Diss / 10

•If the target has been detected and the target coordinates have been measured with the given accuracy, then the corresponding target pip is produced to clarify the target tracing parameters. After that, the costs of refreshment about the detected target tracing are fixed and the control process is assigned to a mode administrator (blocks 11–13, Figure 6.5).

•If after observation of the entire assigned additional target searching directions the target has not been detected, then the algorithm of target tracing end detection is powered on, which produces a decision to continue or to stop target tracking of the considered target tracing (block 15, Figure 6.5). Naturally, in this case, the expenditure of energy is also fixed.

Thus, the considered control algorithm operates interactively in a close way with the algorithms of the main operations of signal preprocessing and reprocessing subsystems within the limits of gate coordinating these algorithms into the unique signal processing subsystem in the individual CRS target tracking mode with the control beam of radar antenna directional diagram.

6.3  SCAN CONTROL IN NEW TARGET SEARCHING MODE

6.3.1  Problem Statement and Criteria of Searching Control Optimality

The problem to control the searching mode is related to a class of observation control tasks from mathematical statistics and is given in the following form. Let the pdf p( x| λ) of the observed random variable X be not given completely and possess some indefinite parameter ν that is selected by the statistician before a next observation of the random variable X. Thus, the following relation takes a place:

In this case, there is a need to choose the parameter ν of the pdf p( x |λ, ν) before observation of the random variable X. As a result, the experimentator controlling the observations affects the statistical features of observed random variables.

In radar, the control process is based on obtaining the radar coverage scanning, selection of angle direction and interval on radar range scanning, shape of radar antenna directional diagram, type of searching signal, and so on. In our case, the control process of scanning is reduced to controlling the radar antenna directional diagram position, the shape and angle dimensions of which are fixed under the condition of realization of the discussed algorithms. Under synthesis of observation control algorithms, it is natural to follow the principle of average risk minimization, that is, minimization of losses arising under the decision error. However, in this case, a choice of the corresponding function of losses is a difficult problem.

As an example, consider the following discussion on generation of the quadratic function of losses widely known from the theory of statistical decisions [12–14]. Let a set of {Ntg;θ1,θ2 ,…,θN}

and the estimation of the total loss will be ∑Niˆ=tg1 v(t, θˆi ). As a figure of merit of the total loss estimation, we can consider the quadratic function of losses in the following form:

If we are able to define the a posteriori pdfs p(θi |X, V), where X is the vector of observed data and V is the control vector under searching, then the average risk can be defined in the following form:

In this case, a minimization of the average risk given by (6.17) must be carried out by selection of the control vector:

It is clear that a realization of such criteria is very difficult, especially within the limited time to scan the radar coverage zone.

The information criterion of efficacy reducing to maximization of average amount of information during scanning and observation of radar coverage was suggested in Refs. [15,16]. By this criterion, the control must be organized in such a way that an increment of information would be obtained at each step of observation. In a general case, a solution by both criteria is difficult. An approximate approach is based on a set of assumptions, of which the following are important:

•The problem is solved for a single target or set of uncorrelated targets.

•The observation space consists of cells, and we can scan a single cell only within the limits

of observation step (the scanning period); in this case, a control process is the integer scalar value υ coinciding with the number of observed cells.

•The υ th cell random value X observed under scanning is a scalar and subjected to the pdf

p0(x) in the case of target absent and the pdf p1(x) if the target is present; These pdfs are considered as known.

6.3.2  Optimal Scanning Control under Detection of Single Target

We start an analysis of optimal control methods to scan with the simplest case, namely, the detection of a single target within the limits of radar coverage consisting of N cells. Assume that there is a single target in the radar coverage or there are no targets. If there are no targets in the radar coverage, then the target can appear during the time τ within the ith cell (i = 1, 2, …, N) with the probability Pτ(i). If there is a single target within the limits of radar coverage, the probability of appearance of new targets is equal to zero. The target can be replaced by cells of scanning area with the probability of passing Pτ (i | j). Under a synthesis of control algorithm, we can consider the time (the number of cycles) from the beginning of scan to the instant of target detection as a figure of merit. The optimization process of the control algorithm is a minimization of this time.

The main result of synthesis shows us that the optimal control algorithm of radar coverage scanning is reduced to get the a posteriori probabilities Pt(i) of target presence in the cells (i = 1, 2,…, N) and to select the cell for next scanning (i = υ), within the limits of which the a posteriori probabilities Pt(i) of target presence are maximal, that is,

In the absence of the a priori data, the initial values of the probability P0(i) at t = 0 are chosen based on the condition that the probability of target presence in any radar coverage cell is the

the cells with the minimal probability of target presence. In accordance with this fact, a sequence of control algorithm operations must be the following:

•At the beginning of scan when the a posteriori probabilities of target presence in the cells are low, the υ th cell with the maximal probability Pt (υ) = max Pt (i) is selected to scan.

•If after regular scanning the probability Pt(υ) is continuouslyi increased, the scanning of the υ th cell is continued until this probability reaches the threshold value C1 chosen based on a permissible value of the probability of false alarm PF. After that, a decision about a target presence in the given cell is made and scanning of this cell is stopped.

•Scanning locator is switched on one no scanned cell with the maximal probability Pt(i). In doing so, if after regular scanning the a posteriori probability of target presence in this cell is decreased, there is a need to select the cell with the maximal value of the probability Pt(i) and scan this cell.

•Scanning the cell with the decreased a posteriori probability of target presence is contin-

ued until this probability is less than the threshold value C2 defined based on a permissible number of cycles required to scan the “empty” cells.

•The cell with the detected target is included in a set of candidates to be scanned after (owing to target moving) the fact when the probability of target presence in this cell becomes less than the threshold C1.

Thus, the control algorithm ensures a sequential addressing to the cells, including the cells with the low a posteriori probability of target presence. The considered algorithm is optimal based on the viewpoint of minimization of the number of scanning cycles T from the beginning of scan to the instant of detection of all targets. In this case, the a posteriori probabilities of target presence are determined by the following formulas:

•In the υ th cell after regular scanning—the formula (6.22)

•In other no scanning cells

Determination of the a posteriori probability of target presence at the instant tN + τ of next cell scanning taking into consideration the movement and appearance of new targets is carried out based on the following formula:

j =1

These formulas take into consideration a peculiarity of situation model within the limits of radar coverage, in accordance with which the cells are independent and changes in the a posteriori probability of target presence in one cell do not change the a posteriori probability of target presence in other cells. The disadvantage of the considered approach in the synthesis of the optimal scanning control algorithm under detection of targets is the fact that there are no limitations on power resources required to scan the radar coverage. Therefore, it is worthwhile to search another approach to solve the problem of scanning optimization within the limits of the radar coverage.

The criterion maximizing the mean of number of the detected targets during the fixed time TS is useful in this sense. In the case when the scanning model of radar coverage is in the form of cells, we have

Ncell

E{TS} = maxϕi ∑Pt (i)PS(ϕi ) (6.27)

i=1

when

i=1

where

ES is the energy during the scanning

Ncell is the number of cells within the radar coverage

Pt(i) is the a priori probability of target presence in the ith cell φi is the energy to scan the ith cell

PS(φi) is the probability of target detection in the ith cell under the condition that the energy φi has been required to do it

The criterion (6.27) allows us to find the optimal distribution of limited power resources by the cells but says nothing about the optimal sequence of scanning all the cells. Naturally, we consider a possibility to combine the criteria (6.21) and (6.27) with the purpose of obtaining the conditions to minimize the time to detect all targets under the limited power resources required to carry out the scanning within the limits of the radar coverage. In this case, the problem of optimal scanning control must be solved in two stages:

•Solving the problem (6.27), there is a need to find the optimal distribution of power resources required to scan the radar coverage expressed, for example, by the number of scanning cycles of each radar coverage cell.

•Using the criterion (6.21) and distribution of power resources by the cells, there is a need to define the optimal sequence of scanning of the radar coverage cells. At the same time, it is permissible to use the queuing theory, in particular, the optimal rule of application service

in decreasing order of the ratio Pt (i)Ni, where Ni is the number of scans in the radar coverage along the ith direction.

A sequence of statements for designing the optimal control algorithm follows:

•Let there be the a priori data about the probabilities PtN (i) (i = 1, 2,…, N) of target presence in each radar coverage cell at the time instant tN.

•Let the power resources to scan the radar coverage be given by the number N0 of scanning at the constant cycle T of searching signals.

•If for each cell of radar coverage the maximal radar range Rmaxi and the effective scattering area Stgi of target (the pulse power Pp and pulse duration τp are known) are given, we are able to define the energy of target return signal Esi under single scanning; the total power spectral density of the interference and noise Σi ; SNR under scanning by a single pulse (the radar receiver is constructed based on the generalized detector) SNRi = qi2 = Esi Σi; SNR under scanning by Ni signals is given by SNRNi = qN2 i = iqi2.

•We assume, for example, that during scanning the ith cell, the target return signal is a noncoherent pulse train with independent fluctuation according to the Rayleigh in-phase

and quadrature components. In this case, the conditional probability of false alarm PF and the probability of detection PD of the pulse train consisting of Ni pulses are determined by the following formulas [15]:

where

Xi is the sum of normalized voltage amplitudes (the signal + the noise) at the GD output Kg is the normalized threshold used under detection of target return signals by GD deter-

mined based on required values of the probability of false alarm PF

•The problem of optimal distribution of power resources by cells within the limits of the

radar coverage for a single scanning is solved by assigning the number of pulses Ni with the purpose to scan each cell, that is,

As a result, we obtain a set of pulse trains of the signals N1, N2 ,…, NNtg to scan all cells within the limits of the radar coverage.

•We define the ratios Pt (i)Ni and arrange them in decreasing order. After that we start a sequential scanning of the cells using the serial numbers of ratios Pt (i)Ni.

Thus, we have a principal possibility to design and realize the optimal control algorithm for scanning in detection of targets within the radar coverage. However, realization of such algorithms in practice is difficult. Difficulties are caused by the following methodological and computational peculiarities:

•The considered cell model of radar coverage is not acceptable in radar. There is a need, at least, to consider a model with a discrete set of scanning directions. At the same time, we should either abandon the cell model of radar coverage or present each direction of scanning by a set of cells that are scanned simultaneously. These approaches are discussed in Ref. [15], but they are very complex to implement in practice.

•The principal problem is to get knowledge about the a posteriori probabilities of target presence in the cells and determination of these probabilities by cell scanning data. The fact is that a computation of the a posteriori probabilities of target presence in the cells after each scanning is very cumbersome and the main thing is that this computation is not matched with the methods to choose the parameters of searching signals and tuning facilities to process these parameters, which are discussed in the previous section. For instance, in this case, it is absolutely unclear how we can control facilities to protect against the passive interferences.

•Preliminary determination of pulse trains to scan each cell of radar coverage by the criterion (6.27) is very cumbersome and not acceptable under signal processing in real time. Moreover, there is a need to know the a priori probabilities of target presence in the cells of radar coverage (directions).

In line with the discussed statements, as a rule, in practice, in designing the control subsystem of signal processing and operation of the complex radar system, we should proceed from the simplest assumptions about spatial searching and distribution of target in this space. First, we assume that the searching space consists of uniform searching zones, within the limits of which the target situation is of the same kind. The intensity of target set in each zone can be estimated a priori based on the purposes of the CRS and conditions to exploit it. The probabilities of target presence in each direction of the selected scanning zone are considered as the same, and the noise regions are assumed to

be localized. Because of this, there are no priority directions to search for the targets, and the zone scanning is carried out sequentially and uniformly with the period TS. In this case, the problem of controlling the scanning is to optimize the zone scanning period, for example, using the criteria to minimize the average time when the target could enter the scanning zone until the target detection under limitations in power resources issued to detect new targets. One of the possible versions of such an algorithm is discussed later.

6.3.4  Example of Scanning Control Algorithm in Complex Radar Systems under Aerial Target Detection and Tracking

As an example, consider a version to organize a scanning control by the corresponding CRS subsystem. In this example, we assume that the power resources must be distributed between the target detection and tracking modes that are realized individually in accordance with the real target and noise environment. At first, we discuss the general statements about designing the scanning control algorithm. After that, we describe the flowchart of the corresponding control algorithm.

First, we note that the power resource of the considered two-mode (in a general case, multimode) CRS required to scan is the random variable depending on the number of tracking targets. For example, at the initial period of operation, when there are no targets to track, all energy of the CRS is utilized to search new targets in the scanning space (the radar coverage). As the targets appear and the number of targets tracking increases, the power required to scan decreases. In doing so, owing to priority of the tracking mode over the detection mode, the power can be very low and close to zero. In the stationary mode of CRS operation, the number of tracking targets also oscillates within wide limits.

The power resources in scanning each direction in the searching mode depend on characteristics and parameters of the targets and the interferences and noise and are also defined by the signal processing algorithm employed in this mode. As noted previously, the minimal expenditure of energy required to scan directions can be reached employing algorithms of sequential analysis under signal processing and two-stage signal detection algorithms. In accordance with the approach considered in this chapter to organize a control of scanning and searching the targets, the number of directional scans under searching is established at each step in passing to a new direction based on analysis of the noise environment and taking into consideration the given probability of detection PD of targets with the known effective scattering area Stgi at the far boundary of the radar coverage. Thus, in our case, the number of directional scans is not a controlled parameter in the searching mode.

In practice, it is very difficult to determine the probability of target presence in each direction of searching space. However, in some clearly defined zones of this space, it is possible to compute parameters and characteristics of the set of the targets crossing each zone boundary, and consequently, the probability of crossing by targets these boundaries based on the a priori analysis of target functions in the performance of target tasks. At the same time, the probabilities of target way to the corresponding zone using any arbitrary direction must be considered as the same, naturally. The far boundaries of searching zones correspond to the maximal radar ranges that are characteristics of each zone under detection of targets with approximately the same effective scattering area Stgi. In accordance with general regulations of optimal control, the scanning process must be considered as a controlled process.

As a rule, the scanned zones possess unequal priorities in service. Priority is related to the importance of the target and with periodicity of searching and computational burdens to carry out a scanning. In particular, the highest priority can be assigned for the ith zone, where the following condition tSi /TSi → min is satisfied. Here tSi is the average time to search the ith zone; TSi is the period of the ith zone scanning, that is, the relative expenditure of energy for scanning is minimal. Priorities of other zones are decreased in increasing order of the relative expenditure of energy for scanning.

A buffer zone should be provided within the limits of scanning space, in which the energy required for the target detection and tracking is balanced. The strict period of scanning is not set in

is the time required to scan the jth priority zone in the kth cycle with the period of Tck ; νj is the

number of scans of the jth priority zone within the limits of the period Tck ; N (jlk) is the number of sweeps during the sth scanning of the jth priority zone at the kth control cycle; Tj is the period of searching pulses under scanning of the jth priority zone. The time to scan the buffer zone during the kth control cycle

is determined by block 4. The searching period of the buffer zone based on data of the kth control cycle

where Nbz is the number of scanned directions within the limits of the buffer zone that is determined by block 5. Henceforth, the computed scan period is smoothed by the totality of some control cycles (block 6). At the same time, as a first approximation, we can assume that the scan period is changed slowly within the limits of a small observation interval, and we can use the formula for exponential smoothing in the simplest form [17]:

where ι is the constant with a sense of the smoothing coefficient lying within the limits 0 < ι < 1. In future, the scan period TˆS(bzk) is compared with the permissible value (block 7), that is,

If the condition (6.39) is satisfied, there is a need to check the buffer zone radar range, that is, whether Rbzmax is a maximal value or not (block 8). If “yes,” then the control algorithm of CRS functioning­ receives a command for the next cycle:

If the condition

is satisfied, block 9 computes the coefficient γk of the buffer zone radar range changing Rbz(k+1) to ensure a restriction on the scan period duration. This coefficient depends on the previous buffer zone radar range Rbz(k) and deviation

that is,

222 Signal Processing in Radar Systems

In this case, block 10 computes the buffer zone radar range as given by

Computation of the buffer zone radar range Rbz(k+1) is also carried out in the case when (6.39) is satisfied and Rbz(k) is not maximal. In this case, first of all, we compute γk; after that we make more exact the buffer zone radar range Rbz(k+1). Evidently, if (6.39) is not satisfied, γk < 1, and if the criterion of optimization (6.44) is not satisfied, γk > 1. Henceforth, we check for all cases a double restriction on the buffer zone radar range. If these restrictions are satisfied, block 11 computes Rbz(k+1). If these restrictions are not satisfied, block 12 checks the inequality γk < 1. If this inequality is satisfied, block 12 yields the equality

Otherwise, the equality (6.44). At this stage the control algorithm cycle is stopped.

There is a need to emphasize that the considered example is one of the possible ways of the scanning control algorithm subjected to analysis and comparison with other versions of scanning control algorithms in the course of designing the CRS.

6.4  POWER RESOURCE CONTROL UNDER TARGET TRACKING

6.4.1  Control Problem Statement

The target tracking mode can be divided conditionally into two stages. The first is the pretracking stage that starts immediately after detection of new target tracing. There is a need to specialize the parameters of new target tracing in such a way that it is possible to evaluate the importance of the target (danger or not danger). In other words, there is a need to define parameters and characteristics of target movement with respect to the defended object, served airdrome of departure/arrival, or other controlled reference points. Additionally, based on obtained target tracking and signal processing data there is a need to define a predictable line of target service, for example, airdrome for air target or air-based, submarine-based, and land-based location. The required accuracy of target information to evaluate the target importance depends on the kind of target and system assigning. To reduce the time required to evaluate target importance, there is a need to select a variable location rate and the value of location rate must be as high as possible.

The second stage is that of stationary target tracking. This stage starts after evaluation of target importance and making the decision about the location of target service line. Since the qualified service and target tracking are possible only after reaching the given accuracy of target tracking estimation at the boundary line, the main task of the second stage is to accumulate information about the target tracking parameters and to extrapolate this information at the required instants, when the target tracking information is in stationary mode. Accuracy of evaluation of the target tracking parameters depends on accuracy of individual measurements of target coordinates and the number of observations (measurements). A choice of signal processing algorithm, in particular, the filtering algorithm, is very essential. In a general case, the problem is to get the required accuracy at minimal expenditure of energy and minimal digital signal processing system loading. Naturally, these requirements look as a contradiction, since increasing accuracy and target tracking subsystem reliability for real targets are possible only under more sophisticated signal processing or filtering algorithm; for example, the implementation of adaptive signal processing algorithms, which leads us to increasing the digital signal processing subsystem loading, or with increasing the frequency of coordinate measurements (increasing the number of observations or measurements), which leads us to an increase in CRS energy expenditure.