The Duration Estimate of the Missing Signal with the Unknown Amplitude

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1 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom Te Durtion Estimte of te Missing Signl it te Unknon Amplitue Oleg V Cernorov * Yur E Korcgin Alener A Mkrov n Alee Gluskov 4 Deprtment of Electronics n noelectronics tionl Reserc Institute Mosco Poer Engineering Institute Krsnokzrmenn st 4 Mosco 5 Russi Interntionl Lortor of Sttistics of Stocstic Processes n Quntittive innce tionl Reserc Tomsk Stte Universit Lenin venue 6 Tomsk 645 Russi Deprtment of Rio Psics Voronez Stte Universit Universitetsk s Voronez 948 Russi 4 Deprtment of Infocommuniction Sstems n Tecnologies Voronez Institute of te Ministr of Internl Affirs of te Russin eertion Ptriots venue 5 Voronez 9465 Russi (*Corresponing utor) Orci ID: Astrct We sntesize te usi-likelioo n mimum likelioo lgoritms for estimting te urtion of te free-form signl it te unknon mplitue Tt te signl m e missing in te receive reliztion of te oservle t is tken into ccount Te crcteristics of te introuce lgoritms re foun Keors: Missing signl Qusi-likelioo estimte Mimum likelioo estimte Signl urtion Signl mplitue Estimte crcteristics Joint etection n estimtion ITRODUCTIO Te estimte of te urtion of te signl oserve ginst noise is importnt for mn prcticl pplictions referring to communiction n loction teor monitoring seismolog etc Tis prolem is consiere in numer of stuies [-4] ere some optiml n usi-optiml lgoritms for te estimtion of te free-form signl urtion re introuce Te signl mplitue is often unknon s ell s te urtion Te estimte of te urtion of te free-form signl form it unknon mplitue is consiere in pper [5] n te mplitue estimte of te signl it unknon urtion in pper [6] Hoever te cse of informtion trnsferre troug n unstle communiction cnnel ie en te signl loss m occur is lso of gret importnce If te presence of te useful signl is not reuire ten e procee to estimting te urtion of te missing signl it unknon mplitue In pper [7] te mimum likelioo n Besin lgoritms re sntesize for te estimte of te urtion of te missing free-form signl it te knon mplitue Te lgoritms for te estimte of te urtion of te missing signl it unknon mplitue re consiere elo THE PROBLEM STATEMET Let te itive mi of te useful signl t st nt () t f t s t () t t n t it te one-sie spectrl n Gussin ite noise ensit is oserve over te intervl T Here re te unknon signl mplitue n urtion corresponentl n f t is priori knon continuous oune function escriing te signl form We presuppose tt te signl urtion possesses te vlues from te prior intervl () ile te signl mplitue is spurious prmeter n oes not nee to e estimte Let te useful signl e present it te proilit less tn p in te reliztion of te oservle t () Ten te iscrete prmeter cn tke te vlue (te signl is present) it te proilit p n te vlue (te signl is not present) it te proilit p p Bse on te oservle reliztion n te ville prior informtion it is necessr to form te estimte of te urtion of te useful signl () tking te prmeters n s te spurious ones THE SYTHESIS O THE ESTIMATIO ALGORITHMS To sntesize te lgoritm for te urtion estimtion e use mimum likelioo (ML) meto [ 8 9] Accoring to te specifie conitions te logritm of te functionl of te 8548

2 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom likelioo rtio (LR) epens on te tree unknon prmeters: L t f t t f t t (4) If te useful signl () is present in te receive reliztion () it te proilit p ten te ML urtion estimte is etermine s te position of te solute (gretest) mimum of te logritm of LR: ere te esigntions re: L rg sup L (5) sup L L L (6) Te lgoritm (5) oes not ccount for te possile signl missing terefore uner p te estimte is not te mimum likelioo one or ll p te estimte (5) is usi-likelioo (QL) one [ 9] In cse te useful signl m e sent te ML pproc is pplie to fin te position of te solute mimum of te logritm of LR (4) ile te current vlues of te spurious prmeters n γ re cnge teir ML estimtes: m rg supsup L (7) rom E (4) e see tt L lgoritm (7) prouces te folloing rule: m L L Terefore te Similrl to [ ] e introuce te certin tresol n te generlize ML urtion estimte: (8) L m (9) L It soul e lso note tt te QL estimte (5) is specil cse of te generlize ML estimte (9) en Accoring to E (9) to fin te ML urtion estimte of te missing signl tere must e foun te mgnitue L n te position of te solute mimum of te logritm of LR L (6) Determining te mgnitue of te solute mimum of te logritm of LR e cn rite on L sup L sup L L L rg sup L Te mimiztion of te logritm of LR te mplitue cn e nlticll implemente or tis purpose e set te L te vrile eul to zero: erivtive of function L t f t t f t t n solve te specifie eution reltive to te vrile : t f t t f t t () B sustituting te mplitue () into te epression (6) e get L t f t t f t t () We no see tt te epression () is nonnegtive for ll τ from te prior intervl () it te proilit Tus P n terefore te estimtion lgoritm (8) is te L egenerte one Ten fter te mimiztion te mplitue it is necessr to ppl te generlize ML estimtion lgoritm (9) Te epressions (5) (9) n () etermine te meter structure Its lock igrm is presente in ig ere te esigntions re: I is te time intervl t integrtor ere t ; E is te retriever sercing position for te input signl mimum itin time intervl (etremtor) B te se line te sceme is selecte te one forming te QL estimte (5) To implement te ML lgoritm (9) te logritm of LR s te function of current time is psse to te pek etector PD its output signl eing te mgnitue of te mimum of te input signl L Te tresol evice (TD) compres te mimum vlue of L it te tresol t te time t n opertes te resolver (RS) ic genertes te ML urtion estimte Eiter tis estimte coincies it te QL one if te tresol is eceee or it is eul to zero if te tresol is not eceee igure : Te lock igrm of te urtion meter 8549

3 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom THE AALYSIS O THE ESTIMATIO ALGORITHMS Let us conuct te nlsis of te sntesize lgoritms for te urtion estimtion We esignte L L s te ecision sttistics () it ( ) or itout ( ) te useful signl in te receive reliztion B sustituting te reliztion () in E () e get L nt f t t f t t () min L f t t n t f t t f t t () Accoring to [] te proilit ensit of te ML urtion estimte (9) cn e presente in te form of p p (4) Here te esigntions re: A A A is te oint proilit ensit of te mgnitue n te position of te solute mimum of te rnom process L ; A L A P sup is te flse lrm proilit; Psup L A is te missing proilit Uner ensit (4) tkes te form of ere p p A (5) (6) te proilit is te proilit ensit of te QL estimte (5) in te presence of te signl () in te receive reliztion () n is te proilit ensit of te QL estimte (5) (pseuo-estimte) in te sence of te signl () o e consier te sttisticl properties of te ecision sttistics () Similrl to [5 ] e introuce te uilir rnom process t f t M t (7) Tis process is te Gussin one it te mtemticl epecttion S M M f t n te covrince function Here K M min min t M M M M min f t t 4 min f t t (8) is te signl-to-noise rtio (SR) t te ML receiver output for te signl it te urtion τ In E (7) e pss to te ne vrile l L L l L L Ten for te rnom process (7) s te function of te vrile l e cn rite on te folloing: Here l M M l l min l l is etermine from te solution of eution l n l is te stnr Wiener process [] B ppling te rnom process (7) te ecision sttistics () s te function of te vrile l cn e presente in te form of ere l l z min l l L l l z z is te SR t te receiver output for te receive signl Let us consier no te cse en te signl is sent in te receive reliztion We epress te proilit ensit A of te mgnitue n te position of te mimum of te rnom process L () troug te proilit ensit l A of te mgnitue n te position of te mimum of te rnom process l l l L (9) 855

4 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom s follos A l A () In turn te proilit ensit l A te form of [] Here l cn e presente in u v l A () u uv A u v P L l u L l l L l ll v is te to-imensionl istriution function of te mgnitue of te mimum of te rnom process (9) In E (9) e crr out noter cnge of vriles: m lnl L m m m lnl L Uner it te rnom process l l possesses te covrince function l l l l m ep m Terefore te rnom process m l l s te function of te vrile m is Gussin Mrkov sttionr rnom process Accoring to [] it stisfies te stocstic ifferentil eution m m m m ritten on in te smmetrize form We multipl te lst m n tking ccount tt eution L m m m m L m e no get te stocstic ifferentil eution for te ecision sttistics L m : L m L m m L m m Tis eution coincies it te similr eution stuie in [] ere te pproimte epression is foun for te toimensionl istriution function of te mimum vlue of te rnom process L m in te form of P P u v P L m u L m v P u P v m m mm u P L m v P L m () ep u ep u u u m u m v ep v ep v v mm v Te proilit ensit m u v m A () u uv A is ssocite it te proilit ensit () te reltion A A L L l m ln (4) B sustituting te function () into te formul () n ten E () into E (4) n E (4) into E () it suseuent integrtion from up to e get ere ln A (5) ep is te flse lrm proilit (5) Let us fin no te proilit ensit (6) A Similrl to E () e epress it troug te proilit ensit l A l of te mgnitue n te position of te mimum of te rnom process L s follos l l z min l l z min l (7) l l l A l A (8) Uner gret SR e cn neglect te lst summn in E (7) n rite on te net epression pproimtel l z min l z min L l l (9) l l In E (9) e crr out te cnge of vriles l zl intervl ere z Te vlue λ possesses te vlues from te Ten e rite on te ecision sttistics (9) s te function of te vrile λ in te form of L l L l L z min min () 855

5 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom Tis function is Gussin rnom process it te mtemticl epecttion n te covrince function S min () min min min K R of min Te correltion coefficient te ecision sttistics () stisfies te conition R R t R t t [8 ] Terefore te rnom process () is te Mrkov one it rift n iffusion coefficients [8 ] k k () If SR is ig enoug ten te position of te mimum of te ecision sttistics () is locte in te neigoroo of te position of te mimum of its mtemticl epecttion [] Te mtemticl epecttion () reces te mimum vlue uner We introuce te vlue z ic solute vlue ecreses it incresing SR n ten rerite te rift n iffusion coefficients () in te form of k k As uner z in te neigoroo of point te ecision sttistics () cn e pproimte Gussin Mrkov rnom process iffusion coefficients k it rift n k () We use tis pproimtion itin ll te intervl of te possile vlues of te prmeter Beteen te vriles λ n τ tere is one-to-one reltion A (8) of te mgnitue n te position of te solute mimum of te rnom process L () cn e epresse troug te Terefore te proilit ensit proilit ensit A of te mgnitue n te position of te solute mimum of te rnom process () mel A A (4) Similrl to [] e cn rite on ere u v u v A (5) u uv A refers to te to-imensionl istriution function of te solute mim of te rnom process u v P u v It is noteort tt uner is escrie te proilit ensit : te rnom vrile ep (6) Te to-imensionl istriution function of te mgnitue of te mimum of te Mrkov rnom process it rift n iffusion coefficients () for te initil conition (6) is foun in []: ep ep epu ep ep ep u v v u v u v u u v u v ep ep u u v ep u v ep ep ep ep v u 8 (7) Sustituting Es (7) into E (5) n ten E (5) into E (4) it suseuent integrting vrile A from up to e fin te proilit ensit 855

6 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom 855 ep ep ep ep ep 8 ep Tis proilit ensit is ssocite it te esire proilit ensit te reltion (8) Accoring to [] te missing proilit (6) cn e otine from E (7) s ep ep ep ep (9) B sustituting te flse lrm proilit (6) te missing proilit (9) n te proilit ensities (5) (8) into te epression (4) e get te proilit ensit of te ML urtion estimte Te ccurc of te estimtes (5) (9) cn e lso escrie te conitionl is n vrince tt for te true urtion vlue re etermine s [] p p V p V p V V As te QL lgoritm (5) is te specil cse of te generlize ML lgoritm (9) uner te crcteristics of its performnce cn e otine from te epressions (5) n (8) tking eul to ere Ten for te proilit ensities of te QL urtion estimte in te signl sence n presence e get ln Here it is esignte: ep 4 ep 4 ep PULSE WITH BEVEL TOP We no specif te otine epressions for te pulse it evel top [] We rite on te function escriing te pulse spe s follos t t f (4) ere te prmeter etermines te pulse top tilt Te multiplier is introuce to provie te inepenence of te energ vlue of te signl of te mimum urtion from te pulse tilt It llos us to compre te ccurc of te urtion estimtes of te signls it te ifferent evel top n ienticl energ We clculte te function (8) it reference to te signl (4): z r (4) ere z r is te SR for te rectngulr pulse

7 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom it te mplitue n te urtion ; is te normlize urtion ile k ere k is te nmic rnge of te unknon urtion vrition In igs tere re presente te epenences of te normlize vrinces V of te QL (5) n ML (9) urtion estimtes of te pulse (4) upon SR z r (4) B te se line te vrince of te QL estimte is son n continuous lines te vrince of te ML estimte Te tresol is clculte eumnn-pirson criterion in terms of te conition p ere α is etermine te epression (6) Curves re plotte for for p curves for p curves p or tis clcultion it is presuppose tt k 9 n te true urtion vlue is foun in te mile of te prior intervl: k k In ig te epenences re constructe uner p 7 n in ig uner p As it cn e seen from igs te QL estimtion lgoritm (5) itout tking into ccount tt te possile signl missing loses in ccurc to te ML lgoritm (9) especill uner te gret SR Te loss in ccurc of te QL estimte in reltion to te ML estimte increses it te proilit of signl sence everteless s it follos from ig in te cse of te lo SR n te proilit p of te signl missing te QL estimte cn provie little gin in ccurc in comprison it te ML estimte () igure : ormlize vrinces of te QL n ML pulse urtion estimtes () COCLUSIO On te sis of te conucte sttisticl nlsis of te lgoritms for te processing of te missing signl it unknon urtion n mplitue tere cn e evlute n influence of prior ignornce out signl presence or sence upon te ccurc of te urtion estimte Te otine results llo us to mke te informe coice of te esire estimtion lgoritm epening on te reuirements for te mesurer implementtion simplicit n te ccurc of estimte B te emple of te reception of te pulse it evel top it is son tt te ccurc of te usi-likelioo urtion estimte cn essentill iel to te ccurc of te pproprite mimum likelioo estimte Besies te mimum likelioo estimtion tkes into ccount te possile signl missing in te receive reliztion ic mens tt in fct it is vrint of te oint etection n estimtion lgoritm ACKOWLEDGEMET Tis stu s finncill supporte te Russin Science ountion (reserc proect o ) REERECES [] Trifonov AP n Sinkov YuS 986 Joint Discrimintion of Signls n Estimtion of teir Prmeters ginst Bckgroun (in Russin) Rio I Svz' Mosco [] Trifonov AP n Korcgin YuE Receiving Signl it Unknon Durtion Izvesti Vssik Ucenk Zveeni Riofizik 45(7) pp [] Trifonov AP n Buteko VK 984 Reception of Signl it Unknon Amplitue n Durtion 8554

8 Interntionl Journl of Applie Engineering Reserc ISS Volume umer 9 (7) pp Reserc Ini Pulictions ttp://ripulictioncom Sumerge in Wite oise Izvesti Vssik Ucenk Zveeni Rioelektronik 7(8) pp 8-4 [4] Cernorov OV Vculik M n Roznov AE A e Clcultion Tecniue of Crcteristics of Multicnnel Mesurer Science Journl of Circuits Sstems n Signl Processing () pp [5] Trifonov AP Korcgin YE n Konrtovic PA Efficienc of Estimting Durtion of Signl it Unknon Amplitue Rioelectronics n Communictions Sstems 54() pp [6] Trifonov AP Korcgin YE Konrtovic PA n Trifonov MV Amplitue Estimtion of Signl it Unknon Durtion Rioelectronics n Communictions Sstems 55(9) pp 85-9 [7] Trifonov AP n Korcgin YuE Durtion Estimte of ing Signl Izvesti Vssik Ucenk Zveeni Rioelektronik 46(7) pp -7 [8] Sosulin YuG 978 Stocstic Signl Detection n Estimtion Teor (in Russin) Sovetskoe Rio Mosco [9] Tikonov VI 98 Optiml Signl Reception (in Russin) Rio I Svz Mosco [] Trifonov AP Korcgin YuE n Konrtovic PA Detection of Signl it Unknon Amplitue n Durtion Riopsics n Quntum Electronics 54(5) pp 54-6 [] Dnkin EB 6 Teor of Mrkov Processes Dover Pulictions Inc e York [] Trifonov AP n Korcgin YuE 5 Optiml Simultneous Detection of Signl n Estimtion of Its Durtion Journl of Communictions Tecnolog n Electronics 5(4) pp 46-4 [] Grznov MI Gurevic ML n Rinin YuA 99 Pulse Mesurement (in Russin) Rio I Svz Mosco 8555

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