Diss / 10

–

Knowing Nwait and σ2Nwait, we can define the buffer memory capacity at the MTI input under the given acceptable probability to loss the request and assuming, for example, that the pdf of the

number of–requests in queue is the normal Gaussian. This approximation makes sense only at high values of Nwait. In this case, the probability of losing the request at the MTI input is defied as [6]

where QBM is the required buffer memory. There is a need to note that the acceptable probability of losing the request at the MTI input PMTIloss must be, at least, on the order less than the acceptable probability of losing the request at the system input. Only in this case the main requirement to the system operation will be satisfied, namely, all queue requests that pass the first phase of queue must be served by MTI.

10.4  ANALYSIS OF “n – m – 1” MTI SYSTEM

The considered system is the three-phase queuing system with losses at the input. The first phase is the n-channel queuing system with losses. Definition of the required number of channels of this system is carried out according to the procedure discussed earlier. The second phase is the m-channel queuing system with waiting. Analysis of QoS factors of this system is the subject of the present section. The input request queue is the simplest one with the parameter γin. We assume that the splitter operates as a counter with respect to the base m sending to the ith channel, i = 1, 2,…, m, the requests with numbers i, i + m, i + 2m. This allocation method of requests is called the cyclic way. The simplest request queue thinned in m − 1 times, i.e., the Erlang request queue of the (m − 1)th order, with the following pdf

comes in at the input of each channel of the one-channel queuing system. The condition of the m-channel queuing system stationary mode is the following:

i=1

At this time we assume that the average request queue time τDS is the same for all channels. Thus, in the considered case, the one-channel queuing system with the Erlang incoming request queue possessing the pdf equal to γinm−1 and the request queue time subjected to the pdf given by (10.14)

and (10.15) with the parameters τ and σ2 is investigated. The result of investigation must be the

DS τDS –

statistical characteristics of the request queue waiting time, namely, the average time twait and the

variance of this time σ2twait.

As noted earlier, the analytical investigation of queuing systems with the incoming request queue different from the simplest one is a complicated problem. A general approach to solving this problem

is discussed in Ref. [5]. However, an implementation of this general approach to specific conditions of

– 2

our problem is difficult, not obvious, and does not lead to final formulae for twait and σtwait. The explicit

solutions for twait / τDS obtained by numerical simulation are shown in Figure 10.8, where the solid lines represent the exponential with shift pdf of request queue time and the dashed lines represent the truncated and shifted normal Gaussian pdf. As we can see from Figure 10.8, with an increase in m the

twait τDS

FIGURE 10.8  Average waiting time versus the number of queuing system channels.

relative average waiting time decreases exponentially. The shift coefficient α affects the average waiting time by an analogous way as for previously discussed systems. At the given parameters γin, m, α, we can compute τDS based on the curves presented in Figure 10.8. We are able to define the average waiting time of request queue and, taking into consideration the average waiting time of request queue, we are able to define the required number of channels n for the first-phase queuing system.

Dependences of the required speed of operation for each mth channel of the second-phase queuing system for several values of–the input request queue parameter γin as a function of the average number of reduced operations N required to process a single request under the loading factor of each mth channel χi = 0.9m are shown in Figure 10.9 for the truncated and shifted normal Gaussian pdf given by (10.15). The required speed of operation is computed by the following formula:

Analysis of (10.47) and dependences in Figure 10.9 show that at the fixed parameters of the input request queue and the already-given number of operations for a single realization of digital signal processing algorithm, the required speed of operation of the second-phase queuing system decreases proportionally to the number of channels in this system.

A set of requests processed by the second-phase queuing system form the incoming request queue for the third-phase queuing system that is a one-channel queuing system with waiting. According to the limiting theorem for summarized flows, the incoming request queue at the third-phase queuing system input is very close to the simplest one with the parameter γin. The request queue time in the third-phase queuing system is constant, as earlier. Therefore, a computation of required characteristics of the three-phase queuing system and buffer memory capacity is carried out by the procedure discussed in Section 10.3.

348 Signal Processing in Radar Systems

If we assume in (10.6) α = τminmemory /στmemory = 2, then the average request queue time in the memory is defined as

To compare the target tracking system of various types we define the following:

•Number of channels (matrices) of the input memory Nmemory

•Average waiting time at the detector–selector input tDSwait

•Average request queue time in the detector–selector τDS

•Required speed of operation of the detector–selector VDSef

•Number of detector–selectors NDS

•Average waiting time at the MTI input tMTIwait

•Request queue time in MTI τMTI

•Number of the buffer memory cells at the MTI input NBM (MTI)

•Required speed of MTI operation VMTIef

•Average request delay by queuing system τΣ

Computational results are presented in Table 10.1, which shows that the system “n – 1 – 1” ensures the minimal average time of signal processing by one target equal to τΣ = 0.16s at the speed of operation of MTI equal to VMTIef = 150 × 103 operations per second and the speed of operation of detector–selector equal to

VDSef = 66.7 × 103 operations per second, i.e., providing a refreshment on nine targets in the course of each scanning. The same average time of signal processing is provided by the system “n – n – 1” when there are six detector–selectors with the speed of operation equal to VDSef = 66.7 × 103 operations per second but at the lesser speed of MTI operation. Obviously, a realization of this version is expensive in comparison with the system “n – 1 – 1.” Other versions of the system can be realized using the detector–selectors with the low speed of operation that can be a clincher in their favor [7]. The system “n – m – 1” can also be realized using the detector–selectors with a low speed of operation. In doing so, the number of detector–selectors is not high (two or three), which is an advantage of this system in comparison with the second type of system. The disadvantage is an increase in the request queue time.

Selection of definite version of the system is defined by computational resources and acceptable time of signal processing. Evidently, the most effective system from considered ones is the system “n – m – 1” with two or three detector–selectors.

TABLE 10.1

Comparative Analysis of Target Tracking Systems

10.6  SUMMARY AND DISCUSSION

Detection and tracking of all targets in the all-round surveillance radar coverage with accuracy sufficient to reproduce and estimate data of the air situation are carried out by the so-called rough channel of global digital signal processing system covering the whole scanned area. The rough channel consists of the following subsystems: the binary signal quantization, the specific microprocessor network for target return signal preprocessing, and the microprocessor network for digital signal reprocessing and control. Employment of binary quantization and simplified versions of digital signal processing algorithms allows us to implement the microprocessor networks and sets in this channel of global digital signal processing system of the all-round surveillance radar with the uniform antenna rotation.

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 it 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.

All considered and discussed versions of target tracking by several MTI systems are threephase queuing systems. Each phase is represented by one or several devices of queuing system connected in series. Generally, the request queuing time for each device is the random variable with pdf available to investigate and define. It is assumed that there is a simple request queue at first-phase input of the queuing system. The request queue coming in at the first device input is immediately served if, at least, one channel of the queuing system is free from service; otherwise, the request is rejected and flushed. The requests carried out by the first device are processed sequentially at the next phases; in other words, the request loss at each next phase or stage is inadmissible. In line with this, the memory device with the purpose storing requests in line should be provided before the second and third phases of the queuing system.

Difficulties under analysis of multiphase queuing systems are the following. At all cases of practical importance, the output stream of phase takes a 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. A 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.

REFERENCES

1.Lyons, R.G. 2004. Understanding Digital Signal Processing. 2nd edn. Upper Saddle River, NJ: Prentice Hall, Inc.

2.Harris, F.J. 2004. Multirate Signal Processing for Communications Systems. Upper Saddle River, NJ: Prentice Hall, Inc.

3.Barton, D.K. 2005. Modern Radar System Analysis. Norwood, MA: Artech. House, Inc.

4.Skolnik, M.I. 2008. Radar Handbook. 3rd edn. New York: McGraw-Hill, Inc.

5.Gnedenko, V.V. and I.N. Kovalenko. 1966. Introduction to Queueing Theory. Moscow, Russia: Nauka.

6.Skolnik, M.I. 2001. Introduction to Radar Systems. 3rd edn. New York: McGraw-Hill, Inc.

7.Hall, T.M. and W.W. Shrader. 2007. Statistics of clutter residue in MTI radars with IF limiting, in IEEE Radar Conference, April 2007, Boston, MA, pp. 01–06.

Part III

Stochastic Processes

Measuring in Radar Systems