

Thus, the design of the CRS is divided into three individual tasks, which are solved independently of each other but satisfying the conditions of continuous interaction and adjustment of parameters to guarantee a fulfillment and reaching the functional purposes of the systems. Problems of energy parameter definition of the systems are outside the scope of this book.

1.4  RADAR SIGNAL PROCESSING SYSTEM AS AN OBJECT OF DESIGN

The systems approach to design assumes the availability of some basic mathematical models and structures of the radar signal processing systems, which must be put into the basis of new development. Under designing the radar signal processing system, the well-known structure of the receiving path of the CRS is considered as the basic. Optimal signal processing algorithms obtained from the statistical radar theory are used as the basic mathematical models. In accordance with the

surveillance radar signal processing concerning information about trajectories of tracking targets. Thus, there is a consecutive sequence in the radar signal processing by stages. Each stage possesses its own real timescale of signal processing that allows us to carry out an autonomous realization of these stages.

The radar signal processing systems are divided into three classes by the method of implementation: (a) analogous, (b) digital, and (c) analog–digital. Now, the radar signal processing systems of the third class are widely used. However, digital signal processing plays a leading role in exploited and designed CRSs owing to flexibility and universality. We can see a tendency of extension to use the digital signal processing techniques in designing CRSs owing to substitution of analog signal processing by digital one. Successes in digital signal processing, which have been achieved now, allow us to use digital signal processing techniques not only in time signal processing of coherent signals but also in spatial signal processing of coherent signals. Henceforth, we consider digital signal processing techniques starting from the matched filtering of coherent signals.

Solving the tasks of designing radar signal processing subsystems under the fixed energy parameters of the CRS, attention is paid to optimization of receiving path in natural and artificial noise and interferences. All tasks of optimal signal processing are solved by methods and procedures of the theory of statistical decisions. Because of this, the QoS indices of radar signal processing subsystems are imported from the theory of statistical decisions. For some cases and examples, these QoS indices acquired characteristic features of a radar.

Independent of any application area of CRSs, the main QoS indices of radar signal processing are

1. Space–time signal processing: the coefficient of energy use given by

where

q2 is the signal-to-interference-plus-noise ratio (SINR) q02 is the SNR [35]

2.Radar signal preprocessing:

a.The probability of signal detection PD

b.The probability of false alarm PF

c.The accuracy of definition of the target coordinates and their estimation, which is char-

acterized by the covariance matrix of estimation errors Kmes, in a general case, and by the variance of estimation error σ2mes

3.Radar signal reprocessing:

a.The probability of target trajectory detection PDtr

b.The probability of target trajectory false alarm PFtr

c.The accuracy of definition of the target trajectory parameters, which is characterized by the covariance matrix of estimation errors of the target trajectory parameters Kmestr

d.The probability of breaking up in the target tracking Pbr

The earlier-listed QoS indices are associated directly and unambiguously with QoS indices of the CRS, as a whole (see Section 1.3). We can observe a direct relation between them: the higher the QoS indices of radar signal processing subsystems, the higher the QoS indices of the system.

Under implementation of digital signal processing, there is a need to take into consideration some limitations in speed of corresponding devices. For this reason, an important and essential QoS index under digital signal processing is the digital signal processing effort defined by the number of operations at a single realization of the radar signal processing algorithm. Data throughput is an important

QoS index under digital signal processing, which can be defined, for example, as the number of target processing by the system simultaneously. Of course, other approaches and methods to estimate the data throughput of radar digital signal processing subsystems are possible [36,37].

In designing CRSs, an initial optimization of signal processing algorithms is carried out by criteria imported from the theory of statistical decisions. For instance, the Neyman–Pearson criterion is the basic criterion in the signal detection theory. The essence of this criterion is the following: we obtain the maximal probability of detection of target return signals (target trajectories) PDop under some limitations in choice of the false alarm probability PF:

PFad is the permissible probability of false alarm

Generally, estimations of signal parameters are optimized by the criterion of minimizing the average risk:

where

av ( ) is the average risk

a is the real value of the signal parameter

aˆ is the estimation of real value of the signal parameter

Operations and functions, which are not related to detection of the target return signals and definition and estimation of target return signal parameters, are optimized by criteria corresponding to maximal effect or maximal value of the corresponding QoS in applying restrictions on the energy supply and hardware implementation.

To optimize the system as a whole, the QoS must include all the main indices, that is, it must be a vector in mathematical form. If m subsystems of the CRS are characterized by individual QoS, for example, q1, q2, …, qm, then the system, as a whole, is characterized by the vector Q = (q1, q2, …, qm). The goal of vector optimization is to choose a CRS that possesses the best-case value of the vector Q. At the same time, we assume that the appropriate QoS vector Q is given already. The theory of vector optimization of a CRS is far from completion. Simple methods that are used allow us to reduce the vector analysis directly or marginally to a scalar one. A simple procedure is used in this book. We consider all individual QoS q1, q2, …, qm except the only one, for example, q1, which is the most essential, as there are restrictions with further conditional optimization of the system by this QoS.

The designing of a radar digital signal processing system is divided into two stages: the designing and construction of the digital signal processing algorithms and the designing of the computer subsystems. The designing of the digital signal processing algorithms employed by the CRS is initiated from making clear the main goal of digital signal processing algorithm construction; how the main functions can be generated into the CRS; definition of basic restrictions; QoS; and designing of the objective function. A sequence of the digital signal processing algorithm design is given as follows:

•Definition of purpose and main functions of the digital signal processing algorithm.

•Designing and construction of logical block diagrams of the digital signal processing algorithms; there is a need to propose several variants.

•Off-line testing and processing of the individual digital signal processing algorithms or logical blocks.

•Simulation and definition of workability and QoS of the digital signal processing algorithm.

•Optimization and construction of the complex digital signal processing algorithm employed by the CRS, which operates in real time. The optimization is carried out by the discrete choice method of trade-off variant from a set of possible ones, which are digital signal processing algorithms corresponding to the given stage of operations. This approach allows us to combine the heuristic procedures and methods based on design-automation systems and optimization of the digital signal processing algorithms.

On the basis of designing and debugging results of a complex digital signal processing algorithm, we are able to define the main parameters of computer subsystems and to state the basic requirements to these parameters with the purpose of realizing all steps of signal processing in the complex radar systems.

Designing of special-purpose computer subsystems (SPCSs) starts from definition of the main parameters and relationships with components of SPCS structure. After definition of functions between parameters of an SPCS as a whole and parameters of structural components of an SPCS, we can formulate the requirements to each elementary structural block of SPCS based on the general requirements to computer subsystems and, by this way, to define the requirements specification to design the CRS as a whole. Relationships between the main parameters and elementary structural blocks of an SPCS are called the parametric balance. The most widely used system balances are the time balance, error balance, memory size balance, reliability balance, balance of costs, and so on [38,39].

Generally, in designing the SPCS structure, the solution can be found using the following variant of the criterion “effectiveness—cost”:

•Providing the minimum time to realize the complex digital signal processing algorithm under given restrictions on the equipment investments

•Providing the minimal equipment investments under given time of realization of the complex digital signal processing algorithm

In designing CRSs, the second variant is preferable. Ultimately, the designing of an SPCS reduces to definition of the computer subsystems number for different functional purposes and conservation of algorithms to interact between computer subsystems.

In conclusion, we consider an example of systematic sequence under the system design of complex digital signal processing algorithms and computer subsystems for radar signal processing systems (see Figure 1.6). Before designing a CRS, there is a need to define the requirements specification, in which the main purposes and requirements, structural block diagrams of subsystems, and the main restrictions and requirements on the parameters at the system output are described.

Consider the main stages of the CRS design, which are presented in Figure 1.6:

•The first stage (block 1) is a formulation of the optimal designing problem, definition of external and internal system parameters and relationships between them, and choice and justification of the objective function of optimal designing. The result is a formalization of the requirements specification.

•The second stage is a decomposition of the general problem of system designing (block 2) on a set of simple tasks of subsystem designing and corresponding representation of the general objective function in the form of superposition of objective functions of the optimal subsystem designing. Success in solving the problems of the first and second stages depends on the level of development of the optimal designing methods, in general, and radar signal processing systems, in particular.

FIGURE 1.6  Stages of CRS designing of the digital signal processing algorithms and computer subsystems.

•The third stage (block 3) is the designing and investigation of digital signal processing algorithms of subsystems and the generalized algorithm of a CRS. Designing and construction of algorithms is accompanied by comprehensive analysis and testing of digital signal processing algorithms on performance and effectiveness by preliminary established criteria. The main procedure of investigations is a simulation. Success in solution of problems of the third stage depends on the level of development of the theory of signal processing in radar systems.

•The fourth and fifth stages (blocks 4 and 5) are the synthesis and selection of equipment and hardware of SPCSs to realize the digital signal processing algorithms in the CRS. Realization of these stages is carried out interacting with previous stages of digital signal processing algorithm design with the purpose to reach the optimal agreement between digital signal processing algorithms and computing facilities by the QoS criterion called “efficiency–cost.” Success in problem solving of the fourth and fifth stages depends on the level of development of the theory of computer systems.

•The final stage (block 6) of system designing is the evaluation of the effectiveness of the constructed generalized digital signal processing algorithm and the computer system as a whole by the general criterion given in the requirements specification or chosen at the first stage of system designing. Completeness and reliability of this stage depend on the level of development of the theory of operation systems. Results of efficiency evaluation are used to make a decision to finish the system designing stage and complementation to technical

designing stage if the requirements specification and required QoS by efficiency criterion are satisfied. In the opposite case, there is a need to change the requirements specification and to repeat all stages of system design again.

Henceforward, in this book, we follow the considered sequence of system designing stages with attention to construction of the digital signal processing algorithms and choice of computer systems for their realization in CRSs.

1.5  SUMMARY AND DISCUSSION

The basic aspects of systems approach to design CRSs discussed in this chapter are the following. The systems approach is considered to design CRSs. The designing process is divided into two

sufficiently pronounced stages—system designing and engineering designing. A conception of complex information and control system integrity, including a CRS, makes specific owing to the idea of backbone communications, for example, structural and control communications. At the stage of system designing, the main factor to consider is the ­structure or architecture of the future complex information and control system, including a CRS. At the stage of engineering designing, the important issue is the choice and development of a fixed totality of CRS elements and communications between them.

Any CRS cannot be imagined without the environment. The main problem is to define an optimal boundary between the system and the environment. The environment, facilities of counteracting forces, level of development of the element base and technologies, economic factors, and human element are the factors affecting the operation of a system. Based on the methodology viewpoint, the definite aspects of the systems approach were defined to design any complex information and control system, including a CRS.

In the design of such systems, we widely use mathematical models, simulation models, and modeling as a process of representation of the CRS by an adequate model with subsequent test operation to obtain information about its functioning. The most characteristic feature of the systems approach is to search for a decision by iterative optimization based on computer-aided designing systems. The main operations of the system design are the definition and generation of the end goals; generation of all possible alternative variants; definition of investments to realize each alternative version of the system structure; designing the models chosen to optimize alternatives and their software implementation; and comparison of alternatives and the decision making.

Automated CRSs are widely used to solve different problems, namely, air-traffic control, military fighter/attack, ballistic missile defense, battlefield surveillance, navigation, target tracking and control, and so on. Based on the purposes and nature of problems, automated CRSs can be classified into two groups: information radar systems and control radar systems.

As an example, the main requirements and QoS (the criterion of effectiveness) of classical radar antiaircraft and missile defense and control systems are discussed. Reasoning from the considered QoS (the criterion of effectiveness) of the antiaircraft defense system, QoS indices of radar subsystems included in the antiaircraft defense system are defined. According to the systems approach in CRS designing, there is a need to take into account the QoS criteria possessing a physical sense, the ability to be determined, and associated with technical parameters of the designed system. In designing CRSs, it is difficult to define a general criterion (QoS) satisfying the requirements mentioned earlier owing to a complicated mathematical model to search for a target. For this reason, it is worth introducing an intermediate QoS instead of the general criterion (QoS) with the purpose of relating the main parameters of the systems and signal processing subsystems that are designed. It has been defined that the SNR is a generalized parameter that can be used as QoS (the criterion of effectiveness) in the design of the CRS and radar signal processing methods.

The main task of the system design is the choice and justification of a structural block diagram of the CRS. In this task, the designing process is based on the existing experience of construction of the system with analogous application. In doing so, we should take into consideration the structural stability of the system to changes in functional purposes and initial premises of design, including motivations stimulating a new design process. In this case, we need to design and construct only a small number of elements and blocks in the future.

The designing problem of the CRS is divided into three individual tasks, which are solved independently of each other but satisfy the conditions of continuous interaction and adjustment of parameters to guarantee fulfillment and enable achieving the functional goals of the systems. They are as follows:

1.Definition of the target return signal processing algorithms to define the parameters of the target return signal required by the user

2.Choice of computer subsystems for signal processing tasks at all stages, from preamplification and signal conditioning to data preparation and radar information output to the user

3.Definition of energy parameters of the CRSs, which is outside the scope of this book

Optimal signal processing algorithms obtained from the statistical radar theory are used as the basic mathematical models. There is a consecutive sequence in radar signal processing by stages. Each stage possesses its own real time scale of signal processing that allows us to carry out an autonomous realization of these stages. Solving the tasks of radar signal processing subsystem designing under the fixed energy parameters of the system, the main focus is on optimization of the receiving path in natural and artificial noise and interferences. All tasks of optimal signal processing are solved by methods and procedures of the theory of statistical decisions.

Independent of the application area of CRSs, the main QoS indices of radar signal processing are

(a) space–time signal processing: the coefficient of energy use; (b) radar signal preprocessing: the probability of signal detection; the probability of false alarm; and the accuracy of definition of the target coordinates and their estimation, which is characterized by the covariance matrix of estimation errors, in a general case, and by the variance of estimation error; (c) radar signal reprocessing: the probability of target trajectory detection; the probability of target trajectory false alarm; the accuracy of definition of the target trajectory parameters, which is characterized by the covariance matrix of estimation errors of the target trajectory parameters; and the probability of breaking up in the target tracking. These QoS indices are associated directly and unambiguously with the QoS indices of the complex radar system, as a whole. We can observe a direct relation between them: the higher the QoS indices of radar signal processing subsystems, the higher the QoS indices of the complex radar system, as a whole.

The designing of a radar digital signal processing subsystem is divided into two stages: designing and construction of the digital signal processing algorithms and designing of the computer subsystems. The designing of the digital signal processing algorithms employed by the complex radar system is initiated from making clear the main goal of digital signal processing algorithm construction; how the main functions can be generated into the complex radar system; definition of basic restrictions; QoS; and designing of the objective function. Designing of computer subsystems with special purpose, which are called SPCS, starts from definition of the main parameters and relationships with components of SPCS structure. After definition of functions between parameters of SPCS as a whole and parameters of structural components of SPCS, we can formulate the requirements of each elementary structural block of SPCS based on the general requirements to computer subsystems and, in this way, define the requirements specification in designing the complex radar system as a whole. Generally, under designing the SPCS structure the solution can be found using the following variant of the criterion “effectiveness–cost.”