15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP
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1 5h Eropan Signa Proing Confrn (EUSIPCO 7 Poznan Poand Spmbr -7 7 opyrigh by EURASIP COMPARISO OF HE COHERE A HE COMPOE MEHOS FOR ESIMAIG HE CHARACERISICS OF HE PERIOICALLY CORRELAE RAOM PROCESSES Ihor Iayv Ihor Javoryj Roman Yzfovyh Phyio-mhania Ini of AS of Urain Lviv Urain hnoogia and Lif Sin Univriy ydgozz Poand ASRAC h ohrn and h omponn mhod for imaing h priodiay orrad random pro probabiii hararii ar onidrd and horm on aympoiay nbiad and onin ima of h man h orraion fnion and hir Forir offiin ar provd h aympoi forma for h bia and h varian of h ima ar invigad and h omparion r for boh mhod ar providd IROUCIO Rrrny and ohaiiy ar hararii far of im hangabiiy of many phyia pro h mhodoogia bai for invigaion of ohai rrrny rr i mahmaia mod in h form of priodiay orrad random pro (PCRP and hir gnraizaion bi- poi- and amo PCRP[5678] h onpa bai for h probabiii mod i h nara gnraizaion of h approah whih on h on hand h prnaion in h form of drminii priodi and amo priodi fnion and on h ohr hand h prnaion in h form of aionary random pro PCRP mod gnraiz h onp of rrrn o h iaion ohaiiy pay an imporan ro h mod nab o ma a mor daid and objiv dripion of ohai rrrny rr hy ind a pia a h abov mniond mod a w a addiiv and mipiaiv mod hir ombinaion and ohr mod hi nab o anayz h ohai pro no ony ing pia mhod for ah a b ao o wor o h gnra mhod for ohai pro anayi m Eξ and orraion fnion h PCRP man ( Eξ ( ξ ( o ξ ( ξ ( m( b (in h a of ra random pro ar priodi fnion of im and if h foowing ondiion ar aifid ( d < b( d < m R hy an b rprnd by h Forir ri: m m b Ζ Ν m iω ( ( m o ω m in ω Ζ iω ( in ω ( ( o ω ( Ν π ω m m im ( ( ( i ( and m ( wih h qanii ( ar ad h orraion omponn [578] (h rm «yi aoorraion fnion» [9] and «offiin fnion» [] for ( ar ao d Priodiiy of hararii on nab o on of h imaion mhod of im hang avraging h raizaion amp va ovr h priod For h priodi fnion imiar avraging giv i va in h im momn ha orrpond o h amp bginning hi gg ha in h PCRP a imiar opraion wi dra h faion omponn ff hi imaion mhod i ofn rfrrd o a ohrn (ao «ynhronizd avraging» [9] L am ha h raiaion ngh qa Ν hn h ohrn ima of h man h man omponn h orraion fnion and i omponn ar a foow: ( ξ ( n [ ] ( n ( d ( o ω d ( ( in ωd [ ξ ( n ( n ] n (4 [ ξ ( n ( n ] ( 7 EURASIP 86
2 5h Eropan Signa Proing Confrn (EUSIPCO 7 Poznan Poand Spmbr -7 7 opyrigh by EURASIP ˆ d ( ( ˆ ( o ω d (5 ˆ ( in ω d h omponn mhod for imaing h PCRP probabiii hararii i bad on h Forir rprnaion ( and ha om advanag ovr h ohrn mhod Fir h of h ingra ranformaion for h givn raizaion ngh giv a mor daid informaion abo propri of h hararii hn w an xp ha in h a of onidrab faion of h PCRP orraion fnion ovr priod h omponn ima hav varian Componn mhod an b d whn w hav apriori informaion abo h nmbr of Forir omponn for h man and h orraion fnion W an ima h paramr by anayzing h pro propri hir phyia ap and h ohrn anayi r I hod b mniond hr ha h varian of omponn ima wi inra wih inraing h nmbr of omponn h omponn ima of PCRP probabiii hararii hav h form: m ˆ ( m ˆ ˆ ( i i (6 (7 i ξ ( d (8 i ˆ ( [ ξ ( ( ][ ξ ( ( ] d (9 and and ar h nmbr of high omponn of h man and h orraion fnion Corraion ohrn and omponn anayi ind imaion of h man m ( orraion fnion b for [ ] and h imaion of h orrponding Forir offiin m and ( W provid h ompx of ohrn anayi in ion h ompx of omponn anayi in ion In ion 4 w ond wih an xamp in whih w inviga h propri of h igna orrponding o a pariar impmnaion of h PCRP pro PROPERIES OF HE COHERE ESIMAES OF HE PCRP PROAILISIC CHARACERISICS h man ima h ima ( i nbiad ba For h varian w hav Em( ˆ m( n m( n [ (] [ ( (] n E E b n n Sin b b( and b n b n hn n [ ( ] b b n ( n aing ino aon h mniond abov w forma h foowing horm: horm Saii ( i h nbiad and onin ima of h PCRP man if h foowing ondiion i aifid im n b n I varian i dfind by xprion ( ( n Proof Condiion (4 i aifid if h m b n inraing wih ampifiaion bing no mor rapid han α α < In hi a ima ( i onin ba i foow from ( ha [ ( ] a I i obvio ha aympoi qaiy ( i aifid if orraion dra wih ag inraing im b b h a ondiion i no nary h ima of orraion fnion h ohrn ima of orraion fnion ha h foowing form: [ ] [ ξ ( n ( n ] n ξ ( n ( n i ( L am ha h raizaion gmn ha h ngh m m i h arg ag for whih h orraion fnion i bing imad If h man i imad ony for [ ] and w ppo for a ohr ( n ( n i an ingr hn h ima ( an b rprnd in h foowing form: n ( n ξ ( n ( ( n ξ ( 7 EURASIP 86
3 5h Eropan Signa Proing Confrn (EUSIPCO 7 Poznan Poand Spmbr -7 7 opyrigh by EURASIP horm h ima of h orraion fnion ( of Gaian PCRP i aympoiay nbiad and onin if ondiion im d (4 ( ar aifid and i bia and varian ar dfind by h orrponding forma: [ ] ε ( [ ε ( oω ε ( in ω ] ε (5 Ν [ ] ( [ α ( oω α ( in ω ] Ν α (6 n ε ( n n n ε ( n (7 n n α ( ( ( n o (8 n n ( α ( ( n o (9 n ar orraion and h fnion ( ( omponn of h pro ( ξ ( ξ ( : E ( ( ξ ( ξ ( b b o Proof Uing raion ( and ( for h bia of h orraion fnion ima w hav ε E b [ ] n b n n x w prn h orraion fnion in h form of Forir ri hn w obain forma (5 and (7 h ε ε ( a if ondiion (4 i aifid and ε [ ] h h ima ( i aympoiay nbiad I i nary o mphaiz ha h bia in (5 for and h varian of h man ima diffr ony in ign For h varian of ima ( w hav in h fir approximaion: [ ˆ ˆ n E b Eb ] n [ b n b( n b n b n ] h xprion wihin h m ar h va of h orraion fnion b aad for n hi fnion i priodi in im and an b rprnd by h Forir ri: b ( ( oω ( inω [ ] Ν Sin b b ( b n b n h varian [ b ] offiin α ( ( (8-(9 h orraion omponn ( ( and ˆ and α an b xprd a (6 and and ar drmind by h prod of orraion omponn of h pro ( (rfr [] for dai hn if h ondiion in (4 ar aifid w obain ε [ b ] and [ ] a If orraion rapidy vanih on h inrva [ ] w hav for h Forir offiin for h bia and varian of h orraion fnion ima for : ε ( ( ε ( α ( α PROPERIES OF HE COMPOE ESIMAES OF HE PCRP PROAILISIC CHARACERISICS h man ima For h man omponn ima (6 w hav h foowing horm: horm Saii (6 i h nbiad and onin ima of h PCRP man if h ondiion ( i aifid h varian of (6 i dfind by foowing xprion: λ τ / r f r r [ (] m i ˆ λ ( ( τ γ ( τ ( τ f ( τ γ ( τ dτ o r ( τ τ i( r d ( ( ( i n τ in τ γ ( τ n h { } a and { } a > i h nmbr of h man omponn Proof of h horm i imiar o ha givn for h horm o hr ha h zrmponn λ in ( i h avrag va of h varian on h priod : 7 EURASIP 864
4 5h Eropan Signa Proing Confrn (EUSIPCO 7 Poznan Poand Spmbr -7 7 opyrigh by EURASIP λ [ ˆ (] [ ˆ (] d m ( τ γ ( τ m Τ τ dτ ( τ f ( τ γ ( τ dτ o r r r I how ha h avrag varian of h ima (6 i mainy dfind by h zrrraion omponn Highr orraion omponn hav impa ino h avrag varian h wigh mipir how h dpndn on h nmbr of orraion omponn h nmbr of omponn in h ranform ( i of h dob ordr of h nmbr of h man omponn So h va of h highr omponn in ( mainy dpnd on h orrponding orraion omponn A w inra h nmbr of h omponn h varian of h man omponn ima ( grow and ha h varian of h man ohrn ima ( a a imi va h ima of orraion fnion For h omponn ima (7 of h orraion fnion whn h orraion omponn ar of h form (9 w an forma h horm: horm 4 h ima of h orraion fnion ( of Gaian PCRP i aympoiay nbiad and onin if ondiion (4 ar aifid and i bia and varian ar dfind by h orrponding forma: ε i [ ˆ b ] ( h ( γ ( d o [ ] β i i ( (4 P β ( ( ( f γ d o (5 im ( im i m h ( m [ ]I and h fnion ( omponn of h pro ( ξ ( ξ ( ar h orraion Proof of h horm 4 i imiar o ha givn for h horm I hod b mniond hr ha h imiar horm for h ohrn ( (5 and h omponn ima (8 (9 of h man and h orraion fnion omponn an b formad 4 SIMULAIO RESULS An imporan par of ndranding h PCRP propri i hir dompoiion ino aionary onnd random pro [4] i ξ ξ ω (6 ( Ζ h ( pro h hararii of h igna ( ξ dno aionary onnd random ξ an hrfor b drivd from h orrponding hararii of h onin ξ ( pro L pify h imaion r on h ohrn and h omponn mhod on h bai of h pro of ohai yi oad whih an b rprnd by PCRP qadrar mod: ξ ( ξ ( oω ( ξ inω hr xi ony h zro h and h ond orraion fnion omponn for hi mod h omponn an b approximad a foow: α o ( ω β ( (7 α 4 o( ω β o( ( ω β ( ( ω β in ( ( ω β ( ( ω β ω β α α ( ( 4 o in ω β α (8 in ω β α α ( ( 4 in o ω β α (9 o h paramr of h PCRP qadrar mod wr dfind on h bai of h ra pro of ohai yi oad hy qa 5 amp α 6 α 6 α 7 β Afr biing h xprion (7-(9 in h xprion ( w hav h xprion ha dfin h varian of h ohrn man ima for any im [ ; ] givn ag m L L and h orraion paramr of h modad pro hi raionhip an b n on fig Crv for h varian of h omponn man ima (-( on im and ag m L i hown on fig Crv for h bia of h ohrn ima of h orraion fnion (7 on im and ag m L i hown on fig Caaion r for h bia of h omponn ima of h orraion fnion ( bj o im and ag m L an b n on fig4 For boh h ohrn and h omponn mhod a rv hav a damping oiaing form h day ra for h priion faor i han ha for h orraion ε fnion hn h raiv imaion rror b ( dra a h ag dra On h bai of h propod xprion h imi ag va max for h givn imaion priion an b fond 7 EURASIP 865
![Cyoaionary in Commniaion and Signa Proing Ed y WA Gardnr IEEE Pr w Yor 994 [] E Gadyhv Priodiay and amo priodiay orrad random pro wih onino im (in Rian Probabiii hory and i Appiaion](/docs-images/79/79558266/images/5-1.jpg "Vo pp 84-89 # 96 [] HL Hrd onparamri im ri anayi for priodiay orrad random pro IEEE ran on Informaion hory vo 5 pp 5-59 989 [4] H Ogra Spra rprnaion of priodi nonaionary random pro")
![IEEE ran on Inf hory I-7 pp4-49 # 97 [5] L Gdzno Priodiay nonaionary random pro (in Rian Radiohnia i Eronia Vo 6 pp 6-64 #6 959 Figr 4 pndn of h bia of h omponn ima of h orraion fnion](/docs-images/79/79558266/images/5-2.jpg "on im and ag m L L5 [6] O Koronivyh h inar dynami ym ndr aion of h random for (in Urainian aovi Zapyy Lvivoho Univry Vo 44 pp 75-8 #8 957 [7] Ya ragan V Rozhov and I Javor yj Mhod of")
![h Probabiii Anayi of Oanoogia Rhyhmi (in Rian Gidromoizda Lningrad 987 [8] Ya ragan and I Javor yj Rhyhmi of Sa Waving and Undrwar Aoi Signa (in Rian aova ma Kijv 98 [9] WA Gardnr h](/docs-images/79/79558266/images/5-3.jpg "pra orraion hory of yoaionary im ri Signa Proing vo pp -6 # 986 [] I Javor yj I Iayv Z Zarzwi and SP roo Cohrn ovarian anayi of priodiay orrad random pro Signa Proing vo 87 pp- # 7 7")
5 5h Eropan Signa Proing Confrn (EUSIPCO 7 Poznan Poand Spmbr -7 7 opyrigh by EURASIP Figr pndn of h varian of h ohrn man ima on im and ag m L L Figr pndn of h bia of h ohrn ima of h orraion fnion on im and ag m L L5 Figr pndn of h varian of h omponn man ima on im and ag m L L o ao ha h ampid of h priion faor for h omponn ima diffr nfod from h ohrn REFERECES [] Cyoaionary in Commniaion and Signa Proing Ed y WA Gardnr IEEE Pr w Yor 994 [] E Gadyhv Priodiay and amo priodiay orrad random pro wih onino im (in Rian Probabiii hory and i Appiaion Vo pp # 96 [] HL Hrd onparamri im ri anayi for priodiay orrad random pro IEEE ran on Informaion hory vo 5 pp [4] H Ogra Spra rprnaion of priodi nonaionary random pro IEEE ran on Inf hory I-7 pp4-49 # 97 [5] L Gdzno Priodiay nonaionary random pro (in Rian Radiohnia i Eronia Vo 6 pp 6-64 #6 959 Figr 4 pndn of h bia of h omponn ima of h orraion fnion on im and ag m L L5 [6] O Koronivyh h inar dynami ym ndr aion of h random for (in Urainian aovi Zapyy Lvivoho Univry Vo 44 pp 75-8 #8 957 [7] Ya ragan V Rozhov and I Javor yj Mhod of h Probabiii Anayi of Oanoogia Rhyhmi (in Rian Gidromoizda Lningrad 987 [8] Ya ragan and I Javor yj Rhyhmi of Sa Waving and Undrwar Aoi Signa (in Rian aova ma Kijv 98 [9] WA Gardnr h pra orraion hory of yoaionary im ri Signa Proing vo pp -6 # 986 [] I Javor yj I Iayv Z Zarzwi and SP roo Cohrn ovarian anayi of priodiay orrad random pro Signa Proing vo 87 pp- # 7 7 EURASIP 866
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