Discrete Time Integrator Comparison for an Angle of Arrival Neural Network Detector on a Transmitter Independent Receiver Network

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1 Procdings of th th WSEAS Intrnational Confrnc on COMMUNICAIONS, Vouliagmni, Athns, Grc, Jul -2, 26 (pp ) Discrt im Intgrator Comparison for an Angl of Arrival Nural Ntwork Dtctor on a ransmittr Indpndnt Rcivr Ntwork NIKOS J. FARSARIS (), PROF. HOMAS D. XENOS (2), PROF. PEER P. SAVROULAKIS (3) () Elctromagntic Radiation Masurmnts Laborator,.E.I. of Crt, GREECE. (2) lcommunications Dpartmnt, Elctrical and Computr Enginring School Aristotl Univrsit of hssaloniki, GREECE (3) Elctronics and Computr Enginring Dpartmnt, chnical Univrsit of Crt GREECE Abstract A Rcurrnt First Ordr Artificial Nural Ntwork ma b usd for targt dtction b Angl of Arrival on a ransmittr Indpndnt Rcivr Ntwork (multimod passiv multistatic radar). In this papr a modl of it is cratd and thn simulation taks plac in ordr to invstigat th prformanc of various intgration and accumulation mthods. It is shown that construction simplicit that is ssntial to militar sstms is rtaind in this mthod in an cas and th optimum intgration mthod can b chosn. Kwords Multistatic Radar, Nural Ntworks, Angl of Arrival, Elctronic Warfar, Digital Signal Procssing.. Introduction. Angl of Arrival mthod is a wll known mthod of transmittr dtction that consists of finding its angular (sphrical) coordinats from at last two rcivrs and thn solving a linar sstm to xtract its coordinats (Gaussian or sphrical) in a standard rfrnc coordinat sstm. his concpt ma b usd also for targt dtction in a ransmittr Indpndnt Rcivr Ntwork (IRN); this is multistatic radar that ma dtct targts illuminatd b transmittrs of opportunit. In ordr to achiv robustnss and rdundanc ndd for a militar sstm plus multimod opration a four rcivr modl is considrd. [] h Artificial Nural Ntwork (ANN) considrd hr is on adaptd from a gnral purpos dsign [2] usd for linar problm solving. It diffrs though consists of two linar variabl gain lars, on linar fixd gain lar an intgrator lar and a loop. It is supportd b argt Matrix Gnrator prprocssing units. (Fig.) 2. h AOA IRN modl dscription. h linar sstm that is cratd is as it has bn said in [] an ovr dtrmind linar sstm. Its gomtrical rprsntation is prsntd in Fig. 2. his dtction mthod can b dscribd as follows: Lt fp 4Hz and ri( xi, i, zi) b th locations of on of th rcivrs ( i {,2,3,4 }) and th targt rspctivl. hn th quations conncting th Cartsian coordinats with th sphrical coordinats masurd on ach rcivr b its monopuls antnna ar givn blow: x xi Ri cosφi sinϑi i Ri sinφi sinϑi () z zi Ri cosϑi With th limination of R i (targt rcivr distanc) it bcoms: x cosϑi cosφi sinϑi cosϑi sinφi sinϑ i z (2) xi cosϑi cosφi sinϑi i cosϑi sinφi sinϑ i zi As it is clar, vn two rcivrs would giv an ovr dtrmind 4X3 linar quation sstm. Rdundanc rasons and rasons of multimod oprations [], [3] is th main rason that at last a four rcivr modl is usd, comprisd b two pairs of rcivrs. his would giv an 8X3 ovr dtrmind quation sstm as shown blow. Ax b (3) With: x x r (4a) z

2 Procdings of th th WSEAS Intrnational Confrnc on COMMUNICAIONS, Vouliagmni, Athns, Grc, Jul -2, 26 (pp ) cosϑ cosφsinϑ cosϑ sinφsinϑ cosϑ2 cosφ2sinϑ 2 cosϑ sinφ sinϑ A (4b) cosϑ3 cosφ3sinϑ3 cosϑ3 sinφ3sinϑ3 cosϑ4 cosφ4sinϑ 4 cosϑ4 sinφ4sinϑ4 xcosϑ zcosφsin ϑ cosϑ zsinφsinϑ x2cosϑ2 z2cosφ2sin ϑ 2 2cosϑ2 z2sinφ2sinϑ2 b x3cosϑ3 z3cosφ3sin ϑ3 (4c) 3cosϑ3 z3sinφ3sinϑ3 x4cosϑ4 z4cosφ4sin ϑ 4 4cosϑ4 z4sinφ4sinϑ4 An on lin mthod of linar quation solving has bn dscribd thoroughl in [2] with a mthod that can b asil adaptd for th AOA mthod. In ordr to solv (3), an nrg function for an stimatd x must b dfind which in this cas is: E ( x).5( Ax b) ( Ax b) (5) h diffrntial quation sstm blow (whr t is in tim units) dscribs gradint dscnt approximation for th minimization of th nrg function E (x) for an initial valu x () (initial valu problm). µ E( x), E( x) A ( Ax b ) x ( ) () x (6) h abov sstm in its analtical form is quivalnt to: n m n j µ jp aip aik xk bi p i k, (7) () xj() xj j, 2,..., n Choic of µ jp must nsur th stabilit of th diffrntial quation and an appropriat convrgnc spd to th dsird solution. It has bn provn that th sstm (6) or (7) is stabl and has a solution that convrgs to a vctor x as t tnds to th infinit as it is: n de E( x) j ( E(x)) M E(x) (8) x j j h abov is alwas tru if M with lmnts µ jp is a (prdfind) positiv dfinit matrix. Furthr analsis of (7) givs: n i ( x ) aik xk bi, i,2,..., m (9a) k m E( x) a ip i ( x), xp i (9b) () p,2,..., n, xj() xj n j E( x) µ ip, j,2,..., n (9c) x p p h rcurrnt ANN, shown in fig., (Appndix) consists of intgrators (as man as th dimnsion of th problm) and wightd input addrs. h wights α and µ ar th lmnts of th matrics A and M, Scond arra lmnts th ar constant. hat maks this ntwork as to construct. In this particular cas, in quations (6) to (9), m 8 and n 3. h ar small valus that mak th construction vn asir. hr ar thr lars in that ANN connctd in fd forward mod. First lar namd "snsor lar" bcaus it snss th actual variabls x i and computs rrors i (x) as th dfind in (9a). Error signals i (x) ar inputs to th scond association lar, which givs th gradint componnts of th sstm. Hr th wights ar approximatl qual to wights of th first lar as (6) dnots. h third, "rspons lar" is consistd of rspons lmnts, which dfin th convrgnc rat. In this cas µ (third lar lmnts) ar considrd constant for simplicit rasons. In ordr to simplif th ntwork, µ ma b considrd lmnts of a positivl dfind diagonal matrix, thus making th limination of th addrs in th third lar possibl. his is th cas simulatd hr in ordr to prov that vn a simplifid dsign can giv th xpctd rsults. Equation 9b thn suggsts that a fdback loop must b constructd with intgrators. Whn discrt intgration taks plac (this is whn digital circuits ar usd for signal procssing), thn th intgrators dfind b thir transfr (n z transform) functions. h ar: For a trapzoidal intgrator: K S ( z+ ) H( z) () 2 ( z ) For a Backward Eulr intgrator: ( ) K S z H2 z () z

3 Procdings of th th WSEAS Intrnational Confrnc on COMMUNICAIONS, Vouliagmni, Athns, Grc, Jul -2, 26 (pp ) And finall a Forward Eulr intgrator: ( ) K H S 3 z (2) z 3. Simulation rsults st sstm was simulatd using Matlab s SIMULINK. his consists of four rcivrs at: x x z.44 z (3) x x z z his st of rcivrs is placd on a rough land surfac as rcivr altituds dnot. Coordinat axs xxand ' ' dnot position from Wst (ngativ) to East and from South to North rspctivl whil axis zz ' dnots altitud (hight) placmnt. PRF dfins th sampl tim of th simulation and th trials hav bn mad at fp 4 Hz. hus th sampl tim is S 2.5µ sc. h targt is a tri-sonic aircraft manouvring on hard turns. All th dfind first ordr intgrators ar trid. Simulation taks plac for sconds. For th trapzoidal intgrator a compromis must b don about th constant K (intgrator gain) Vr larg gains giv most inaccurat rsults, but on th othr hand vr small gains giv a vr low convrgnc rat thus making thus making initial acquisition of targt mor difficult. 4 For xampl whil K givs accurat rsults (Fig.3 Fig.4 Fig.5) and convrgs in 6 sampls whn K thn rrors of th ordrs of 2m Forward Eulr intgration mthod must b avoidd. Although intgrators of this tp wr thoroughl tstd with diffrnt intgrator gains th all faild to acquir th targt. Backward Eulr Intgration has givn th most accurat rsults. (Fig.7, Fig.8). h ordr of rrors is in on millimtr, whil this mthod convrgs in th first 3 sampls. 4. Conclusions Although Artificial Nural Ntworks giv accurat rsults, an nginr must b carful in vr dsign In ordr to achiv narl optimal rsults. In th cas prsntd hr onl changing a small part of th dsign ma giv totall diffrnt rsults. For th spcifid problm Backward Eulr intgration mthod givs prfct rsults (rrors ar onl of acadmic importanc) whil rapzoidal Intgration ma giv accptabl rsults if intgration constant is compatibl with th PRF of th transmitting radar or th sampling priod if it is a CW transmittr. Rfrncs: [] Nikos J. Farsaris, homas D. Xnos and Ptr P. Stavroulakis: racking of Manuvring and Ballistic argts via Masurmnts akn b a ransmittr Indpndnt Rcivr Ntwork Using First Ordr Rcurrnt Nural Ntworks. WSEAS ransactions of Communications Vol. 4, Issu 8. August 25 [2] A. Cichocki, R. Unbhaun: Nural Ntworks for Optimization and Signal Procssing. John Wil & Sons Ltd 993 [3] Nikos J. Farsaris, homas D. Xnos, Ptr P. Stavroulakis: racking of Manuvring and Ballistic argts with a ransmittr Indpndnt Rcivr Ntwork using im Diffrnc Of Arrival and a Scond Ordr Rcurrnt Artificial Nural Ntwork. WSEAS ransactions of Communications Vol. 4, Issu 7. Jul 25

4 Procdings of th th WSEAS Intrnational Confrnc on COMMUNICAIONS, Vouliagmni, Athns, Grc, Jul -2, 26 (pp ) Appndix. Figurs and Diagrams. Fig..Simulation modl: Fig.2. Gomtr of th AOA problm

5 Procdings of th th WSEAS Intrnational Confrnc on COMMUNICAIONS, Vouliagmni, Athns, Grc, Jul -2, 26 (pp ) X,Y,Z () - Rang rror (m) Fig.3. Computd Cartsian coordinats in sampls Fig.6. Convrgnc. (rapzoidal Intgrator, K -4 ) X,Y,Z (mm) - X,Y, Z (mm) Fig.4 Errors in coordinats in mm (rapzoidal Intgrator, K -4 ) Fig.7 Errors in coordinats in mm. (Backward Eulr Intgrator, K) rang rror (mm) Rang rror (mm) Fig.5. Rang rror in mm. (rapzoidal Intgrator, K -4 ) Fig.8. Rang rror in mm. (Backward Eulr Intgrator, K)