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									               SUMMARY OF THE THESIS

 FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN
          ELECTRICAL ENGINEERING




   CONTRIBUTIONS TO THE
    DEVELOPMENT OF THE
 ADVANCED ALGORITHMS FOR
   AIR TARGETS TRACKING

                    Submitted by Ph. D. Student
          Augustin Sperila, Major, Electrical Engineer
                      Romanian Air Force HQ


                   Thesis under the supervision of
Gheorghe Gavriloaia, Colonel (R), Professor, Electrical Engineer
               Military Technical Academy, Bucharest




                       Bucharest, May 2005
         Augustin Sperila – Contributions to the development of the advanced tracking algorithms
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                                    CONTENT:

Chapter one                    Introduction
Chapter two                    The actual knowledge on the multisensor data fusion
                               when measurements' origin is uncertain
Chapter three                  Original contributions
Chapter four                   Tracking the air targets when measurements’ origin is
                               uncertain
Chapter five                   Multisensor data fusion
Chapter six                    Simulation set-up and results
Chapter seven                  Overall conclusions
Chapter eight                  Ph. D. Student’s Resume
Chapter nine                   The author’s list of publications


KEY WORDS: maneuvering air targets, data association, avoiding tracks
coalescence, BLUE data fusion.




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                                1. INTRODUCTION

      Effective track fusion against cluttered background and missed detections
has been a major challenge in the target tracking community. Although there
have been several algorithms proposed for performing the distributed track
fusion when the measurements’ origin is unsure, no general benchmark has been
given to date, which would be able to discriminate between their effectiveness
against the tracking scenario and system’s parameters. But, no matter the fusion
method is used, it has to deal with two major issues. Firstly, it has to prevent the
local estimators losing tracks in the presence of missed detections and false
measurements; and secondly, it has to account for the local estimates correlation
induced by the propagation of the common noise in the cinematic model.
Additionally, the problem is made more difficult by the need to search for the
best trade-off between the maximum allowed target maneuverability and the
system’s capability to maintain the tracks against a heavy clutter density.
      Supposing that the problem of maintaining the correct tracks identity at
the local processors was solved in an appropriate manner, the use of the Best
Linear Unbiased Estimate (BLUE) fusion rule appears to be very promising for
performing tracks fusion but meanwhile, one should have made provisions for
allowing the maximum possible targets maneuverability.
      The aim of this work is to investigate the effectiveness of a simplified
approach in employing the BLUE fusion rule which is avoiding the explicit
dependence of the cross-correlation between the local estimates on the history of
detections for each track. This approach will be tested together with an altered
version of the Jonker-Volgenant-Castanon (JVC) assignment procedure for
avoiding tracks coalescence, optimized at the expense of a small reduction in
target maneuverability during tracks interaction. The alteration will adapt the
JVC procedure to the Probabilistic Data Association Filter (PDAF), in order to
balance the need for both targets maneuverability and false alarms handling.
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      The herewith IMM-PDAF(altered)-JVC algorithm proposed for avoiding
tracks coalescence for maneuvering targets will be validated by extensive
simulation trials, on a program built under the MATLAB environment. Then, an
assessment of the improvement in the precision of the estimate will demonstrate
the power of the BLUE fusion rule when it is employed following the equivalent
approach described in chapter five.




         2. THE ACTUAL KNOWLEDGE ON THE
          MULTISENSOR DATA FUSION WHEN
        MEASUREMENTS' ORIGIN IS UNCERTAIN

       During the past two decades, among target tracking community there was
much work done for refining the known fusion rules, by that making them to
effective in a stressful environment. The stressful environment would, at
minimum, comprise strong interference between trajectories when those ones
cross each other, moderate-to-high miss-detection and false alarms probabilities
and, highly maneuvering-capable targets. To the knowledge of the author, none
of the works performed to date addressed all the listed issues in a coherent
manner.
       However, by employing the Chong's fusion rule in conjunction with the
Joint Probabilistic Data Association Filter (JPDAF), the resulting algorithm
should have the capability to appropriately deal with al the listed issues. In one
of the cited papers, a sound mathematical development offers the equations for
what is called there the "distributed JPDAF algorithm", but the analysis is
confined to the performances of the algorithm in improving the estimation
precision on straight trajectories, with a small plant noise covariance. Moreover,
despite the heavy processing burden, no indications were given about the rate of
lost tracks, and the fusion with incomplete communication rate is approximated
only for small noise covariance. On the other hand, there is another newer
(1999), more powerful data fusion rule than the Chong’s one, namely the Best
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Linear Unbiased Estimate (BLUE) rule, whose optimality criterion is more
restrictive. Indeed, the BLUE fusion rule is using weighting coefficients
optimized so that at any time, the Hessian of the estimation error covariance is
minimized (for the Chong’s fusion rule, only the trace of the error covariance is
imposed to yield the minimum).
       Starting from its strength, it would be highly desirable to implement the
BLUE fusion rule in conjunction with a dedicated algorithm for managing the
incertitude concerning the measurements’ origin. The problem which arises here
is that for iteratively computing the local estimates’ weighting coefficients, one
should have the possibility to rigorously account for the cross-correlation
induced in the local estimation errors by the propagation of the common plant
noise. But, for any known approach able to deal with the measurements’ origin
uncertainty, the cross-covariance of the local estimation errors will become
explicitly dependant on the local processors measurements’ validation
configuration. This problem is the main drawback in any attempt to perform
data fusion on the background of missed detections and false alarms.
       In a reference paper, that difficulty was overcome by using the Chong’s
fusion rule, which is the only one fusion rule which has not to explicitly account
for the local estimates’ cross-correlation, but this was done at the expense of
using the very complex and computationally demanding JPDAF algorithm. As
far as it was widely recognized that the memory requirements for JPDA grow
exponentially with both the number of targets in track and false alarm density,
its ability to correctly solve the problem of correct data association for
interacting targets would make its employment as basic tracking algorithm for
BLUE fusion at least questionable in what concerns the worthiness.
      The reasons which make it unworthy are twofold: on one hand, the
previously addressed JPDAF issues will raise too much the local processing
complexity, with no guarantee that the (interacting) tracks lost will be
completely eliminated and, on the other hand, by using JPDAF the cross-
covariance of the local estimates will be over-expanded, asking for the
transmission of too much data from the local processors to the fusion center.
One possible mid-way solution will be to perform BLUE fusion having the less-
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complex PDAF (the earlier, single-target version of JPDAF) algorithm as the
basic local estimator, but in this case the problem of lost tracks caused by
trajectories’ interaction will have to be addressed separately.
       A (partial) approach for dealing with this problem was given in one of the
reffered papaers, namely the JVC algorithm. It is a global nearest neighbor
optimization technique which is used for avoiding tracks to be lost when they
interact, but by taking into account only the missed detections, not also the false
alarms. Along this attempt, the author strained to investigate the possibility to
adequately adapt the JVC algorithm in order to handle both missed detections
and false alarms, on the expense of a small reduction in the targets allowed
maneuverability during tracks interaction. The resulting PDAF(altered)-JVC
algorithm will then be used as a basic local estimator for performing the BLUE
data fusion.




                   3. ORIGINAL CONTRIBUTIONS

       The main scope of this work is to adapt the JVC algorithm in order to
handle both missed detections and false alarms in conjunction with PDAF,
meanwhile making provisions for allowing to the maximum possible extent the
targets maneuverability. This leads to a variant of the what could be called the
“IMM-PDAF(altered)-JVC” algorithm.
      This algorithm will be used as a basic estimator for the BLUE fuser, in
order to avoid tracks coalescence when they cross each other and meanwhile
allowing for the maximum targets maneuverability. Further, an efficient
approach to compute the cross-covariance of the local estimation errors of the
BLUE fuser is proposed. This approach eliminates the dependence of the cross-
covariance on the local measurements’ validation configurations, by using an
equivalent Kalman filter to perform the transition between the actual predicted
and updated local estimation errors.

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       The equivalence is established via the Information Filter equations, by
considering the information reduction at the local processors due to the
measurements’ origin uncertainty is actually caused by the update made with a
less-precise sensor, when the measurements origin is not questionable. The
effectiveness of the resulting BLUE/IMM-PDAF-JVC algorithm will be
extensively trialed by simulation, in order to define both its capability to avoid
the tracks coalescence and to improve the fused estimate’s precision, allowing
for important conclusions about its worthiness for real life applications.




      4. TRACKING THE AIR TARGETS WHEN THE
        MEASUREMENTS ORIGIN IS UNCERTAIN

        For managing missed detections, clutter, and maneuvering targets the
employment of the IMM-PDAF(altered)-JVC instead of IMM-NN-JVC is
needed, because it allows both tracking maneuvering targets against heavy
clutter and making the JVC technique effective by its very precise estimates.
IMM-PDAF is a combination between the IMM and PDAF algorithms, but it
fails to track closely spaced targets. That’s why the JVC algorithm is used to
yield a unique optimal pairing between the tracks and the measurements, based
on a cost function. The cost function weights are the probabilities that the
normalized innovations squared exceed the actual values:

                                                               
                                        cij  P  2  Dij  1  P  2  Dij                (1)

       The assignment becomes a linear optimization problem, where all feasible
track-to-measurement pairs for the current validation configuration are first
mapped, than added together into the cost function, via weighting coefficients.
Thgere are sought the variables x ij so that:
                            n m               n           m
         x ij  arg min   x ijcij  x ij  1 ;  x ij  1 ; 0  xij  1         i, j     (2)
                             i 1 j1         i           i
                 i 1, n ; j1, m
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        IMM-PDAF replaces the Kalman filters in the IMM module with PDAF
filters. The difference is that the modes probabilities are formed using the PDA
“likelihood” function:

                                           
          j k   p Zk  / M j k , Zk 1 

                                                                                         
                                                      PD m k                                  (3)
                 Vk    mk
                                   
                              1  PD  Vk1 m k            N z t k ; Hx jk / k 1 , Sk
                                                                       ˆ
                                                      m k t 1

       At a first glance, the use of the IMM-JVC assignment procedure in the
PDAF context seems to be trivial. Indeed, the JVC algorithm would assign the
common validated measurements to the tracks such as, all but the measurement
which do not belong according to JVC decision to the track under consideration
will be used for updating the track by the IMM-PDAF algorithm.
       By trials performed using the JVC-driven assignment, a quite high track
lost rate was obtained. The main reason for that was the bias introduced by the
wrong assignment decided in the JVC algorithm when the number of the
common validated measurements was equal to one, more important at low
clutter densities.
       In order to eliminate the possibility the tracks being misled by incorrect
decisions made in JVC when there was only one commonly validated
measurement, in those situations the measurements were discarded for both
tracks in the current update iteration. If there were any other measurements
validated for the two tracks, they were used for the current update. If the
common validated measurement was the only measurement validated for any of
the two tracks, that estimate was predicted for the track, but the measurement
would update the associated covariance, in order to avoid an artificial increase in
the size of the following validation gate.
       At high clutter densities, the combined effect of the clutter drift and tracks
interaction bias was an important generator of tracks lost. That effect was
minimized by shrinking the size of the validation gates, by computing the
“innovation spread” in the covariance update only for the measurement which is
the closest to the prediction:

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                                                                       
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                                        q k  arg min i k i k
                                                            T
                                                                                            (4)
                                                   i 1, m k

                                                        
                                       k  q k 1  q k  q k  Tk
                                                                  q                        (5)

                        Pk / k  I  1  0 I  K k H k Pk / k 1  K k  k K T
                                                                                    k       (6)

      Those above alterations in the basic PDAF were the main reasons for the
degradation in the allowed target maneuverability mentioned in the first section.




                  5. MULTISENSOR DATA FUSION


       The BLUE fusion rule seeks its fused estimate as:
                                                      T
                                               x k  Wk X k
                                               ˆ                                            (7)

                                             X k  [ x (1) ,...,x (1) ]T
                                                     ˆk         ˆk                          (8)

                                                    (         (
                                             Wk  [Wk1) ,...,Wk1) ]T                        (9)

       The weighting coefficients are:
                                                                            1
                                                           m  
                                      W r  
                                                    m
                                         k          C1   Cij1 
                                                       rj                                  (10)
                                                   j1    i, j1
                                                                 
                                                                  

with


                                   k
                                             ˆ
                                             
                                                k          ˆk        T
                                  Cij   E  x i   x k x  j  x k 
                                                                         
                                                                                          (11)


      With PDAF, the current cross-covariance of local estimation errors will
be obtained dependent on the current measurements validation configuration,
which is computationally demanding and communication bandwidth consuming.

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       This issue could be overcome by considering that at each step, the update
was made on an equivalent classical Kalman Filter, in the absence of incertitude
concerning the measurements’ origin. The Kalman Filter should have the
covariance of measurements errors set such as, it would be capable to make the
transition between the actual (PDAF-like obtained) predicted and updated
estimates and covariance matrices. Thus, the recursion of the cross-covariance
between the local estimates is obtained via Information Filter formulae, as:

                         (                            
                                                       1 
                   Cij1  Pki/)k FPki1 / k 1FT  Q  
                    k
                            
                                      ()
                                                          
                                                                                           (12)
                                                                      
                                                                       1    T
                                                   
                           FCij FT  Q Pk j/k FPk j1 / k 1FT  Q 
                              k
                                                                         

   The above formula is no more dependent on the current validation
configuration and can be easily used with complete/incomplete communication
rate and without or with feedback.




            6. SIMULATION SET-UP AND RESULTS

       In order to validate the proposed method, a 2D scenario was simulated in
the MATLAB™ environment. Two identical radars having the measurements
errors variances of 40 meters in range and 0.2 degrees in bearing are used for
updating the tracks of two targets moving with 300 m/s each, whose trajectories
cross at a difference angle of 15 degrees in the middle of the simulation period.
The simulation lasts for 120 seconds. The radar scan period is 2 seconds. The
radar detection probability was varied between 0.8 and 1 and the clutter density
was trialed in the range from 0.001 to 0.3 returns per square km.
     The above parameters reconstruct the basic constraints of two very
demanding scenarios found in literarture. The basic IMM-PDAF algorithm was
used when the validation gates do not overlap, and the IMM-PDAF(altered)-

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JVC when tracks interact each other. Each IMM-PDAF has two “nearly-
constant velocity” models: a benign one with the noise variance of 3 m/s 2 and a
“noisy” one, tuned for allowing the tracking of a target which maneuvers at 7g
at very low clutter density (0.001 returns per sq. km).
      The average maximum g-factor of the considered target plotted via trials
against the clutter density is shown in the next figure:

                                     Target max. g-factor
   10
    8
    6
    4
    2
    0
              0           0.1          0.2              0.3

                                           Clutter density, ret. / sq. km
                                    Pd=1        Pd=0.9        Pd=0.8

                                             Figure 1

       The noise variance for the IMM maneuvering model was optimized by
trials depending on the clutter density. A successful sample run of the
simulation, plotted for PD=0.9 and clutter density equal to 0.1 returns/km2 is
shown in Fig. 2:




                                          Figure 2
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       The output of the simulation runs for assessing the IMM-PDAF-JVC
capability to avoid track coalescence, averaged over 100 runs, are summarized
in the table 1 below:


                                            Table 1
                        IMM-PDAF-JVC Track Lost Percentage (%, 100 Runs)



        PD                                  Clutter density (ret/km2)
                          0.001               0.1                0.2                0.3
         1                  1                  1                  1                  1
       0.95                 2                  2                  2                  2
        0.9                 4                  4                  4                  3
        0.8                13                 11                  9                  -



        First of all, one should note that due to the JVC hard assignment
procedure, tracks are no more lost by their convergence, as in the standard
IMM-PDAF, but by shifting the tracking from one trajectory to another. Another
fact is that, when the tracking scenario is symmetrical (the targets approach each
other with the same speed), the JVC assignment decisions are confused by the
symmetrical disposal of the measurement errors, and the JVC-driven assignment
will fail in 4 percent of the trials when PD=0.9. If the targets speeds are slightly
different, the discrimination by the JVC becomes much more efficient. The
percentage of tracks lost will decrease to below one when targets speed
difference equals 15 m/s.
       Also, the unacceptable decrease in the JVC assignment performance when
PD falls below 0.8 excludes this probability of detection from the instrumented
range of the application. Extensive trials were performed in order to determine
how much the alteration proposed for the basic IMM-PDAF affects the
maximum target maneuverability under the most stressful conditions. The trials
output is shown in Fig. 3, and Fig. 4 is a successful sample run for clutter
density equal to 0.2.
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                                   Target max. g-factor (PD=0.9)
           8
           6
           4
           2
           0
                        0            0.1           0.2             0.3
                                           Clutter density, ret. / sq. km
                                              IMM-PDA               IMM-PDA-JVC




                                                      Figure 3




                                                    Figure 4

    The trials run for assessing of the effectiveness of the proposed variant for
BLUE fusion rule provided the following averaged results over 100 runs:
                                                     Table 2
                   Errors in BLUE fusion rule, no feedback, communication rate 1/3 (meters)



                                            Clutter density (returns/km2)

 PD             0.001                        0.1                      0.2                     0.3

        Local    Fus        Inc    Local     Fus     Inc    Local     Fus    Inc    Local     Fus    Inc

  1      81.2    56.8       67.9    80.6    55.9    64.9    78.2      56.1   64.3   77.2      54.3   63.3

0.95     79.7    56.6       66.2    78.6    55.4    63.9    78.9      56.3   64.7   80.4      56.8   64.2

 0.9     80.6    57.3       66.2    80.7    57.0    65.8    80.5      55.4   64.8   81.3      56.9   64.9

Local - local estimate; Fus: fused estimate, complete communication rate; Inc – fused estimate, incomplete

communication rate.
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                                                                 Table 3
                                          Errors in BLUE fusion rule, feedback (meters)



                                                     Clutter density (returns/km2)

  PD                 0.001                               0.1                       0.2                  0.3

             Local             Fus           Local             Fus         Local         Fus    Local         Fus

   1          61.3            48.9            60.9             48.4        59.5          48.3   59.0          48.6

 0.95         61.6            49.7            61.6             49.6        60.2          48.4   59.8          47.8

 0.9          63.8            50.5            63.2             50.8        62.7          51.5   61.4          51.0

Local - local estimate; Fus: fused estimate, feedback.


       From the above tables, one could observe that the errors are constant over
the whole clutter range and they grow very slowly as a function of the decrease
in detection probability. With complete communication rate, the increase in
precision for the fused estimate over the local one is roughly 30 percent, equal to
that one in which Chong’s fusion rule and JPDAF were used; in the current
approach, the advantage over the previously mentioned one is the linear
dependency of the computational burden on the number of tracks (JPDAF made
that dependency exponential). The precision improvement in the fused estimate
falls from 22 percent to 12 percent, when communication rate decreases from
1/2 to 1/5. When feedback is used, the precision improvement is about 40
percent over the non-feedback local estimates (the 30 percent improvement
reported with Chong-JPDAF was obtained with feedback too).




                                   7. OVERALL CONCLUSIONS

       An altered version of the IMM-PDAF-JVC algorithm was optimized via
computer simulations for avoiding tracks convergence for closely spaced
trajectories. The simulations were carried out in stressful conditions, which
considered both missed detections and moderate-to-heavy clutter.

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      The algorithm was proved as being effective in avoiding tracks
convergence, on the expense of a small reduction in the allowed target
maneuverability during tracks interaction.
       The proposed IMM-PDA-JVC algorithm was then used as the basis for of
a simplified implementation of the BLUE fusion rule. The flexibility of that one
allowed its implementation with complete/incomplete communication rate and
feedback. The overall improvement in the precision of the fused estimates over
the local ones was better than one reported with the Chong-JPDAF due to the
more demanding optimality criterion defining BLUE fusion rule than one used
in the Chong’s one.




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                               AUGUSTIN SPERILĂ
                                Major, Electrical Engineer, Ph. D. Student

OBJECTIVE

               Obtaining the Ph. D. degree in Electrical Engineering

EXPERIENCE

               September 1993 – June 1999: Commissioned as 2nd Lt. (EE) with Romanian Air Force, GBAD
               Unit (promoted as Capt., August 1995)

               Deputy Commander, SAM Battalion

                  Conducting logistics management

                  Planning and supervising scheduled maintenance

                  Participating in SAM live firings

                  Searching for improved maintenance procedures in order to increase equipment availability

               June 1999 – March 2002: Air Force Staff, Technical Development Branch, Logistics Division

               Staff Officer

                  Providing technical expertise in GBAD and Radar fields
                  Drawing operational requirements for equipment related to major procurement programs

                  Acting as consultant for Air Force Technical Standardization Board

                  Keeping track of and promoting Air Force Technical Library

                  Publishing articles in engineering international symposia

               March 2002 – November 2002: Air Force Staff, Standardization Branch

               Specialist Officer

                  Participating in the Drafting Board for 14 National Technical Military Standards

                  Keeping track of Air Force Standardization Library

                  Drawing policy and monitoring NATO technical STANAGs implementation

               November 2002 – February 2005: Air Force Staff, International Military Cooperation Office

               (promoted as Maj., June 2004)

               Staff Officer

                Registering as Ph. D. student and participating in the doctoral training program (undergoing all

                   due examinations and partial dissertations)

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                Preparing working, administrative and protocol arrangements for international activities

                Participating in international operational seminars, working parties and exercises

                Getting english language proficiency certification SLP 3.3.3.3 NATO STANAG 6001

                Publishing articles in technical national journals and participating with papers in engineering

                  international symposia

               February 2005 - present: Air Force Staff, Technical Supply Section, Logistics Division

               Staff Officer

                Keeping track of and facilitating excess equipment disposal

                Providing technical expertise in GBAD and Radar matters

                Preparing the full Ph. D. Thesis and Dissertation

                Publishing articles in technical national journals and participating with papers in engineering

                  international symposia

EDUCATION

                    1988–1993      Military Technical Academy, Bucharest, Romania

                    B.S., Electronics for Air Defence Systems, graduated second in file.

                    June 2004      Radar System Design Course, Military Technical Academy Bucharest,
                                   Romania (Co-sponsored by Technical University Delft and Thales

                                   Netherlands)

INTERESTS

                    History, computers, traveling, bodybuilding.




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          THE AUTHOR’S LIST OF PUBLICATIONS

Papers in international conferences and symposia:

   1.    Gheorghe Gavriloaia, Adrian Stoica, Augustin Sperilă, „An ad-hoc
         method for avoiding tracks coalescence in PDAF for tracks fusion” sent
         to „Telsiks 2005”, Niş, Yugoslavia, September 2005;
   2.    Gheorghe Gavriloaia, Augustin Sperilă, “BLUE data fusion rule
         implemented on an altered version of IMM-JVC-PDAF”, accepted for
         publication in NAV-MAR Conference, Constanta, June 2005;
   3.    Gheorghe Gavriloaia, Augustin Sperilă, “The BLUE sensor fusion rule.
         A benchmark”, Technical Military Academy Symposium “Modern
         technologies in the XXIst century”, Bucharest, November 2003;
   4.    Gheorghe Gavriloaia, Augustin Sperilă, “Matching a m out of n track
         initiation algorithm to system’s parameters”, XXXIVth International
         Scientific Symposium of METRA, Bucharest, May 2003;
   5.    Augustin Sperilă, “Discrimination gain for sensor resource allocation”,
         Communications 2002 IEEE International Conference, Bucharest, May
         2002;
   6.    Augustin Sperilă, “Tracking maneuvering targets. Interacting multiple
         models or jump/pruning models for Kalman filter banks?”, XXXIIIrd
         International Scientific Symposium of METRA, Bucharest, May 2002;
   7.    Augustin Sperilă, Ioan Burduşel, “An approach for enhancing the
         positional estimate of an air target within a radar data fusion system”,
         Communications 2002 IEEE International Conference, Bucureşti,
         decembrie 2002;
   8.    Ioan Burduşel, Constantin Căliman, Augustin Sperilă, “Considerations
         concerning the employment of the strategic wargames”, XXXIst
         International Scientific Symposium of METRA, Bucharest, September
         1999.



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                                 Technical Military Academy
                                              18
         Augustin Sperila – Contributions to the development of the advanced tracking algorithms
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Articles in Romanian Armaments Directorate’s Journal “Tehnica Militara” :

1.     Gheorghe Gavriloaia, Augustin Sperilă, “Aplicarea eficientă a regulii de
       fuziune a celui mai bun estimat liniar nedeplasat pentru datele radar”,
       Revista Academiei Tehnice Militare, nr. 1/2005;
2.     Gheorghe Gavriloaia, Augustin Sperilă, “Evitarea pierderii însoţirii
       ţintelor aeriene la intersectarea traiectelor pe fondul clutterului”, Tehnica
       Militară, Supliment Ştiinţific, nr. 2/2004;
3.     Gheorghe Gavriloaia, Augustin Sperilă, “Însoţirea ţintelor aeriene
       manevriere cu modelul lui Singer pentru filtrul Kalman”, Tehnica
       Militară, Supliment Ştiinţific, nr. 1/2003;
4.     Augustin Sperilă, “Detalii privind alegerea modelelor şi iniţializarea
       filtrelor Kalman, Tehnica Militară, nr. 3/2002;
5.     Augustin Sperilă, “Un model de filtru Kalman extins, pretabil pentru
       aplicaţii de control al traficului aerian”, Tehnica Militară, nr. 1/2002




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                                 Technical Military Academy
                                              19

								
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