Information Fusion White Noise Deconvolution Smoother for Time-Varying Systems

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1 Informaon Fuson Whe ose Deconoluon mooher for Tme-Varyng ysems Xao-Jun un Yuan Gao and Z- Deng Absrac Whe nose deconoluon or npu he nose esmaon problem has mporan applcaon bacground n ol sesmc eploraon. Based on he opmal nformaon fuson rules eghed by marces dagonal marces and scalars n he lnear mnmum arance sense hree dsrbued fused he nose deconoluon smoohers are presened for he lnear dscree me-aryng sochasc conrol sysems h mulsensor and colored measuremen noses. The accuracy of he fuser h he mar eghs s hgher han ha of he fuser h scalar eghs bu he compuaonal burden of he fuser h he mar eghs s larger han ha of he fuser h scalar eghs. The accuracy and compuaonal burden of he fuser h dagonal mar eghs are beeen boh of hem. They are locally opmal and globally subopmal. Ther accuracy s hgher han ha of local he nose esmaors. They can handle he he nose fused flerng and smoohng problems. In order o compue he opmal eghs he ne formula of compung he local esmaon error coarances s gen. A one Carlo smulaon eample for a Bernoull-Gaussan npu he nose fused smooher shos her effeceness. Inde Terms Tme-aryng sysem colored measuremen noses mulsensor nformaon fuson he nose esmaors deconoluon reflecon sesmology I I. ITODUCTIO n ol sesmc eploraon [-4] an eplose s deonaed belo he earh s surface so ha he sesmc aes generaed and are refleced n dfferen geologcal layers. The ol eploraon s performed a he reflecon coeffcen sequence hch can be descrbed by Bernoull-Gaussan he nose hch s he npu sgnal of receers. Esmang npu he nose s called deconoluon hch has been of mporan applcaon alue for fndng and dscoerng he ol feld and deermnng s geomery shape. Whe nose esmaors also occur n many felds ncludng communcaon sgnal processng and sae esmaon. endel [-3] and ormylo [4] presens he opmal npu he nose esmaors h applcaon o ol sesmc eploraon based on he alman fler bu he measuremen he nose esmaors as Ths or s suppored by aonal aural cence Foundaon of Chna under Gran FC X. J. un s h Deparmen of Auomaon elongang Unersy 58 arbnchnae-mal: s@hlu.edu.cn. Y. Gao s h Deparmen of Auomaon elongang Unersy 58 arbnchnae-mal: gaoyuan@hlu.edu.cn. Z.. Deng s h Deparmen of Auomaca elongang Unersy 58 arbn Chna correspondng auhor phone: e-mal: dzl@hlu.edu.cn. no presened. Deng Zhang u and Zhou [5] presens a unfed he nose esmaon heory based on he modern me seres analyss mehod hch no only ncludes npu nose esmaors bu also ncludes measuremen he nose esmaors. Bu s lmaon s ha can only sole he seady-sae he nose esmaors bu canno sole he opmal he nose esmaon problem for me-aryng sysem. The general and unfed he nose esmaon heory based on alman flerng for me-aryng sysems has been presened [67]. ecenly un [8] ges he opmal nformaon fuson he nose fler eghed by scalars based on alman predcor bu doesn' sole he nformaon fuson he nose smoohng problems. un's he nose fused fler s no suable for applcaons. For eample for mulsensor sysem h uncorrelaed noses un's he nose fused fler becomes zero hose accuracy s loer. In order o mproe he fuser accuracy e mus consder he he nose fused smooher. Alhough un [9] presens a he nose fused smooher bu s suable only for me-aryng sysems h he measuremen nose and only ges a he nose fused smooher h mar eghs and doesn presen he seady-sae he nose fused smooher. The mulsensor nformaon fuson has receed grea aenon n recen years due o eense applcaon bacgrounds hch has dely been appled o many feld ncludng gudance defence robocs negraed nagaon arge racng G posonng and sgnal processng. For alman flerng-based fuson o basc fuson mehods are cenralzed and decenralzed or dsrbued or eghed fuson mehods dependng on heher ra daa are used drecly for fuson or no []. The cenralzed fuson mehod can ge he globally opmal sae esmaon by drecly combng he local measuremen daa bu s dsadanages are ha may requre a larger compuaonal burden and hgh daa raes for communcaon. The dsrbued fuson mehod can ge he globally opmal or subopmal sae esmaon by eghng he local sae esmaors. Ths mehod has consderable adanages: can faclae faul deecon and solaon more conenenly and can ncrease he npu daa raes sgnfcanly. The eghed fuson approach s an mporan dsrbued fuson approach. I s que mporan ho o selec he opmal eghng rules and ho o compue he cross-coarances among local esmaon errors. The eghed leas squares 647

2 W fuson rule as presened by Carlson []. The mamum lehood fuson rule as gen by m []. The opmal fuson rules eghed by marces dagonal marces and scalars hae been presened n lnear mnmum arance sense [834]. oce ha he opmal fuson esmae s relaed o he performance nde of opmzaon and s hn a resrced local lnear space. All hese fuson rules ge locally opmal esmaors hch are globally subopmal compared h he cenralzed fuson esmaor. o far he nformaon fuson s manly focused on he sae esmaon problems bu he nformaon fuson concernng he npu he nose s seldom repored hch has an mporan applcaon bacground n ol sesmc eploraon [-4]. In order o oercome he aboe drabac and lmaon he unfed and general opmal nformaon fuson he nose deconoluon esmaors eghed by marces dagonal marces and scalars are presened for me-aryng sysems h mulsensor and colored measuremen noses n hs paper hch ncludes un [8]'s resuls as a specal case. They can handle he he nose fused flerng smoohng and predcon problems. In order o compue he opmal eghs he formula compung he local esmaon error coarances s presened hch s compleely dfferen from un s formula n [9]. As a specal case he dsrbued fuson seady-sae he nose esmaors are also presened. II. OBE FOUATIO Consder he dscree me-aryng lnear sochasc conrol sysem h sensors Φ B u z η η A η ξ 3 here s dscree me m n s he sae z are he measuremens u s he r non conrol npu and ξ are he he m noses η are he colored measuremen noses and Φ and are me-aryng marces h A compable dmensons. Assumpon. and ξ are ndependen he noses h zero mean and arance mar and m p ξ respecely. Assumpon. The nal sae h mean µ and error arance mar s uncorrelaed h and ξ. The opmal nformaon fuson he nose deconoluon smooher problem s o fnd he opmal lnear mnmum arance fuson he nose deconoluon smoohers ˆ > eghed by marces dagonal marce and scalars based on he local he nose deconoluon smoohers ˆ respecely. Inroducng ne measuremens y z A z B u 4 ubsung -3 no 4 e hae y Φ A ξ 5 eng Φ A 6 ξ 7 and combnng and 5 e hae Φ B u 8 y 9 The ne sysem 8 and 9 are he lnear sochasc conrol sysem h correlaed he noses and and h correlaed measuremen noses ha s E E [ ] δ [ ] δ here Ε s he epecaon he superscrp denoes he ranspose and δ s he ronecer dela funcon δ δ. III. DITIBUTED FUIO WITE OIE DECOVOUTIO ETIATO emma [4]. For he mulsensor me-aryng sysems -3 h he assumpons and he h sensor subsysem has he local alman predcor for he sae as ˆ Φ ˆ B u ε ε y ˆ ξ or ˆ p ˆ B u p p p p y Φ 3 p [ Φ ] ε 4 ε 5 here predcon error arance mar sasfy he cca equaon Φ Φ [ Φ ][ ] [ Φ 648

3 ] 6 h nal alue µ ˆ. roof. The proof of emma s gen n [4] hch s omed. Theorem. For he mulsensor me-aryng sysem -3 h he assumpons and he cross-coarance marces among local predcon errors are gen as ] [ p p p p 7 here or p p p p p p 8 specally hen 8 becomes 6 here e defne h he nal alue 9 roof. From [4] e hae he predcon error equaon p p here s uncorrelaed h and. Usng yelds 7. emma [4]. For he mulsensor me-aryng sysem -3 h he assumpons and he h sensor subsysem has he local opmal he nose deconoluon esmaors ˆ < ˆ ε here e defne ε D ε D p > ε 3 here e defne: D p 4 he local error ˆ and s arance ] [ Ε s gen as ε < 5 roof. The proof of emma s gen n [4] hch s omed. Theorem. For he mulsensor me-aryng sysem h he assumpons and he cross-coarance marces among local esmaon error cross-arances ] [ 6 here and e defne 7 p 8 r I δ p δ δ 9 p p 3 pecally for e hae a dsnc formula from 5 o compue > as follos ] [ 3 roof. From [4] e hae p p hch yelds p p ] [ p 3 so e hae p ε p [ p 33 From e hae ε 34 ubsung 33 no 34 e hae 649

4 p p p p 35 Combnng he erms n 35 for and respecely e hae or 36 [ ] 37 ong ha s uncorrelaed h and. Usng and 37 e oban 6-3. Theorem 3. For he mulsensor me-aryng sysem -3 h he assumpons and hree dsrbued opmal nformaon fuson he nose esmaors as ˆ Ω ˆ 38 ˆ < 39 For he fuser h mar eghs e hae [ Ω Ω ] e e e 4 4 n n here e I n I ] and he fused error arance mar [ n m s gen as [ e e] m 4 For he user h scalar eghs Ω ω e hae [ ω ω ] e e e 43 r 44 here r denoes he race of mar e [ ] and he fuson error arance mar s s gen as s ω ω 45 For he fuser h dagonal mar eghs e hae Ω dag ω ω 46 [ ω ω ] e e e r here e [ ] and are he h dagonal elemens of n. The race of he fused error arance mar d s gen as n d [ e e] r 49 Denong he cenralzed fuson error arance mar as c e hae he accuracy he relaon c m r r r s r r 5 roof. Applyng he hree opmal fuson formulas eghed by marces dagonal marces and scalars n [9] e drecly oban Theorem 3. The equaon 5 shos ha he obaned hree eghed fusers are locally opmal bu are globally subopmal and her accuracy s loer han ha of he cenralzed fuser and s hgher han ha of each local esmaor. The accuracy of he fuser h mar egh s hgher han ha of he fuser h scalar egh and he accuracy of he fuser h dagonal mar egh s beeen boh of hem. IV. TEADY-TATE FUIO WITE OIE DECOVOUTIO ETIATO For he me-aran sysem -3 h consan marces Φ Φ B B A A ξ ξ e hae and. If eery local sensor subsysem has seady-sae alman esmaors e can oban he nformaon fuson seady-sae fused he nose deconoluon esmaors hch can reduce he on-lne compuaon burden. Theorem 4. For mulsensor me-naran sysem -3 h assumpons and he local seady-sae alman predcor s gen as ˆ Φˆ Bu ε 5 ε y ˆ 5 d p 65

5 or here ˆ ˆ Bu y 53 p p Φ 54 p p [ ΦΣ ] ε 55 ε Σ ε Σ Σ sasfes he seady-sae cca equaon Φ Σ Φ [ ΦΣ ][ Σ ] ] [ ΦΣ 59 and he local seady-sae opmal he nose deconoluon esmaors are gen as ˆ < 6 ˆ here e defne ε 6 ε 6 D p ε 63 p D 64 p h defnon. The seady-sae smoohng error arance marces Ε[ ] are gen as ε 65 < 66 The seady-sae smoohng error cross-coarance marces are gen as Σ [ ] 67 p δ p p I r p δ δ 69 p p 7 pecally for e hae a formula o compue as follos Σ [ ] 7 hch s dfferen from 65. roof. For he seady-sae alman flerng e hae ha p p p p Σ Σ ε ε as. Tang n emmas - Theorems -3 e drecly oban Theorem 4. Theorem 5. For mulsensor me-naran sysem -3 h assumpons and 3 he hree dsrbued fuson seady-sae fed-lag smooher s gen by ˆ Ω ˆ 7 ˆ < 73 here ˆ are compued a Theorem 4. The eghs Ω are compued a Theorem 3 here Ω ω ω Σ and are replaced by Ω ω ω Σ and respecely. We hae he seady-sae accuracy relaon m d r c r r r r 74 here m d and denoe he seady-sae s error arance marces for fusers h mar eghs dagonal mar and scalar eghs respecely denoes he seady-sae error arance mar of cenralzed fuser. roof. Tang n Theorem 3 e sraghforard oban Theorem 5. c V. IUATIO EXAE Eample. Consder he 3-sensor dscree me-aryng racng sysem h colored measuremen nose Φ 75 z η 76 η A η ξ cos π Φ.5 cos π [.3.cos π ] A..5cos s π A..5cos π A 3.3.5cos π 78 here.cos π b g s Bernoull- Gaussan npu he nose[] b s Bernoull he nose sasfyng b λ b λ λ. 4. g s a Gaussan he nose h zero mean and arance mar g σ.3 and s ndependen of b. and ξ are 65

6 ndependen Gaussan he nose h arances he noses h zero means and arance marces ξ ξ ξ σ. σ. σ. 3 respecely. The problem s o compare he accuracy of local he nose decongoluon smoohers ˆ 3 fused he nose decongoluon smooher ˆ 3 and cenralzed fuser s ˆ 3. The smulaon resuls are shon n Fg.-Fg.8 c and Table. In Fg.-Fg.5 e can see he comparson beeen and ˆ 3 θ 3 c. In Fg.6 and Table e θ can see ha he accuracy of fuson smoohers s hgher han ha of each local smooher and he heorecal accuracy relaon 5 holds. 3 one Carlo runs are carred ou and he means square error E cures s shon n he Fg.7 here he E alue a he me s defned as m E m here 3 ˆ 3 3 c ˆ 3 s he h sample of he sochasc process ˆ 3 m m 3 s he sampled number. From Fg.7 e see ha he accuracy of he fuser s hgher han ha of each local smooher; he accuracy of he cenralzed fuser s hgher han ha of he eghed fuser. The comparson beeen he E and he heorecal races of he esmang error coarcance s shon n Fg. 8 from hch e can see ha he E alues can be consdered as he sampled mean of he heorecal alues. 和 3 /sep Fg. npu he and local opmal he 和 nose smooher ˆ /sep Fg. npu he and local opmal he nose smooher ˆ 3 /sep Fg.3 npu he and local opmal he nose smooher ˆ 3 3 /sep Fg.4 npu he and fused smooher eghed by calars ˆ 3 /sep Fg.5 npu he and cenralzed fused r 3 和 r s 3 和 33 和 3 和 c smooher ˆ 3 c 5 5 /sep Fg.6 Comparson of local and fused heorecal races of smoohng error arance marces sensor sensor sensor3 fuson eghed by scalars cenralzed fuson 65

7 .4.4 E Fg.7 The means square error E cures of local and fused he nose deconoluon smoohers n 3 one Carlo runs he heorecal alue and E alue Fg.8 Comparson of he heorecal race of smoohng error arance mar and he E alue for fuser eghed by scalars sensor sensor sensor3 fuson eghed by scalars cenralzed fuson E alues heorecal alues Table. Comparson of local heorecal races r 3 and fused heorecal races r s r r r r s r c I. COCUIO Based on he opmal nformaon fuson rules eghed by marces dagonal marces and scalars n he lnear mnmum arance sense hree dsrbued fused he nose deconoluon smoohers hae been presened for he lnear dscree me-aryng sochasc conrol sysems h mulsensor and colored measuremen noses. In order o compue he opmal eghs he ne formulas compung he cross-coarance among local smoohng errors hae been presened. They are locally opmal and are globally subopmal. A one Carlo smulaon eamples shon he accuracy relaons among local fused he nose deconoluon smoohers. The one-carlo smulaon resuls sho ha he accuracy dsncon of hree he nose fusers s no obous so ha employng he he nose fuser h scalars eghs s suable for real me applcaon. The proposed resuls oercome he lmaons and drabacs n some references. ACOWEDGET Ths or as suppored by aonal aural cence Foundaon of Chna under Gran FC The auhors sh o han he reeers for her consruce commens. EFEECE [] J.. endel Whe-esmaors for sesmc daa processng n ol eploraon [J]. IEEE Trans on Auomac Conrol ol. 977 pp [] J.. endel nmum arance deconoluon [J]. IEEE Trans on Geoscence and emoe ensng ol pp 6-7. [3] J.. endel Opmal esmc Deconoluon And Esmaon-Based Approach. e Yor: Academc ress 983. [4] J.. endel ormylo J e fas opmal he-nose esmaors for deconoluon [J]. IEEE Transacons on Geoscence Elecroncs ol pp 3-4. [5] Z.. Deng.. Zhang. J. u and. Zhou Opmal and self unng he nose esmaors h approach o deconoluon and flerng problem. Auomaca ol pp [6] Z.. Deng Yan XU Whe nose Esmaon Theory Based on alman Flerng ACTA AUTOATICA IICA ol. 9 3 pp

8 [7] Z.. Deng Unfyng and unersal opmal he nose esmaors for me-aryng sysems Conrol Theory and Applcaon ol. 3 pp [8].. un ul-sensor nformaon fuson he nose fler eghed by scalars based alman predcor Auomaca ol. 4 pp [9].. un. ulsensor opmal nformaon fuson npu he nose deconoluon esmaors. IEEE Trans on ysems an and Cybernecs-ar B: Cybernacs ol pp [] X.. Y.. Zhu J. Wang and C. Z. an Opmal lnear esmaon fuson-par. Ⅰ: Unfed fuson rules. IEEE Transacons on Informaon Theory ol pp 9-8. []. A. Carlson Federaed square roo fler for decenralzed parallel processes. IEEE Transacons on Aerospace Elecronc ysems AEol pp [].. m Deelopmen of rac fuson algorhm. rocessng of Amercan conrol conference aryland 994 pp [3].. un and Z.. Deng ul-sensor opmal nformaon fuson alman fler Auomaca ol. 4 pp [4] Z.. Deng Yuan Gao n ao Yun Gang ao e approach o nformaon fuson seady-sae alman flerng Auomaca ol. 4 pp [5] Z.. Deng Opmal Esmaon Theory h Applcaons- ondelng Flerng and Informaon Fuson Esmaon arbn: arbn Insue of Technology ress pp