Diss / 10

Receiver

Memory for signals in gate

Signal preprocessing and selection

FIGURE 10.2  Nontracking MTI structure.

gate, as an example. Under optimal binary signal processing within the limits of two-dimensional (by radar range and azimuth) gate, the logarithm of likelihood ratio takes the form [3]

where

dij are the values of binary signals, “1” or “0,” in ijth gate cell; i = 1,…, n and j = 1,…, m are the number of discrete gate elements by the radar range and azimuth, respectively

ϖ(ij | i0 j0) is a set of the weight coefficients, with which the signals 1 and 0 are added in the gate cells under formation of the likelihood ratio

The concrete form of the function ϖ(ij | i0 j0) depends on the shape of amplitude envelope of two-dimen- sional signal that is processed. If we consider the signal model in the form of two-dimensional pdf surface in the polar coordinate system, the coordinates r and β, the weighting function takes the following form:

where

C1 and C2 are the constant values

r and β are the gate sampling intervals by the coordinates r and β

δr and δβ are the half signal bandwidth by the coordinates r and β at the level exp (−0.5) i0 and j0 are the coordinates of the maximal signal amplitude envelope

The two-dimensional likelihood ratio surface, which is the initial likelihood ratio to detect and select the target return signal pips within the limits of gate, is obtained as a weighting the binary target return signals by the weight function (10.2). Moreover, in this case, the two-dimensional likelihood ratio surface peaks contain all information about the signal presence and signal coordinates.

The detection and selection problem of a single target within the limits of the gate is assigned in the following way based on information contained in the likelihood ratio maxima. First, we take the hypothesis that there is only a single target within the limits of the gate. In the considered case, the event, when several targets are within the limits of the gate, is possible but is improbable. Using the relief of the two-dimensional likelihood ratio surface with M peaks of different heights, there is a need to define the maximum (the peak) formed by the target return signal and, if a “yes,” there is a need to define the coordinates of this two-dimensional likelihood ratio surface maximum. The amplitudes of peaks Zl (l = 1, 2,…, M) and their coordinates ξl and ηl with respect to the gate center are used as the input parameters, based on which the decision is made. If the hypothesis about the statistical independence of the two-dimensional likelihood ratio surface peak amplitudes is true and the coordinates of the two-dimensional likelihood ratio surface maxima are known within the limits of range where the target return signal is present, the optimal detection–selection problem of the target return signal pips within the limits of gate is solved in two steps [4].

Step 1: There is a need to select the two-dimensional likelihood ratio surface maximum with the number l* with quadratic form

where

–

Z is the average amplitude of the two-dimensional likelihood ratio surface in the signal domain σ2Z is the amplitude variance of the two-dimensional likelihood ratio surface in the signal domain

Step 2: Compare the obtained quadratic form Ql with the threshold defined based on the acceptable probability of error decisions and, in the case of exceeding the threshold, issue a decision about the target pip detection. The coordinates of the two-dimensional likelihood ratio surface maximum are considered as the coordinates of the detected target pip within the limits of gate.

The described optimal algorithm is difficult to realize in practice owing to the high work content

to estimate the average value – of amplitude and amplitude variance σ2 of the two-dimensional

Z Z

likelihood ratio surface in the signal domain. Therefore, the following simplified algorithms of target pip detection and selection are used in practice:

•Algorithm of detection and selection using the two-dimensional likelihood ratio surface maximum—at first, there is a need to define the amplitude of the two-dimensional likelihood ratio surface maximum within the limits of gate and compare with the threshold. Target detection pip is fixed if the amplitude of the two-dimensional likelihood ratio surface maximum exceeds the threshold and the coordinates are defined by a position of the two-dimensional likelihood ratio surface maximum within the limits of gate.

•Algorithm of detection and selection using the two-dimensional likelihood ratio surface maximum amplitude exceeding the threshold and possessing a minimal deviation with respect to the gate center.

The first algorithm possesses the best selecting features evaluated by the probability of detection PD and selection at the fixed probability of false alarm PF. For each case, there is a need to carry out a detailed analysis and find the trade-off, taking into consideration the requirements of the problem solution quality and available computational resources with the purpose of selecting the acceptable signal detection and selection algorithms within the limits of the target tracking gate of the nontracking MTI.

course of target trajectory coordinate measurement, random error of coordinate extrapolation, and dynamical errors caused by maneuvering targets. Errors of measurements and extrapolation by each individual coordinate are subjected to the normal Gaussian pdf with zero mean and known variance.

As noted in Chapter 4, under stable target tracking, the physical target tracking gate dimensions by each coordinate are minimal, i.e., Zmin {Z = {r, β}}. The target maneuver, regular and random deviations of target return signal power, and misses of target pips on target trajectory track lead to an increase in the physical target tracking gate dimensions in comparison with minimal ones. At the same time, the probability of event that the physical target tracking gate dimensions are minimal or close to minimal is the highest. Distribution corresponding to described process of changes in dimensions of the physical target tracking gate can be presented in the following form:

where σ2Z is the variance of changes in the physical target tracking gate dimensions by the coordinate Z. In this case, the pdf of request queuing time takes the following form:

Sometimes the physical target tracking gate dimensions are chosen as constant values determined at the maximum total error of extrapolation. For this case, the request queuing time by memory will be constant.

When the simplest signal processing algorithm for definition of the maximal weighted sum of the target return signals is used in the detector–selector, the signal processing operations are the following:

•Definition of the weighted sum amplitude of the target return signals for each physical target tracking gate cell

•Sequential comparison of amplitudes of the target return signals for each physical target tracking gate cell with the purpose of choosing the maximum one

•Comparison of selected amplitudes of the target return signals for each physical target tracking gate cell with the threshold and making a decision about detection of the target pip within the limits of the physical target tracking gate

In this case, the analysis time is defined by dimensions of the physical target tracking gate. Furthermore, we consider the case of two-dimensional physical target tracking gate. According to (10.5), the distribution of normalized dimensions of the two-dimensional physical target tracking gate by each coordinate is defined as

Since generally, the exact definition of the request queuing time pdf is impossible, we use two types of approximation given by (10.14) and (10.15) in further analysis of the detector–selectors. Finally, in the considered case we think that the time required to complete all signal processing operations is a constant value.

Now, consider and discuss the quality of service (QoS) of the multiphase queuing system. We can think that QoS is based on the probability of failure under a service of the next request as a function of input memory capacity and the average time of request processing by the queuing system:

where

τi is the average request queuing time during the ith phase

ti wait is the average waiting time for request queue before the ith phase or stage

Taking into consideration these QoS indicators under the known number of operations that are required for a single request queuing, we can define and estimate the required speed of microprocessor network operation realizing each phase of queuing systems.

Difficulties under analysis of multiphase queuing systems are the following. At all cases of practical importance, the output stream of phase takes more complex form in comparison with the incoming request queue. In some cases, the output request queue can be approximated by the simplest incoming stream with the same parameters. Then we can use analytical procedures and methods of the queuing theory to analyze the next phase or stage. If this approximation is impossible, then the only method to investigate the stream is the simulation. Rational combination of analytical and simulation methods and procedures allows us to solve the problem of the three-phase signal processing system analysis using the MTI system for any design and construction version.

10.2  ANALYSIS OF “n – 1 – 1” MTI SYSTEM

10.2.1  Required Number of Memory Channels

Since according to the operational conditions of MTI system the digital signal processing within the limits of the physical target tracking gate can be started only after filling out all memory cells of this gate, the request queuing time in memory is equal to the time of scanned angle by radar antenna corresponding to the azimuth dimension of the physical target tracking gate. This time is distributed according to (10.6). In this case, the average memory request queuing time is given by

The acceptable probability of request losses in memory is given and equal to, as a rule, Ploss = 10−3 −10−4. It is assumed that the request queue at the memory input is the simplest with the density γin that is given. The density γin is assigned based on the possible number of targets liable to tracking.

Using the Erlang formula [4]

Denoting χDS = γ in τDS, we obtain finally

where

υDS = τDS−1

χDS is the loading factor of the detector–selector

Formula (10.26) allows us to determine the detector–selector QoS factors as a function of its loading and relative shift ν of the pdf.

If the request queuing time τDS is distributed according to (10.15), then

Formula (10.32) allows us to determine also the detector–selector performance at the corresponding pdf of request queuing time. In particular, (10.26) and (10.32) show that an increase in relative shift of the pdf of request queuing time at the same loading factor χDS leads to a decrease in the average relative waiting time. As ν → ∞, the limiting value of this time tends to approach the value corresponding to the constant request queuing time. In this case, the request queue is stored in the input memory device. Computing the memory capacity by (10.30), there is a need to take into consideration the average waiting time for request queue in the detector–selector adding the average request queuing time for queuing requests in the memory device: