University of Maribor. Faculty of Electrical Engineering and Computer Science. Aleš Holobar

Transcription

1 Uvety of Mabo Faculty of Electcal Eee ad Copute Scece Aleš Holoba BLIND DECOMPOSIION OF CONVOLUIVE MIXURES OF CLOSE-O-ORHOGONAL PULSE SOURCES APPLIED O SURFACE ELECROMYOGRAMS SLEPA DEKOMPOZICIJA KONVOLUIVNIH MEŠANIC SKORAJ OROGONALNIH IMPULZNIH IZVOROV UPORABLJENA PRI POVRŠINSKIH ELEKROMIOGRAMIH PD e Mabo Octobe 4

2 Auto: tle: Aleš Holoba Bld decopoto of covolutve xtue of cloe-to-otooal pule ouce appled to uface electoyoa UDC: 6.39:643.3 Keywod: Copyt: dtal al poce bld ouce epaato ult-cael bld decovoluto pule ouce te-fequecy aaly e-ode tattc w-lce Newto-Gau optzato vee coelato atx baed etod MIMO yte detfcato uface electoyoapc al uface EMG decopoto oto ut evato pule ta oto ut acto potetal Syte oftwae laboatoy Faculty of Electcal Eee ad Copute Scece Uvety of Mabo Slovea

3 Uvety of Mabo Faculty of Electcal Eee ad Copute Scece Aleš Holoba Bld decopoto of covolutve xtue of cloe-to-otooal pule ouce appled to uface electoyoa Cottee ebe: Pof. d. Io ča caa Pof. d. Daja Zazula upevo Pof. d. Robeto Melett upevo Pof. d. adej Bajd ebe Pof. d. Goazd Lešjak ebe Mabo 6 t of Octobe 4

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5 BLIND DECOMPOSIION OF CONVOLUIVE MIXURES OF CLOSE-O-ORHOGONAL PULSE SOURCES APPLIED O SURFACE ELECROMYOGRAMS UDC: 6.39:643.3 Keywod: dtal al poce bld ouce epaato ult-cael bld decovoluto pule ouce te-fequecy aaly e-ode tattc w-lce Newto-Gau optzato vee coelato atx baed etod MIMO yte detfcato uface electoyoapc al uface EMG decopoto oto ut evato pule ta oto ut acto potetal Abtact: I t doctoal detato bld decopoto of covolutve xtue of cloe-tootooal pule ouce al addeed. ee ovel decopoto appoace baed o te te-fequecy aaly e-ode tattc ad alebac appoac ae developed. Futeoe eceay codto fo te ecotucto of coplete pule equece ad te pule epoe ae vetated ad a toou ad detaled tudy of te facto fluec t pefoace uc a oe ad o-otooalty of ouce ae caed out. Altou deved te cae of oe eaueet ta ouce ovedeteed yte te alebac appoac exteded to ltly udedeteed yte wt oe ouce ta eaueet. I cotat wt ote decopoto tecque te popoed appoace wok well alo te cae of ot copletely otooal ouce al. Fally all te popoed oluto ae appled to te uface electoyoapc EMG al. e te be wt a ovevew of ext etod ad tecque fo bld ouce epaato. e etod fo bot ultplcatve ad covolutve cae ae ctcally aeed ad utually copaed. Next a bef toducto to te pyoloy of te ua ucle ve. I ode to povde te ba fo evaluato of te decopoto eult o te eal uface EMG al te popete of te oto ut MU evato pule ta ad eeato of te oto ut acto potetal MUAP at te ed-plate ae explaed. e facto fluec te ape ad apltude of MUAP detected at te k uface ae alo detfed. Aftewad te decopoto etod fo bot ta-ucula ad uface EMG al ae ctcally evaluated ad te fluece of te upepoed MUAP tuded. Fally te aued data odel of uface EMG al toduced ad t a ltato ad aupto clafed. Suface EMG al ae odelled a a ult-cael lea ft-vaat ultpleput-ultple-output MIMO yte.

6 I te ecod pat of t detato tee dffeet appoace to bld ouce epaato of te covolutve xtue of eeal pule ouce al ae deved. Ftly te ove-deteed cae aued te cae wt oe eaueet ta ouce ad two ovel appoace toduced. e ft oe utlze te We-Vlle te-fequecy dtbuto ad eable te ecotucto of bot te pule ouce al ad te coepod MIMO yte epoe. e ecod appoac eable autoatc ecotucto of te MIMO yte epoe ad baed o e-ode cuulat. Next te decopoto exteded to te udedeteed cae te cae wt oe ouce ta eaueet ad a copletely ovel appoac to bld decovoluto of pule ouce al o called vee coelato atx baed etod deved. e toduced appoace ae teted o bot ytetc ad eal uface EMG al. Ftly te pact of te ube of actve MU te f fequece dept te ucle tue etc. a well a te fluece of oe ae evaluated o te ytetc al. e eult pove te upeoty of te vee coelato baed etod. Apply t to te ove-deteed cae at a al to oe ato SNR alot all ulated MU evato pule ta ae copletely ecotucted. I te ude-deteed cae wt ube of ouce exceed te ube of eaueet by facto.4 appoxately a alf of te ulated MU ae copletely detfed. e pefoace alo dop wt te SNR. At SNR of db appoxately 3 % of te MU detfed at SNR db ae ecotucted. I all cae te decopoed evato pule ta exbt a pefect atc wt te efeece ytetc ouce al. e ote two etod ae fcatly le effcet. Fally all tee decopoto appoace ae appled to te eal uface EMG al ecoded du a oetc 5 % ad % cotacto of te doat bcep bac ucle of 9 ealty you ale ubject. Aa te vee coelato baed etod pove to be upeo. Altoete 3 ad 56 MU evato pule ta ae copletely ecotucted fo te 5 % ad % ucle cotacto eaueet epectvely. e ecotucted MU f patte ae copaed aat vaou pyolocally duced ltato ad pove to be aeeet wt expectato ad caeful vual aaly.

7 SLEPA DEKOMPOZICIJA KONVOLUIVNIH MEŠANIC SKORAJ OROGONALNIH IMPULZNIH IZVOROV UPORABLJENA PRI POVRŠINSKIH ELEKROMIOGRAMIH UDK: 6.39:643.3 Ključe beede: dtala obdelava alov lepa ločtev zvoov večkaala lepa dekovolucja pulz zvo čaovo-fekveča aalza tattke všj edov w-eze Newto- Gauova optzacja etoda z vezo koelacjke atke detfkacja teov MIMO povšk elektooa dekopozcja povšk elektooaov otoča eota evacjk vlak pulzov akcjk potecal otoče eote Povzetek: V doktok detacj obavavao potopke za lepo ločevaje kovolutv ešac pulz zvo alov vpeljeo t ove etode k ooočajo popolo ekotukcjo zvo alov oz. odzvov poaez ešal kaalov fltov v teu MIMO ultple-put-ultple-output. P zpeljav ajpej pvzaeo da je etev več kot zvoov peučo pa tud ožot dekopozcje ko je etev aj kot zvoov. Ovedoto še vplv šua eotooalot zvo alov azeja ed števlo etev števlo zvoov. Razkavo pčeo z opedeltvjo cljev lepe ločtve alov peledo obtoječ etod. Oedotočo e pedve a ločtev kovolutv ešac zvo alov opšeo pa tud ajpooteje upoabljee etode za ločtev ultplkatv ešac. Opae etode ktčo ovedoto j edebojo pejao. V adaljevaju a katko opšeo aatojo fzološke oejtve elektč alov v človešk šca poebe poudaek pa aeo opu ataka potovaja pooa akcjk potecalov v šče tkvu. Razpavo adaljujeo peledo obtoječ etod za dekopozcjo alov EMG. Pedtavljee etode ktčo ovedoto avedeo jove pedot oejtve. Nazadje defao pooje pod kate lako večkaale etve povškea EMG odelao kot te MIMO koč vzoč čaovo epeeljv tek odzv. V adaljevaju zpeljeo t ploše potopke za lepo ločtev koaj otooal pulz zvo alov. Najpej obavavao kovolutve ešace z več etva kot zvo opšeo dva ova potopka za jovo ločtev. Pv teelj a čaovo-fekveč aalz du pa a tattka všjea eda. Dekopozcjo ato azšo a kovolutve ešace z več zvo kot etva. V okvu alebajk

8 ptopov tattk duea eda azvjeo popoloa ov kocept ločevaja kovolutve ešace t.. etodo z vezo koelacjke atke. Upešot zpelja potopkov peveo a uet eal povšk al EMG. P podajaju ezultatov vpeljeo uteze kvattatve ee za vedoteje opa potopkov peučo vplv šua azeja ed števlo etev števlo zvoov. Rezultat a uet al potjujejo upeoot etode z vezo koelacjke atke etode IC. Če je etev več kot zvoov upe etoda IC p voke azeju al-šu SNR popoloa ekotuat koaj ve evacjke vlake pulzov ula ME. V poddeteae teu kje števlo zvoov za fakto 4 peea števlo etev upe etoda IC popoloa ekotuat pblžo polovco ve ula ME. Učkovtot dekopozcje pada tud z azeje al-šu. V povpečju upe etoda IC p SNR db ekotuat pblžo 3 % tt ME k o ble ekotuae p SNR db. Rekotua evacjk vlak pulzov pa e tud p zk azej al-šu koaj popoloa ujeajo z efeeč uet vlak. Peotal dve etod ta aj učkovt. Metoda IC e zkaže kot ajbolj učkovta tud p eal povšk al poet ed zoetčo kotato 5- -odtoto kčtvjo doate šce bcep bac p devet lad zdav poku oeba oškea pola. Skupo etoda IC popoloa ekotua evacjke vlake 3 ME p 5-odtot šč kčtv oz. 56 ME p -odtot šč kčtv. Rekotua evacjk vlak pulzov zpoljujejo ve fzološke oejtve.

9 Ackowledeet Foeot I would lke to tak y eto Pof. Daja Zazula fo patece uppot udace ad ecouaeet ove te lat 8 yea ad epecally touout te coue of y PD wok. My cee attude alo exteded to y coeto Pof. Robeto Melett fo all udace uppot ad valuable dcuo o y eeac topc. I would epecally lke to tak D. Dao Faa Maco Gazzo Maco Pozzo ad all te ebe of te LISN laboatoy at Poltecco d oo Italy fo te uppot u te SEMG ulato ad te de ad pleetato of te expeetal potocol. I would alo lke to tak y cottee ebe Pof. Goazd Lešjak ad Pof. adej Bajd fo te valuable ueto ad expete. I a ceely ateful to all of y colleaue fo te Syte Softwae Laboatoy at Faculty of Electcal Eee ad Copute Scece Mabo wo ave ve e excellet uppot ad fedp fo all tee yea. Specal tak o to Matjaž Dvjak fo all ueto ad te coecto e a cotbuted to t detato. I would alo lke to tak Pof. Zvoko Fazac. H uque ad fted atttude to ay apect of lfe a alway bee ad wll alway be a eat ouce of pato fo e. Lat but ot ay eaue te leat I ut tak y paet ad y te fo te to uppot ad pato. I ca eve tak eou y beloved Katja otly fo e oal uppot patece ad udetad. wok wa uppoted by te Slovea Mty of Educato Scece ad Spot Cotact No. S-796-/3/ ad Poae fud P-4 ad by te Euopea Saed Cot Poject NEW Neuoucula aeet te Eldely Woke Cotact No. QLK ak you all. Aleš Holoba

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11 able of cotet Lt of ybol v Lt of abbevato..... v. Itoducto Explaato of fequetly ued yoy Bld Souce Sepaato Sepaato of lea ultplcatve xtue Sepaato by axzato of o-gauaty Sepaato baed o axu lkelood Sepaato of te al Idetfcato of udedeteed MIMO yte Sepaato of lea covolutve xtue Maxu lkelood baed appoace Sepaato te-fequecy plae Refoulato of covolutve xtue to ultplcatve oe 5 3. Electoyoa Pyoloy of ua ucle Moto ut acto potetal EMG decopoto Aued data odel of uface EMG al Exteo of te covolutve MIMO vecto fo Abute of bld ouce epaato Decopoto of covolutve xtue of ouce oe eaueet ta ouce Metod baed o te-fequecy dtbuto Dcete te-fequecy dtbuto Sepaato of eeal ouce... 3

12 4..3 Bld epaato of pule ouce Decopoto of cloe-to-otooal ouce Noe fluece W-lce ad e-ode cuulat He-ode tattc Coae etato of yte epoe: w-lce Nolea optato of coae etate Ifluece of o-gaua oe ad ootooal ouce Appedx 4A: Devatve of cuulat Ivee coelato baed etod Decopoto of otooal ouce Decopoto of o-otooal ouce Sepaato of te upepoed ouce Noe fluece Noe te pace of ouce Decopoto of ude-deteed yte Appedx 5A: Ivee of daoally doat atx Appedx 5B: Pobablty of ovelapped pule Appedx 5C: Aveae ube of pule te poduct of te upepoed pule ta of ouce Appedx 5D: Popete of otooal pojecto atx Suface EMG decopoto eult Reult o ytetc uface EMG al Metod baed o te-fequecy dtbuto Metod baed o e-ode tattc Ivee coelato baed etod Reult o eal uface EMG al Ivee coelato baed etod Metod baed o te-fequecy dtbuto Metod baed o e-ode tattc... 3

13 7. Dcuo Copao of te eult o ytetc uface EMG al Copao of te eult o te eal uface EMG al Coputatoal coplexty Cocluo. 35 Refeece Boapy... 39

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15 Lt of ybol # A Mooe-Peoe peudovee of atx A B C jk C wte atx cuulat atx calculated fo -t j-t ad k-t eaueet td-ode cuulat atx of -t eaueet deote te fxed eaueet D f patal te-fequecy dtbuto atx of exteded ouce D f patal te-fequecy dtbuto atx of exteded eaueet xx δ Dac pule F total ube of pule a pule ouce φ l keel of a te-fequecy dtbuto G γ 3 j H et of dce deot te ouce tat ae ultaeouly actve at ve te oet cadal ube of et G kewe cetal value te td-ode cuulat atx of ouce S j x atx j l j-t yte epoe of aued MIMO yte Id actvty dex K exteo facto te ube of delayed epetto added to eac eaueet κ H H codto ube of atx H L M axal let of yte epoe aple ube of oal eaueet v

16 N p R ube of oal ouce pobablty tat te pule of two abtay ouce ovelap ube of ubet of ouce touout te te we aue te ouce ca be clafed to R ubet eac cop Q ouce ad tat te ouce fo te ae ubet caot ovelap wt eac ote. v j j-t eleet of atx R R R x R Q coelato atx of ouce coelato atx of exteded eaueet coelato atx of exteded ouce ube of ouce cottut eac ubet of ouce ee explaato fo R vecto of pule ouce exteded vecto of pule ouce S a S c atx cop all at-caual pat of all auto-cuulat ad all co-cuulat tat ae baed o two dffeet eaueet atx cop all caual pat of all auto-cuulat ad all cocuulat tat ae baed o two dffeet eaueet j j-t pule ouce σ v G W oe vaace otooal pojecto atx upepoto of ouce wc ae ultaeouly actve at ve te oet atx of cuulat wet ω oe vecto ω oe detected -t eaueet v

17 x vecto of all oe-fee eaueet x exteded vecto of oe-fee eaueet x -t oe-fee eaueet y -t oy eaueet y vecto of oy eaueet y exteded vecto of oy eaueet v

18 Lt of abbevato AP BSS EMG FIR HOS..d IC IPI MIMO ML MU MUAP MVC SFD F Acto potetal Bld ouce epaato Electoyoa Fte pule epoe He-ode tattc Idepedet detcally dtbuted Ivee coelato Ite-pule teval Multple-put-ultple-output Maxu lkelood Moto ut Moto ut acto potetal Maxu volutay cotacto Spatal te-fequecy dtbuto e-fequecy v

19 . Itoducto Dtal al poce a becoe oe of te ot potat tool duty pocee eeac actvte ad ay feld of cece. It alo oe of te fudaetal buld toe of te ode ocety. Moe ad oe of eveyday actvte ely o popt ad pece foato wc alway edated by te al. Sal poce ca be foud eveywee fo te wde aea of telecoucato ecooy eoloy ada ad oa applcato exploato of pace edce ad boloy dow to te coco ad quatu ecac. e al tat te foato about te wold we lve ad cucally fluece ou kowlede ad udetad of te exteal wold. Altou te al ae ofte tout to be veted by te ua tey ae qute coo te atue telf. ouout te evoluto vaou lv be leaed ow to poce dffeet al fo te exteal wold. e ot obvou exaple of uc a atual al poce ae te ua ee of t ad ea. ey ape ou eveyday lfe. Yet we ae oly be to udetad te coplcated pocee ou ba bed te. Le obvou but ot le potat ae te electcal al ua ucle. ey cotol te oveet ad ve u te feedback about te ucle actvato. Moeove we ca ealy tack te elatve poto of ou a eve bldfolded. ea te ua ba ae qute capable of calculat te vee keatc of ou extete ve jut te electcal al fo te ucle. Code te ubject vetated t detato pobably te ot teet featue of ou ba te ablty to epaate te oud of dffeet ouce out of te copoud xtue. peoeo ca ealy be obeved at te pate. Eve tou tee ae eveal ultaeouly peak peo we ca ealy focu o oe of te ad oe te oud fo te ote. Wtout eve ealz we ae pefo oe of te vey dffcult al poce tak o called bld ouce epaato. Fo te cetfc pot of vew te al poce a wde eeac feld. It clude oe ad atefact eoval fequecy aaly ad yte patte ecoto cod ad decod peec ad ae aaly etc. Accodly ay dffeet teoe ad tool a fo pla flte te-fequecy ad te-cale aaly ecod- ad e-ode tattc HOS to ceptal aaly ad eual etwok wee developed te pat. Oe of te ay tool wc ae beco cealy potat o called Bld Souce Sepaato BSS. It oal to ecotuct te oal ouce out of te

20 ultplcatve o covolutve xtue. Altou be a elatvely you cetfc feld t a bee uccefully appled to ay dffeet aea. I t detato te epaato of lea covolutve xtue addeed. Altou tee wee ay attept te pat o eeal etod applcable to abtay covolutve xtue of optoal al ext. e appoace developed o fa focu o pecfc xtue ad explot te foato offeed by te axu lkelood etato [ ] ecod- ad e-ode tattc [ ] ad te-fequecy ad te-cale aaly [ ]. Suppo utually depedet ouce tey uually eable oly te ecotucto of pule epoe wle te ouce ae eeally ecotucted oly up to a ukow flte effect [7 84 6]. I ou wok we lt te dcuo to te lea covolutve xtue of pule ouce. e latte ca be odelled a te-doa equece of pule.e. te te al wt te eey cocetated te epaated te oet pule. Except a coo aupto o tattcally depedet ouce o foato about te pule ouce o te coeffcet of te x flte uppoed. Howeve we aue eveal cocuet eaueet of te ae ouce al ae ve.e. o called ultple-put-ultple-output MIMO yte. e aued data odel coepod to boedcal ad ot of te telecoucato [ ] al ad a bee exploted by ay appoace [ ]. Howeve tey all uffe fo at leat two ajo dawback. Ftly te ajoty of decopoto tecque uppoe te ube of eaueet eate ta te ube of ouce. Fo te pactcal pot of vew t a ly ctcal aupto a tee ae ay pycal ad pyolocal eao peak aat t [56 3]. Secodly a etoed above te ouce al ae oly patally ecotucted. o be exact te tate-of-te-at decopoto tecque wok well oly we te ube of ouce all ad te cotbuto pule epoe do ot ovelap fcatly. I t detato tee ovel appoace fo bld decopoto of covolutve xtue of pule ouce al ae toduced. e poble oute towad a pefect ecotucto of pule ouce al ad te pule epoe ae exaed ad te fluece of oe ad o-otooal tattcally depedet ouce al ae evaluated. e detfcato of ltly udedeteed MIMO yte yte wc te ube of ouce ltly exceed te ube of eaueet alo tuded. e ovel decopoto appoace ae teted o uface electoyoapc SEMG al [9 3 46]. e latte cope a lae ube of actve ouce wt elatvely ot pule epoe. Naely du volutay cotacto of keletal ucle all fuctoal oup of ucle fbe called oto ut MU cotact ycoouly poduc a eauable electcal al called oto-ut acto potetal MUAP. We ecoded by a aay of electode at low ucle cotacto level te MU cotbut to te SEMG al ca be teated a cloe-to-otooal pule ouce al woe co-coelato do ot exceed appox. 5- % of te axal poble

21 value [ ]. e SEMG al ca teefoe eve a a pefect exaple of covolutve xtue of pule ouce al vetated t detato. Fally te applcato of developed tecque to te SEMG al ceae te value of ou wok a a ajo tep towad a coplete ecotucto of cottuet MUAP ta ade ee Secto 3.. o uae two dffeet cetfc feld; bld ouce epaato ad uface EMG decopoto ae tuded. Altou all decopoto appoace ae baed o eeal pule al we fequetly ue te uface EMG al to exeplfy ad llutate te effcecy of appled oluto o to ply deotate te coplexty of covolutve xtue ude vetato. e a aupto ad expected ltato ae uaed te follow te: I te cae of cloe-to-otooal pule ouce al ad oe eaueet ta ouce a coplete bld ecotucto of covolutve x atx ad ouce pule ta out of te covolutve xtue poble. We te ube of ouce exceed te ube of eaueet oly a lted ube of ouce al ca be copletely ecotucted. Poce te uface EMG al te pule epoe coepod to te oto-ut acto potetal wle te pule ouce al tad fo te oto-ut evato pule ta. Elte te decopoto aupto poblte ad ltato at leat tee oup of ypotee ca be toduced. ey epaately adde te expectato te cae of o-otooal ouce al te cae of oe ouce ta eaueet ad te peece of oe:. I te cae of oe eaueet ta ouce a coplete ecotucto of pule ouce al ad te pule epoe out of te covolutve xtue poble. We te ube of ouce exceed te ube of eaueet oly te pojecto of ouce al to te ubpace of eaueet ca be obeved. A a eult te pule epoe te covolutve x atx tefee ad oly a lted ube of ouce al ca be copletely ecotucted.. Suppo utually depedet ouce al te fluece of covolutve x atx ca be elated by te ecod- ad e-ode tattc. We poce o-otooal ouce al te pule epoe tattcal oet ad e-ode cuulat tefee wat toduce addtoal eo to te decopoto poce. A a eult te ecod- ad e-ode tattc ca be ued to epaate oly otooal o at leat cloe-to-otooal ouce al woe pule do ot ovelap fcatly. Sla ltato ca be ecouteed we u te-fequecy aaly. 3

22 3. I oy evoet oly a lted ube of ouce pule equece ca be copletely ecotucted. e exact ube deped o te al-to-oe ato ad o te powe of ouce al. e latte te cae of pule al coepod to te ube of pule ad coequetly to te ouce f ate. e ouce wt e f ate ae teefoe expected to be oe uccefully ecoed ta te ouce wt lowe f ate. Ao all te popoed appoace te etod baed o HOS ae expected to be upeo oe ejecto. detato oazed 8 capte. Ft Capte ovevew of te tate-ofte-at te bld ouce epaato outled. May oluto fo te ultplcatve cae ca be tatfowadly eealzed to te covolutve oe. Hece te dffeet appoace to te decopoto of bot ultplcatve ad covolutve xtue ae ctcally evaluated ad utually copaed. I Capte 3 te a pcple of te aatoy ad pyoloy of ua ucle ae befly uazed. e decpto ve ould oly eve a a ba fo te evaluato of te eult o eal SEMG Capte 6. Hece te a focu o te eeato of acto potetal ucle fbe MU f fequece ad te flte effect of te ubcutaeou tue. Next a ovevew of te tate-of-te- at EMG decopoto tecque ve alo wt te a po ad co. Fally te codto ude wc te uface EMG al ca be odelled a lea tevaat MIMO yte ae elteed. I Capte 4 ad 5 tee ovel appoace to te decopoto of covolutve xtue of cloe-to-otooal pule ouce al ae deved. Ft Capte 4 te detfcato of ovedeteed MIMO yte yte wt oe eaueet ta ouce tuded ad two ovel decopoto etod toduced. e ft oe baed o te-fequecy aaly wle te ecod oe utle te e-ode tattc. I Capte 5 te decopoto exteded to ltly udedeteed MIMO yte. Baed o alebac appoace ad ecod-ode tattc a ovel cocept of bld ouce epaato te o called vee coelato baed etod toduced. I Capte 6 te toduced decopoto etod ae teted o bot ytetc ad eal uface EMG al. e fluece of oe ad o-otooal ouce al ae evaluated ad te pact of dffeet MU paaete tuded wle Capte 7 te decopoto eult ae utually copaed. We coclude ou wok Capte 8 wee te pcpal ypotee ae evaluated ad te ltato of te toduced appoace ae dcued. 4

23 . Explaato of fequetly ued yoy We ave explaed tat two dffeet aea bld ouce epaato ad uface EMG decopoto wll be tuded t detato. Altou we wll ty to tctly epaate te te bac teoloy of bot aea wll be ued. Fo exaple we decb te EMG al we wll be talk about te evato pule ta ad acto potetal. O te ote ad dev ovel decopoto appoace we wll ue oe eeal te uc a pule ouce al ad yte epoe. Wle te evato pule ta ca cetaly be odelled a pule al we wll pefe to ue oe pecfc ae weeve poble. Geeal otato wll oly be ued Capte 4 ad 5 ode to te te eealty of te deved decopoto etod. I te equel te ot coo te fo bot aea bld ouce epaato ad uface EMG decopoto ae clafed. ey ould ot be udetood a defto. ey ae jut explaato of te ot fequetly ued yoy. Pule ouce: a ouce woe al ca occupy oly bay value ad a wee a deote a abtay ozeo eal ube. Saple wt value a ae efeed to a pule. I te equel te te pule ouce wll fequetly be ued a a yoy fo ouce pule al. Altou t ca duce oe ucetate t qute coo coveto te feld of bld ouce epaato. Ite-pule teval IPI: te dtace betwee two ucceve pule a ouce pule ta. Supepoto of al: te eleet-we u of al. I t detato two dffeet upepoto te upepoto of pule ouce ad te upepoto of oto ut acto potetal wll be tuded. Cloe-to-otooal: te pae cloe-to-otooal wll be ued to deote te ouce woe co-coelato do ot exceed % of te axal poble value. I te cae of pule ouce t ple oly up to % of pule of two abtay ouce pule ta ca ovelap. Syte epoe: te cotbuto to a le eaueet caued by a le f of a pule ouce. It cooly efeed to alo a a yte pule epoe ad defed oe detal Secto 3.3. I te equel te oto ut acto potetal wll be odelled a pecal fo of yte epoe. Moto ut MU: a all fuctoal oup of ucle fbe wc ae evated by te ae alpa oto euo ad cotact ycoouly Secto 3.. ouout t detato MU wll be udetood a a o of te uface EMG al wt t evato pule ta a a cloe-to-otooal pule ouce al ad t acto potetal a a yte epoe. 5

24 Ievato pule ta: te te equece of acto potetal alpa oto euo wc ae tafeed tou euoucula jucto to all te ucle fbe of te ae MU. A tey te te epoe of MU acto potetal oto euo ae cooly efeed to a pule. ouout ou wok te evato pule ta wll be odelled a cloe-to-otooal pule ouce. Fo te bld ouce epaato vewpot t a vey coae appoxato becaue te pule euo ae at leat a few lo ad occupy oe ta two dffeet value we apled wt fequecy. Nevetele oly te MU epoe MUAP to evato pule ta ae eaued by te uface electode wle te ape of euo potetal do ot pact te ape of MUAP. eefoe acto potetal a oto euo ca be coceptually eplaced by tue bay pule. e evato pule ta ad MUAP ae decbed oe detal Secto 3. ad 3.. Acto potetal AP: a apd cae te taebae potetal a euo o ucle fbe teed by electc cecal o ecacal exctato of te fbe ebae. It uually eveal lo ad popaate alo te euo o ucle fbe fo t o towad te ed of te fbe. Moto ut acto potetal MUAP: a upepoto of flteed acto potetal of all ucle fbe cop te coepod MU a detected a le EMG eaueet. e ubcutaeou tue betwee a patcula ucle fbe ad a eau electode act a a low-pa flte ad atteuate te le fbe AP. e MUAP ape deped o ay pyolocal paaete te MU dept ucle tue te ube of MU fbe te coducto velocte etc. o te popete of te detecto yte te type of patal flte te-electode dtace te ape of electode etc. a well a o te elatve poto of le fbe wt epect to te detecto pot te patal flte poto. e ape ad te apltude of MUAP alo vay a a fucto of te dtace fo te evato zoe. A a eult te MUAP of dffeet MU a well a MUAP of te ae MU eaued at dffeet poto pace wll dffe fcatly. I t detato te MUAP wll be cooly odelled a pecal fo of yte epoe. Electoyoa EMG: a te ecod of upepoed MUAP equece fo all MU wc ae actve detecto volue of eau electode. Spatal flte: a cofuato of pck-up electode ued to eaue a le EMG eaueet. ee ae ay poble electode cofuato patal flte. e plet oe a upola electode F.. a. Sle dffeetal SD patal flte F.. b cot of two electode wle te eaueet foed by ubtact te al eaued o bot electode. Double dffeetal DD flte depcted F.. d ad cope tee electode. e EMG eaueet foed by ft add toete te al fo two de electode ad aftewad ubtact twce te al fo te cetal electode. Spatal flte ca alo be clafed accod to te oetato wt epect to te ucle fbe. Lotudal patal flte F.. b ad d ave te electode aled wt te ucle fbe wle te cae of taveal 6

25 flte F.. c ad e te electode ae placed taveal to te oetato of ucle fbe. y a b c d e x Fue.: Spatal flte cooly ued fo uface EMG ecod: a upola electode b lotudal le dffeetal flte c taveal le dffeetal flte d lotudal double dffeetal flte ad e taveal double dffeetal flte. Electode ae depcted by ey ccle wle te coepod ube deote te wet wt wc te al fo electode ae ultpled befoe tey ae ued up to yeld te fal EMG eaueet. e ucle fbe ae aued aled wt te y ax. 7

26 8

27 . Bld ouce epaato Bld ouce epaato BSS [ ] beco a cealy potat tool te fled of al poce. Fo te lat decade t a bee uccefully appled to te aea of ada [3 4 79] ad oa al aaly [78 6] peec ecoto [84 6 8] telecoucato [5 45] epaato of ec al [5 79] ae poce [ ] ad lately to te aaly of boedcal data [ ]. It oal to ecotuct te oal ouce ve jut te ultplcatve o covolutve xtue. e x poce uually deoted by te o called x atx wc of coue ukow. I te cae of ultplcatve xtue te x atx cot of cala facto wet. I te cae of covolutve xtue te x atx cope pule epoe of dffeet cael covolutve flte [79 6]. Uually eveal eaueet of te ae ouce ae avalable wc eult MIMO data odel [ ]. capte povde a ot ovevew of te tate of te at te bld epaato of lea xtue. e a focu o covolutve xtue altou te etod epaat te ultplcatve xtue ae alo decbed. ee ae at leat two eao fo tat. Ftly ay etod fo epaato of covolutve xtue pove to be tatfowad eealzato of te oluto developed fo te ultplcatve cae. Secodly evey covolutve xtue ca alway be efoulated to te ultplcatve oe ope a poblty of olv t by te ultplcatve BSS tecque. e decopoto tecque decbed t capte ae clafed wt epect to te bac decopoto pcple. I te cae of ultplcatve xtue Secto. te etod baed o te axzato of o-gauaty axu lkelood etato ad te tuctue of ouce ae dcued. Altou te a focu o te epaato of ovedeteed MIMO yte te etod epaat te udedeteed MIMO yte ae alo decbed. Secto. outle poble oute towad epaato of lea covolutve xtue. e etod baed o axu lkelood Foue tafo ad efoulato of te covolutve xtue to te ultplcatve oe ae befly uazed. Majo aupto alo wt po ad co of eac oup of etod ae alo peeted. 9

28 . Sepaato of lea ultplcatve xtue I te cae of lea ultplcatve xtue te x cael ae aued to be eoyle.e. te xtue ae uppoed to be tataeou deped oly o te cuet value of te ouce wle o delayed veo of te ae ouce ca be xed toete. I te aued data odel te ouce ae ft ultpled by te ukow facto wet ad aftewad ued up to fo te obeved eaueet: x. j j wee x deote te -t eaueet j ukow x facto ad j tad fo te j-t ouce. We cooly uppoe te facto j cotat te. Aote coo aupto tat te ube of obeved eaueet exceed te ube of ouce. U te atx otato. ca be ewtte a j x H. wee x [ x... x M ] ad [... N ] deote te vecto of eaueet ad ouce epectvely wle H tad fo te ukow M N x atx: N N H..3 M M MN ee ae ay appoace to te bld epaato of ultplcatve xtue. e decopoto tecque ae baed o utual foato ad Kullback-Lebe dveece [ ] o ecod- ad e-ode tattc [ ] o te-fequecy [ ] ad te-cale aaly [ ] o alebac [ ] ad eoetc appoace [ ] ad fally o te axu lkelood etato [ ]. e ajoty of te uppoe depedet ad detcally dtbuted..d. ouce te ouce aple ae odelled a depedet equally dtbuted ado vaable [ ]. It a bee ow [ ] tat te epaato of..d ouce poble oly f at ot oe ouce a Gaua dtbuto. e ote BSS etod code te cae of tepoally coelated ouce [ ]. Idetfablty uc a appoac ated eve we te al ae oally dtbuted. e td poble oute to ultplcatve BSS to explot te equalte te ouce aple dtbuto.e. o-tatoate of ouce [ ]. I te equel te ot fequetly ued oup of etod fo epaato of lea ultplcatve xtue of ouce ae coaely outled. e detaled ovevew ca be foud [ ].

29 .. Sepaato by axzato of Noauaty Vat ajoty of te ultplcatve BSS etod ae baed o te cetal lt teoe wc tate tat te dtbuto of a u of..d. ado vaable ted towad a Gaua dtbuto. I ote wod uppo te aple of ouce..d. o- Gaua ado vaable te cetal lt teoe uaatee tat te lea xtue oe Gaua ta ay of te oal aple. Moeove te xtue becoe leat Gaua exactly we t equal to te aple of oe of te ouce. Hece te epaato poce we look fo uc a epaato atx wc axze te o- Gauaty of te eaueet. Altou te exact dtbuto of te eaueet ae uually ukow we ca alway etate te out of te al aple. e o-gauaty of te eaueet ca alo be etated dectly. Oe of te ot fequetly ued eaue utlze e-ode cuulat [ 66 67]. ey ae kow to be zeo fo all te Gaua ado pocee. fact effectvely utlzed by te epaato etod wc typcally ue te adet baed optzato ode to uecally axze te value of td- o e-ode cuulat of eaueet. e ecod poble eaue of o-gauaty eetopy [84]. It defed a a dffeece betwee te etopy of te obeved ado vaable ad te etopy of te Gaua ado vaable wt te ae ea ad vaace. Be oeatve ad zeo f ad oly f te obeved ado vaable Gaua t vey atual eaue fo te optato. e a dawback of te etod tat ue cuulat-baed o eetopy-baed eaue tat tey do ot dtu ao te ouce. I te optato poce te lobal axu foud wc ple oly oe ouce ca be uccefully ecotucted. o ecotuct eveal ouce te epaato alot ut be u eveal te u dffeet talzato pot. It eay to ee tat t ot a elable oluto. Moe cotet etod cobe te eult of te decbed appoace wt dffeet otooalzato alot wc utlze te utual otooalty of te wteed vecto of eaueet [8 83]. Wt te elp all te ouce ca be ecotucted... Sepaato baed o axu lkelood A ecod vey popula cla of ultplcatve BSS etod cope te axu lkelood ML etato. Gve te pobablty dety of te ouce ad te x atx te lkelood fucto of te xtue ca be deved fo te well-kow eult of dety of a lea tafoato [84 6]. Uually due to t alebac plcty te loat of te lkelood uecally optzed. ake o dffeece ce te axu of te loat obtaed at te ae pot a te axu of te lkelood. ML-baed etod ply ty to fd te x atx ad te ouce pobablty dete wc ve te et pobablty fo te obevato. e uecal axzato of te lkelood fucto uually pefoed by dffeet adet-baed alot [ ].

30 e lo-lkelood fucto of te xtue eeal fucto of bot te eleet of te x atx ad te pobablty dete of te ouce. e foe ae codeed te paaete of te aued data odel ad ca ealy be optzed. e optzato of te ouce pobablty dete uc oe dffcult poble becaue eeal te etato caot be educed to te etato of a fte paaete et. ee ae two poble way to avod te opaaetc dety etato [ ]:. Soete te dete of te ouce o at leat te faly t be kow advace. I t cae t po foato ca be ued te optzato poce ak te lo-lkelood fucto depedet of te x atx oly.. e ukow dete ca alo be appoxated by a faly of dete tat deped o a fte ad all et of paaete. Uually vey ple ad eeal paaetezato of te dete yeld atfactoy eult. e cla of ML-baed epaato etod cope ay dffeet oluto. Ao te te etod baed o te foax pcple [ 7] etopy ad Kullback- Lebe dveece [8 8 8] ad utual foato [3 7 84] ae te ot popula. e a dawback te coputatoal coplexty...3 Sepaato of te al e BSS etod codeed o fa odel te ouce aple a..d o-gaua ado vaable. Meau o-gauaty of te xtue tey ca oly be ued fo epaato of xtue wt at ot oe Gaua ouce. I te cae of two o oe Gaua ouce te decbed appoace fal [ ] ad we ae foced to eek fo a alteatve oluto. Ofte te obeved ouce ae te al. A uc tey cota uc oe tuctue ta te ple et of tctly depedet ado vaable. tuctue ca te be ued to pove te epaato o eve to ake t feable. Actually explot te te tuctue alo Gaua ouce ca be epaated. I te pat two lae fale of te te-tuctue baed etod evolved. e ft oe explot te te depedece ao te aple eac ouce. I ote wod te aple of eac ouce ae odelled a depedet detcally dtbuted ado vaable. e lae ajoty of te epeetatve of t faly eploy o called jot ultaeou daoalzato of eveal coelato atce at dffeet te la [ ]. e ecod faly of te te-tuctue baed etod take advatae of te otatoaty of te ouce. Hee te ouce aple ae odelled a depedet ado vaable wt te-depedet otatoay dtbuto. Mot of te epeetatve of t faly ely o te-fequecy aaly ad o te fact tat te x atx vaat to te Foue tafo [ ].

31 Due to te plcty ad plct capablty of epaat te Gaua ouce te te-tuctue baed etod aed a lot of teet ove te pat few yea. May effcet oluto wee popoed fo bot ultplcatve ad covolutve xtue. Oe of ay poble oute towad te epaato of covolutve xtue decbed Secto Idetfcato of udedeteed yte So fa we uppoed te lea ultplcatve xtue ave oe eaueet ta ouce. e decbed epaat etod aed to etate te x atx wle te ouce wee ecotucted by ply apply te vee of te x atx to te eaueet. o loe te cae we te xtue wt oe ouce ta eaueet ae poceed. I uc a cae te x atx caot be veted ay oe ad te exact value of te ouce caot be ecotucted eve f we kow te x atx exactly. Hece te epaato of te udedeteed yte a dffcult poble ad ca be copaed to te ateatcally well udetood poble of olv a udedeteed yte of equato. Altou ftely ay poble oluto ext we ca uually a atfactoy eult by lt te popete of te ouce [ ]. Sepaat te udedeteed yte we face two lae poble. Ft we ave to etate te x atx ad ecod we ave to etate te ukow ouce al. Suppo te x atx kow te ouce ca be etated by apply t peudo-vee to te eaueet [68 9]. Altou ple t ot te optal oluto. Muc oe cotet appoac baed o te axu lkelood etato [ ]. e poble wt te ML etato tat t caot be expeed aalytc fo ad ut be optzed uecally wc ly ceae t coputatoal coplexty. I eeal te x atx ukow ad ut be etated fo te eaueet. Aa te ot popula appoace ae baed o te ML etato. Ao te o called Mote Calo etod baed o te Bayea etato ad uecal teato ae beco cealy potat. e ot popula etod of t faly utlze o called Gbb apl to etate te pobablty dete [5 59]. e dawback tat tey ae vey coputatoally dead. Muc oe attactve ae te o called otooalzato etod [84 96] wc ae baed o a ate uealtc aupto of qua-otooal colu of te x atx. 3

32 . Sepaato of lea covolutve xtue ee ae at leat two eao wy te ultplcatve xtue ae elatvely ae pactce. Ft eau te obeved al wt te aay of eo te fte popaato peed of te al duce dffeet te delay of te ae ouce dffeet eaueet. Secod due to te ult-pat popaato te obeved edu eac eaueet cota dffeet te-delayed veo of te ae ouce. call fo te covolutve xtue odel.e. te odel wee te cotbuto of eac ouce to eac eaueet odelled by a covoluto. ak MIMO yte to codeato te -t eaueet ca be expeed a N L j j l x l l...m.4 j wee j ad j l deote te ukow ouce ad fte covoluto flte epectvely. e oal bed te epaato of covolutve xtue to ecotuct te covolutve flte j l ad te ouce j ve jut te eaueet x. A te ultplcatve cae te aple of te ouce ae aued to be..d. ado vaable wt ukow pobablty dete. e epaato of covolutve xtue appea te lteatue ude ay dffeet ae: ultcael bld decovoluto bld detfcato of MIMO yte depedet copoet aaly ICA wt covolutve xtue ultcael bld detfcato etc. e appoace developed o fa ae baed o ML etato [ ] te-fequecy aaly [ ] ad efoulato of te covolutve xtue to te ultplcatve oe [ ]. Suppo utually depedet ouce tey uually eable oly coae etato of te covolutve flte wle te ouce ae eeally ecotucted oly up to te ukow flte effect [ ]. ecto povde a quck ovevew of te ot popula decopoto tecque... Maxu lkelood baed appoace A aleady explaed te pevou ecto te ML baed etod ae of te ot popula oe. Be baed o te pobablty dety of te ouce tey ae eetly capable of captu te po kowlede about te ouce. Moeove tey ofte povde te upeo pefoace ad ca ealy be exteded to te udedeteed cae. e a dawback te coputatoal coplexty. It ould ot coe a a upe tat te ML etato ca alo be exteded to te covolutve cae. Actually te popoed oluto ae oe o le dect eealzato of te oluto developed fo te ultplcatve cae [ ]. Be baed o uecal optato of a lae ube of ukow paaete tey eetly uffe fo a coputatoal coplexty ad low 4

33 coveece. Altou ueou attept to peed te up wee ade te pat te effectve oluto wee popoed oly fo te plet covolutve odel poble te o called le-put-le-output SISO yte... Sepaato te-fequecy plae e laet ad pobably te ot effectve cla of covolutve BSS etod utle te Foue tafo. It baed o well kow fact tat te covoluto te te doa becoe te poduct betwee te Foue tafo te fequecy doa. I ote wod u te te-fequecy tafoato te covolutve xtue ca be tafoed to a et of ultplcatve xtue te te-fequecy doa [ ]. e a dawback of te te-fequecy baed etod tat we ave to etate te x atx fo evey poble fequecy b. ot a tval poble due to te plct deteacy of te ultplcatve BSS poble Secto Naely o atte ow ucceful te decopoto we copaed to te oal ouce te ecotucted ouce ca alway appea adoly peutated ad eve ultpled by te ukow cal facto. Recotuct te ouce te fequecy doa te peutato ad te cale of te ouce eeally dffeet fo evey fequecy b. o be able to ecotuct te ouce te te doa we ut ft fd te way ow to fd te fequecy copoet tat belo to te ae ouce. poble a bee kow a a peutato poble ad a bee tevely tuded te pat [9 57 8]. e ot effectve oluto utlze te pectal cotuty of te ouce [4 7] foato about te decto of aval DOA [85 8] ad vaou cobato of te aboveetoed oluto [84 57]. Nevetele te peutato poble ea epecally we decopo lae covolutve MIMO yte. A a eult te tefequecy etod ave oly bee appled to elatvely all ovedeteed MIMO yte wt o oe ta ultaeouly actve ouce...3 Refoulato of covolutve xtue to ultplcatve oe By add te delayed epetto of eac ouce to te vecto of ouce [... N ] evey covolutve xtue ca be tafoed to te ultplcatve oe ee Secto 3.3 ad olved wt oe of ay poble oluto etoed Secto.. Altou te ube of ouce ly ceaed t a vey attactve oluto a tee o peutato poble. e a dawback of uc a oluto le te tattcal popete of te ewly foed ouce wc at leat eeal cae ae ot tattcally depedet ayoe. pove to ave a cucal pact o te epaato pefoace. Altou ay attept to ovecoe t etback wee ade te up-to-date etod eable oly ecotucto of ouce up to te ukow flte effect [ 7 75]. 5

34 6

35 3. Electoyoa Electcal actvty of ucle a bee ude te cetfc vetato fo cetue. Aleady te 8 t cetuy Galva dcoveed a up popety of ucle wc cotact we elcted by te electcal tulato. He alo owed te ucle epoe ca be eaued te fo of electcal potetal. I 99 Ada ad Bok ued cocetc eedle electode to eaue te electcal potetal de te lve keletal ucle. Stude tat followed evealed te ecet bed te electcal al te ucle. It wa oo ealzed ow te ucle fbe ae oazed to te bac buld block of ucle o called oto ut MU ad ueou tude of te electcal popete eeed. Fo a lo te te EMG ecod ave bee baed o te expeve ad vave eedle electode. It wa oly te ecod alf of te pevou cetuy we te vetato of o called uface EMG SEMG.e. te EMG eaued by te electode placed upo te k uface eeed Baaja 974 Deluca 984 Melett ad LoCote 995. Due to te plct o-vave atue uface electode ave ueou advatae [ ]. ee fcatly le dcofot wat allow fo ulted epetto of te lo-te electoucula oto exactly te ae place. Futeoe ecod of uface EMG expeve ad ve lobal foato about ucle actvty [ ]. Nevetele te SEMG a t ade t way to te clcal pactce yet. aly due to te poo opolocal foato caued by te flte effect of te ubcutaeou tue. Moeove tee uually a lae ube of actve MU te detecto volue of a uface electode cau te detected al to be ly coplex. May attept to eace te le MU foato SEMG wee ade te pat evetele te decopoto of te SEMG al ea a vey delcate poce. capte povde a ot ovevew of a pcple bed te ucle actvte. e a focu o te eeato ad extcto peoea of te acto potetal ucle fbe ad o te flte effect of te uoud tue. ee decpto wll eve a a ba fo evaluato of decopoto eult Capte 6. I Secto 3. te tate of te at te decopoto of bot eedle ad uface EMG befly outled wle Secto 3.3 te ateatcal odel of SEMG wc wll be ued t detato foulated. 7

36 3. Pyoloy of ua ucle Hua ucle cot of to 5 lo ad 5 to 9 µ t ucle fbe wc ae te tedo eo attaced to te boe. e oetato vae ubtatally betwee te ucle. Altou te ucle fbe ca be aled wt te decto of te tedo tey oe ofte fo a o-zeo ale wt te. Bede te ucle fbe te ucle tue cope vaou ote tuctue: eve blood veel coect tue etc. ee alo a laye of ubcutaeou fat epaat te ucle fo te k. Eac ucle fbe evated by a le oto euo wc tat te cotol coad fo te cetal evou yte a fo of te f pule ta. Noally eveal ucle fbe ae evated by te ae euo fo a bac fuctoal ut of te ucle o called oto ut MU. e ube of fbe eac MU vae codeably betwee te ucle. e allet MU ca be foud te eye ucle ad cope le ta fbe. e laet MU belo to te lae le ucle ad cobe oe ta fbe [37]. e ze of MU vae codeably alo wt te ae ucle. It potat to otce tat te fbe belo to te ae MU ae ot adjacet to eac ote but ate pead pace wt te adu of appox. to 5. Code te te eeded to aceve te peak teo te ucle fbe ca be clafed to two djuctve clae: low o type I ad fat o type II. e fbe type deteed by te cetal evou evato. Coequetly all te fbe fo te ae MU ae of te ae type wle all ucle cope all type of MU. e aea of euoucula jucto betwee te ucle fbe ad oto euo called evato zoe. Altou te fbe alway evated appoxately te ddle betwee bot tedo te locato of te evato zoe ca vay codeably wt epect to te ucle belly te fbe ae uually ot paallel to eac ote. Evey pule oto euo te a cotacto of MU fbe. Du a volutay cotacto te wole equece of pule o called evato pule ta ae tatted by te oto euo. e ube of pule pe ecod.e. MU f ate deped o te type of ucle fbe ad o te ucle cotacto level. At low cotacto level te all ad low MU of type I ae ft actvated. e f ate vae betwee 6 ad Hz. We te cotacto foce ceaed lae MU of type I a well a te fat MU of type II ae actvated. e f ate adly exceed -3 Hz ude te oal codto. By cea te f ate te cotacto teed by eac dvdual pule oto euo ae eed toete to fo cotat ad oot ucle cotacto. 3.. Moto ut acto potetal Evey pule te oto euo te a bef cae te taebae potetal of te ucle fbe o called acto potetal AP. aly caued by te cae of te ebae peeablty to odu Na o wc ae located te pace 8

37 betwee te ucle fbe. e Na o ete te fbe ad caue te depolazato of te taebae potetal wc cae t fo te eatve appox. -7 V to te potve oe appox. V. Aftewad te ebae peeablty to odu o deceae wle te peeablty to potau o K ceae. Outflow of te K o caue te epolaato of te taebae potetal. Fally u te Na /K pup te Na ad K o ae actvely tapoted out of ad to te ucle fbe epectvely [47]. A fbe ad teefoe a oto ut caot be actvated aa utl te AP due to a pevou actvato alot copleted. well kow a a efactoy peod. I addto du utaed ucle cotacto le MU fe at ot -3 pule pe ecod. I ae cae tee ay be o called doublet tat two f of te ae MU epaated by oly -5. e AP te ucle fbe tated at te euoucula jucto ad popaate bot decto towad te tedo eo wee t adually uppeed. e AP apltude appox. V duato 3 to 5 ad t popaato/coducto velocty 3 to 5 / deped bot o te fbe daete ad o t type. e le fbe AP ca be eaued alo te cloe vcty of te ucle fbe. e pave electc eda uoud te ucle fbe act a a low-pa flte wc deceae te AP apltude ad ceae t duato. Moeove te cotbuto fo dffeet ucle fbbe ca be tout ued toete [ ]. A a eult eau te electcal potetal at a abtay poto pace te AP fo dffeet fbe wll appea upepoed bot te ad pace. Aa due to te flte effect of te uoud tue te cotbuto fo te fbe tat ae cloe to te eau te wll be toe [38]. Uually te detected AP fo all te fbe belo to te ae MU ae tout to be ued toete fo o call oto ut acto potetal MUAP. e upepoto of te le fbe AP ca ow be eplaced by te upepoto of MUAP. It vey potat to ealze t ot ele but a vey pactcal coveto wc ly educe te ube of te copoet te detected al wle tll eabl a tudy of te MU popete. It alo potat to ealze te ape of MUAP ca vay eoouly wt te poto of te detecto pot. MUAP fo te ae MU detected at dffeet poto pace wll dffe ubtatally fo eac ote. Not oly wll tey ave dffeet te delay due to te fte peed of te MUAP popaato alo te fbe te flte effect of te uoud tue a well a te dffeece te le fbe coducto velocte wll caue te to dffe ubtatally ape let ad apltude. Eve be dffeece ca be expected betwee te MUAP of dffeet MU. A aleady explaed te MU dffe bot te ube ad te type of ucle fbe. Fat MU wll ave ote ad e AP ta te low oe. Altou be abuou ad plctly depedet o te ay dde facto te te MUAP well etabled te feld of boedcal al poce a well a te fled of euopyoloy. 9

38 3. EMG decopoto Decopoto of te EMG al to cottuet MUAP ta a poved to be a vey potat clcal ue. Recotucted equece of evato pule povde te ba fo eeac tude ad clcal exaato of MU cotol popete ecutet tatee te-pule teval IPI vaablty ot te MU ycozato ad yoelectcal afetato of fatue to ae jut a few aea of vetato. e ext copute-aded EMG decopoto etod ave bee aly focued o te ta-ucula EMG al [ ]. Be baed o patte ecoto [43] ad clute te te doa o patal flte ad o te-fequecy ad te-cale aaly tey typcally explot te dffeece opoloy of MUAP ad cope two tep. Ftly dvdual MUAP detfed fo te aw te-ucula EMG o fo te te-fequecy/te-cale tafo. Secodly te detfed MUAP clafed ad aed to te bet ft cla of pevou decopoed MUAP ape. e ot fequetly ued eaue clude te coelato ad apltude let ad ape of MUAP. I te cae of ultcael EMG te decopoto eult fo dffeet EMG ecod ca be utually copaed wc fcatly pove te decopoto effcecy. Ufotuately ot of te etod fal we a lae ube of MUAP becoe upepoed te coplete decopoto poble oly f te ube of ultaeouly actve MU elatvely all. O te ote ad eve at low cotacto level uffcetly lae ube of MU cotact aycoouly to poduce ly coplex tefeece patte. Mot ecet patte ecoto decopoto tecque ake ue of te oluto fo ap teoy ad atfcal tellece to cope wt te coplex upepoed MUAP patte [ ]. e a oal to educe te et of poble MUAP cobato ode to peed up te decopoto of te detected patte. ey buld upo dffeet po about te MU f patte to pedct te ot pobable oluto. Atfcal tellece baed etod ave aote ce popety. ey cotatly eae te ow eult ad lea fo te take. A a eult tey ae capable of adapt to te te-vay popete of te detected al. e eult ae qute po ad te ot ucceful appoace ave aleady bee tafeed to te eveyday clcal pactce [95 59]. Ufotuately te ue etcted to te eedle EMG wle te cae of uface EMG al te decopoto fal. I te cae of eedle EMG te cotbuto of MU ae elatvely to wt te apltude of appox. V ad ot about. O te cotay te cae of uface EMG te flte effect of ubcutaeou tue atteuate te opolocal dffeece of MUAP deceae te MUAP apltude to appox. µv wle te MUAP duato ly ceaed up to 5. Moeove te cae of eedle electode we ca electvely obeve te acto potetal of oly a few actve oto ut o eve of a le ucle fbe. I te SEMG cae o te ote ad we deal wt eveal te of actve MU ad te eauable cotbuto fo ote ucle ot be ude te clcal vetato wat ofte efeed to a ucle co-talk [57]. Coequetly uface EMG cope ly coplex patte [65] ad te appoace

39 popoed fo te decopoto of eedle EMG fal. Nevetele te o-vave atue of SEMG ad t wde potetal to offe lobal foato about te ucle actvty opeed ueou applcato aea. e foato extacted fo te SEMG al be exploted eveal dffeet clcal tude aly coceed wt te t of te ucle actvato [54 6] te EMG apltude odulato [36] cae te fequecy cotet of te EMG al [7 55] ad te coducto velocty etato [53]. May attept to eace te le MU foato ad to uppe te co-talk SEMG wee ade te pat. A aleady etoed te ft appoace wc wee baed o te etod fo epaato of eedle EMG oo poved to be effectve. Meawle te SEMG acquto becae a ate developed ad atued eau tecque povd all te equed elablty obute ad epeatablty. Flexble -dety D aay of electode ave bee developed ope te poblte of ecod up to a few te o eve uded of te SEMG eaueet ultaeouly. Futeoe wt te developet of te copute powe oe optcated ad coputatoally coplex epaato tecque capable of poce ad cob te foato fo uc a lae ultcael ecod becae feable. Ao te ot popula oe ae cetaly te etod baed o eual etwok [9 6]. e ecod cla of te uface EMG decopoto appoace cope te tefequecy ad te-cale baed etod. e foe becae a popula tool fo extacto of foato out of te dyac EMG.e. te EMG al ecoded du te dyac ucle cotacto. I uc a cae te ecoded al ca be odelled a otatoay ad te te-fequecy aaly ee to be qute a atual tool fo te. ak te lae ube of actve MU te te-vay f ate ad te coplex actvato ad deactvato patte to accout [5] alo te al ecoded du te oetc cotacto ca be odelled a otatoay. Nevetele te tefequecy baed etod poved to ave lted ucce we t coe to te epaato of uface EMG al. O te ote ad te etod baed o te-cale aaly ty to fd te ote wavelet wc would pefectly eeble te popete of te oe of te detected MUAP [9 48]. e cotucto of uc a pefect wavelet ly ctcal. Ft t deped o te ape of detected MUAP ad ecodly t ly te-cou. Moeove eve f we fd a pefect ote wavelet t ly ulkely tat t wll be able to fcatly uppe te cotbuto fo ote MU. Coequetly te waveletbaed decopoto aed lttle popety. Oly ecetly oe oe optcated decopoto etod ave beu to eee. Explot te advaced bld ouce epaato tecque Secto tey povde ew t of clcal teet a tey fo exaple ake te way towad te eoval of atefact [ ] co-talk uppeo [5] ad te detfcato of te EMG cottuet MUAP [4 6 7]. Nevetele te decopoto of te SEMG al to te cottuet MUAP ta eve at low ucle cotacto ea a vey delcate poce. Altou te aaly of dvdual MU ad te f patte vey

40 potat te clcal electoyoapy [ ] o appoac developed o fa eable te coplete ecotucto baed o SEMG al oly. 3.3 Aued data odel of uface EMG al EMG ecod ca be odelled a a ult-cael lea te-vaat yte a lo a tey ave bee take table cotollable ad tatoay eaueet eo. ple a oetc ucle cotacto ad avod of te appeaace of fatueduced cae. Deal wt apled ult-cael EMG ecod te dcete ft-vaat MIMO odell ot feable. Eac ouce cael put uc MIMO yte codeed a MU evato pule ta te te ucle wle te yte epoe cael coepod to te MUAP a captued by a patal flte pck-up electode. e dvdual EMG eaueet epeet te odel output. e ape of yte epoe te aued MIMO yte ae of o potace. Hece ay popety of MU e.. t dept te ucle tue te ube of fbe ad te coducto velocte etc. a well a te popete of te detecto yte te patal flte ued te-electode dtace etc. fluec te ape of MUAP ca be odelled te yte epoe. A explaed Capte we wll ofte tecae te te MUAP ad yte epoe. Altou t te bot expeo wll deote te ae t we wll pefe to ue te te yte epoe weeve we talk about te BSS ad te te MUAP weeve we talk about te EMG. Let ow j l deote te j-t yte epoe.e. te MUAP of te j-t MU a detected te -t eaueet. Fute uppoe all te yte epoe MUAP of let L aple f te yte epoe dffe let tey ca alway be padded by to eac te ae let. e te cotbuto of te j-t ouce MU to te -t eaueet ca be odelled by te follow covoluto: L l l 3. j l j wee j tad fo te j-t ouce. ak to accout te leaty of te odelled MIMO yte te -t eaueet ca be wtte a a u of cotbuto fo all te ouce MU: N L x l l...m 3. j j l j

41 wee au ultple put ad output vecto otato follow: x [ x... x M ] fo te tapoed vecto of M uface EMG eaueet ad [... N ] fo te vecto of N pule ouce. We te fluece of oe codeed we et: y x ω 3.3 wee y [ y... y M ] deote te tapoed vecto of M oy eaueet ad ω [ ω... ωm ] tad fo te oe vecto. e aued data odel depcted F. 3.. N l l l l N l N l M l M l MN l... x M x x ω ω y y ω M y M Fue 3.: Aued covolutve MIMO yte wt ouce j yte epoe j l eaueet x oe ω ad oy eaueet y. e opeato of covoluto depcted by ectale addto wt ccle. We ave explaed te ape ad te let of te yte epoe deped o ueou paaete detecto yte te dept of te MU te ucle tue te ube of ucle fbe cottut te coepod MU etc.. But te pyolocal popete of te ucle uaatee te yte epoe ae lted te apltude ad let: L l <...M j...n 3.4 j l wc ple te tablty of te odelled MIMO yte. We ca fute aue te aued MIMO yte caual: 3

42 j l l<...m j N. 3.5 ouout t te we wll ofte uppoe te yte put j utually depedet wt te co coelato equal to zeo: H R l da[... N N ] 3.6 wee R deote te coelato atx of ouce cojuate tapoe of te vecto of ouce da[] tad fo te daoal atx wt t paaete o te * daoal ad j j l j j fo te autocoelato fucto of ouce j wt * j a a coplex cojuate value of j. Deal wt te boedcal al te ouce ae ot tctly otooal wle te co coelato adly exceed te to 5 % of te axal poble value. Wee eeded te o-otooalty of ouce wll be caefully outled. e addtve oe ω [ ω... ωm ] cooly odelled a tatoay tepoally ad patally wte zeo-ea Gaua ado poce depedet fo te ouce: H H E[ ω ω ] σ δ I 3.7 wee E[:] tad fo ateatcal expectato δ fo te Dac pule delta fucto σ fo te oe vaace ad I deote te detty atx. 4

43 3.3.. Exteo of te covolutve MIMO vecto fo Covolutve elatop decbed 3. ca alway be expeed te atx fo: wee H tad fo te o called x atx of ze aple epoe j l : x H 3.8 M NL wc cota te ut H M N MN 3.9 wee [ ] L 3. j j j j deote te L vecto of j-t yte epoe. e exteded vecto of ouce take te follow fo: j ]. 3. [... L... N... N L Suppo oe eaueet ta ouce a potve tee K ca be foud wc atfe: KM > N L K. 3. Exted te vecto of oy eaueet y by K- delayed epetto of eac eaueet: y ]. 3.3 [ y y... y K... ym... ym K Exted te oe vecto ω te ae ae 3.8 evolute to y H ω

44 e x atx H ow of ze KM N L K ad tll cope te yte epoe j l : H H H M H H N MN 3.5 wt K L K atx H j j j j j L j L j L j. 3.6 j L e exteded vecto of ouce ow yeld: ]. 3.7 [... L K... N... N L K I 3.8 ad 3.4 we tetoally ued te ae otato fo te two dffeet x atce. toduced oe level of abuty ou otato. Howeve ode to keep te equato ple ad clea te x atx wll alway be deoted oly by H. We eeded te facto K wll be defed explctly. e exteo of Eq. 3.8 to te ove-deteed yte of Eq. 3.4 ot a ovato. I fact t a coo appoac we deal wt te covolutve xtue of ouce ad a bee extevely ued te pat [46]. Fo te claty eao we befly explaed oly t a dea. 6

45 3.3.. Abute of bld ouce epaato e oal bld ouce epaato to detfy te x atx H ad to ecotuct te ouce ve oly te vecto of eaueet x. Howeve te bld epaato de two plct abute [8 9 46]. BSS Abuty : It poble to detee te ode of te ouce becaue wt bot te vecto of ouce ad te x atx H ukow ay peutato of ouce ad colu H a oluto of 3.4 [46]. BSS Abuty : Wtout ay fute foato t poble to detee te vaace eee of ouce becaue ay cala ultple ca be excaed betwee a abtay ouce ad te coepod colu of te x atx H: K N L x H α 3.8 α wee α deote a abtay cala ultple ad tad fo te -t colu of x atx H. Altou dffeet cala facto α cae te value of bot te exteded ouce ad te coepod colu H we wo t cae te deotato. We ae o to ue te deotato H ad eve tou oe ecto te value of te cala facto α wll be exactly defed. If ot explaed dffeetly touout te detato bot abute wll be elected. 7

46 8

47 4. Decopoto of covolutve xtue of ouce oe eaueet ta ouce I pevou capte we toduced te a pcple of bld ouce epaato ad ave a bef ovevew ad evaluato of te tate of te at te decopoto of covolutve xtue. We fute decbed te popete of ua ucle explaed te poce of eeato coducto ad extcto of oto ut acto potetal ad dcued te dffeet uface EMG ecod tecque patal flte. We detfed te a poble uface EMG decopoto ad befly evaluated te ext decopoto tecque. Fally we focued o te ult-cael uface EMG ad u te teoloy of te coucato teoy poted out te codto ude wc te uface EMG ca be odelled a a lea ft-vaat MIMO yte. I t capte ad te ext we oe wll deve tee ovel decopoto appoace. We wll ft exae te ovedeteed cae wt oe eaueet ta ouce ad decbe two dffeet way towad ou fal oal te decopoto of covolutve xtue of cloe-to-otooal pule ouce. e ft appoac deved Secto 4. baed o te te-fequecy aaly ad utle te oluto toduced by A. Belouca ad M. A [4]. We befly dcu te wok ad ow ow te dea ca be exteded to te cae of pule ouce ode to etate bot te x atx H ad te ouce up to te ukow cala facto. e ecod decopoto alot baed o e-ode tattc ad deved Secto 4.. We deotate ow e-ode cuulat ca be appled to a MIMO yte to et a vey obut etato of te yte epoe j l. e td appoac te o called vee coelato baed etod teated te ubequet capte wee we dcu te udedeteed cae te cae wt oe ouce ta eaueet. Fo eac ovel appoac te fluece of oe ad o-otooal ouce alo aalyed. It of potace to ealze tat te ovel appoace aue eeal pule ouce ad do ot deped o te type of te x atx. ey oly uppoe te x atx tatoay ad te aveae te-pule dtace eate ta te let of te yte epoe. e latte cetaly tue te cae of EMG al wee te MU f fequecy uually well below 3 Hz wle te aveae let of MUAP ca be 9

48 etated to -5. e latte uaatee all tee appoace ca be appled to uface EMG al a deotated Capte 6. Let u ow aue te ube of ouce doe ot exceed te ube of eaueet ad deve te ft decopoto appoac te o called etod baed o tefequecy dtbuto. 4. Metod baed o te-fequecy dtbuto e-fequecy F aaly oe of te ot fequetly ued tool te fled of al poce. It alo play a potat ole bld ouce epaato becaue t eable u to epaate ouce wt Gaua dtbuto. Altou t ot fequetly ued fo epaato of ultplcatve xtue [ ] t beco oe ad oe potat alo we t coe to te epaato of covolutve xtue [ ]. Vey effcet appoac to epaato of ultplcatve xtue of otatoay ouce wa toduced by A. Belouca ad M. A [4]. e etod explot te dffeece eey locato of ouce F doa ad baed o te jot daoalzato of patal atce wc ae cotucted fo te te-fequecy dtbuto of all poble pa of eaueet. Sla appoac wa alo ueted fo te covolutve cae [7 84]. By exted te vecto of eaueet wt te delayed epetto of eac eaueet Subecto 3.3. evey fte covolutve xtue of ouce ca be ewtte te atx fo. But a decbed te equel t toduce oe ovel dffculte wc pevet te exact ecotucto of covoluted ouce. e ecto oazed fve ubecto. I Subecto 4.. we befly defe te te-fequecy dtbuto. Subecto 4.. uaze te a dea of te appoac tat wa toduced by A. Belouca ad M. A alo wt t a po ad co. e exteo of te appoac to te cae of pule ouce ve Subecto 4..3 ad We tudy te fluece of o-otooal ouce ad toduce oe eaue ad pocedue fo oveco te dtub fluece. We coclude ou dcuo Subecto 4..5 wee te fluece of oe caefully outled alo wt oe poble couteeaue. 3

49 4.. Dcete te-fequecy dtbuto e dcete F dtbuto of Coe cla fo al x defed a [37]: D x x f φ l x l x l e l * 4π f l 4. * wee ad f deote te ad fequecy dce epectvely x coplex cojuated value of x ad φ l tad fo te keel tat caactee te F dtbuto. Co F dtbuto of al x ad x j defed a [37]: D f φ l x l x l e. 4. x x j l * j 4π f l U defto 4. ad 4. we cotuct te o called patal F dtbuto SFD atce of exteded eaueet x [4]: D x x f Θ l x l x l e l H 4π f l 4.3 wee H deote atx cojuate tapoe. D xx f a fou-deoal data tuctue woe j-t eleet equal te dcete co- te-fequecy dtbuto of eaueet x ad x j : [ f ] D f D x x fo j...n. 4.4 j x x j Nelect te oe ad tak to accout te leaty of te aued MIMO yte we ca wte: Dx x f HD f H 4.5 wee fo evey fxed dex pa f D f deote te SFD atx of exteded ouce. I te equel te off-daoal eleet of D f atce value of co-f dtbuto wll be efeed to a cote wle te daoal eleet of D f wll be called autote. A a eult atce D f wll be daoal f ad oly f all te cote wll be equal to zeo [73]. 3

50 4.. Sepaato of eeal ouce e bld epaato appoac toduced by A. Belouca ad M. A [4] baed o aupto of oe eaueet ta ouce ad cope two tep. I te ft tep te exteded vecto of eaueet x defed ubecto 3.3. wteed by te KM N K L atx B wc atfe * B l x x B BRx B BHRH B I 4.6 wee R tad fo te coelato atx of exteded ouce ad coelato atx of exteded eaueet. Deot te atx quae oot of R x fo te R by R t obvou fo 4.6 tat U BHR a utay atx of ze N K L N K L. e wte tep a teefoe educed te x atx H of ze KM N K L to te utay atx U of ze N K L N K L. e B atx ca be cotucted a a atx quae oot of te vee of te R x atx [3 4]. Fo te wteed vecto of exteded ouce z te leaty of te odelled MIMO yte tll old: z Bx BH UR. 4.7 Multply te D xx f atce by B yeld te wteed SFD atce of exteded eaueet [4 7]: D zz f BD f B UR D f R U. 4.8 xx Eve we te ouce ae otooal te exteded ouce do ot eed to be ad te atce R ad R D f R ae eeal block daoal. Accod to 4.8 ay wteed SFD atx D zz f block daoal te ba of te colu of U ad te utay atx U ca be eteved by block jot daoalzato of et of D zz f atce dffeet F pot f [ 7]. e alot of jot block daoalzato a eealzato of well-kow Jacob tecque fo daoalzato of Heta atce ad decbed oe detal []. Kow te atce B ad U te x atx H ca be etated a [3 4]: # H B U 4.9 3

51 wee # deote Mooe-Peoe peudovee. Howeve due to t eet abuty te jot block daoalzato tecque toduce a ukow block daoal utay atx D [ 7]. eefoe te oal ouce ca be ecotucted oly up to te ukow flte effect [7]: H e U Bx DR 4. wee atce D ad R ae ukow Bld epaato of pule ouce A. Belouca ad M. A codeed eeal otooal ouce wt ootooal delayed epetto of eac ouce. O te ote ad te eey of pule ouce cocetated aoud te ukow te oet pule. Futeoe deal wt te boedcal al uc a te EMG te uppe lt of te MU f fequecy Secto 3.. uaatee alo te exteded ouce ae otooal a lo a te axal delay eaueet doe ot exceed te al te-pule teval. Code te BSS abuty te coelato atx of exteded ouce ca ow be wtte a: H R l I. 4. elate te ukow atce R ad R ad 4.. e oly ukow left te block daoal atx D 4. wc toduced by te jot block daoalzato [ 7]. Block daoalzato ca ow be eplaced by jot daoalzato [5 3] wat would eable ecotucto of ouce up to a cala facto. Howeve to be able to ue te jot daoalzato we ut eue te SFD atce D f wc ete te ecotucto of ukow atx U ae alo daoal. Geeally peak te D f atce ae eve te cae of pule ouce block daoal. e eao dde te keel φ l of F dtbuto wc ued to aveae te ouce te ad ece pead te foato aoud pule. Exclud te keel φ l 4. yeld te o called We-Vlle F dtbuto D f WV f x l x e. 4. x x j x x j l * j 4π f l But ew poble ae. e We-Vlle pecta uffe etvty to te cote value of co F dtbuto wc aa ake te ouce SFD 33

52 atce block daoal. Naely te co We-Wlle dtbuto of te -t ad j-t ouce a abtay te pot k a uato of all te pule fo te -t ad j- t ouce wc atfy te follow elato: 4.3 k p j q wee p q ad p j q deote te te oet wc te p-t q-t pule of te -t j-t ouce appeaed. I ote wod te co F dtbuto WV j f dffe fo zeo evey te oet wc le betwee te abtay pule of te -t ad j-t ouce. We ca educe te ube of pule cotbut to WV j f by k te calculato of te We-Wlle dtbuto 4. to te fte teval [-aa]: D x x j f PWV x x j f a x l x la * j e 4π f l 4.4 wee a deote a abtaly lae potve tee ube. Sett te lt a to zeo all te cote ae left out ad te D f atce be to ow te daoal tuctue. e F dtbuto 4. ow yeld: * j D f PWV x x. 4.5 x x j x x j Hav eued te daoalty of te D f atce we ca ue te jot daoalzato [ ] of a et of PWV xx atce to ecotuct te utay atx U ad te ouce up to a cala facto: wee A ukow daoal atx. H U Bx A Decopoto of cloe-to-otooal ouce I te pevou ubecto we uppoed te ouce tctly otooal. ple eveal etcto tat ae ad to eet ealty. Poce te EMG al fo exaple te evato pule ta wll be depedet oly at vey low level of ucle cotacto. We te cotacto level ceae te evato ta becoe oe ad oe coelated up to % o 5 % ad te pule be to ovelap. Moeove by exted te vecto of eaueet Subecto 3.3. we ave vtually ceaed te ube of ouce fo NL to NLK- wc fute deceae te aued otooalty of ouce. 34

53 Let G { j... } j p deote a et of dce of ouce wc all te a ve te oet tey all ae te ovelapped pule at. D f atx wll be fa fo daoal a t wll ave p eleet dffeet fo zeo all autote ad all cote at te poto tat ae cotaed G. A lae ube of uc atce wll caue te jo daoalzato to ecotuct a wo utay atx U. o be exact u te eedecopoto f U ΛU we ca wte D Λ Λ D f UD f U zz UU Λ ΛU Λ U 4.7 wc pove tat te daoalzato of D f poduce te utay atx UU Λ. zz ee yet aote codto tat a to be et we u te jot daoalzato. o uaatee te uquee of te ecotucted utay atx U te et of SFD atce ete te jot daoalzato ut cota te cotbuto of all actve ouce clud te delayed epetto [6 73]. But to uaatee te daoalty te colu-pace of atx U te atce ete te jot daoalzato ut cota te cotbuto of a le ouce oly. ple tat we ut daoalze at leat N L K SFD atce D f oe fo eac ouce. zz I ote wod to uaatee te ecotucto of oal x atx ad coequetly te ecotucto of oal ouce we ut be able ot oly to dtu betwee te daoal ad odaoal D f atce but alo to dtu te cotbuto of dffeet ouce. Wt utay atx U ukow we ca oly ely o te foato cotaed te D f atce. Let D f deote te daoal SFD atx of ouce wt a le ozeo autote d at te -t daoal poto.e. wt cotbuto fo te -t exteded ouce. Bea d tat U utay we ca deve te follow elato: j j zz Dz z f Dzz f UD f D f U f j. 4.8 j Deote by D f te SFD atx of ouce wt cotbuto of te -t ad j-t j exteded ouce. D f wll ave fou ozeo eleet d d j d j ad d jj at te poto j j ad jj epectvely. We deve D j z z zz j f D f UD f D f U 4.9 ad j j j j Dz z f Dzz f UD f D f U

54 Accod to ad 4. te SFD atce of ouce ad ece alo te SFD atce of wteed eaueet wt a cotbuto fo a le ouce fo te otooal ba of N L K -deoal pace. Defe te follow eaue: M M tace MM 4. tace M wee tacem deote te u of daoal eleet of atx M te atx tace. We ca te ealy ceck te follow elato: j zz j tace D f D f D zz f D zz f d 4. tace D f zz zz ad D j zz j zz j zz j zz j tace D f D f f D zz f d tace D f jj 4.3 wee d ad d jj tad fo te -t ad jj-t eleet of D f epectvely. e quet fo D zz f atce wt cotbuto fo le but dffeet ouce a ow educed to uc eae poble of fd te otooal ba of te N L K -deoal pace. a vey potat eult a te cteo fucto 4. eable u to cotol te D zz f atce ete te jot daoalzato ad coequetly uaatee te uquee of atx U. We ut tll eue te et of te D zz f atce ete te jot daoalzato cota te atce wt cotbuto fo all te ouce. Suppoe ow we fal to fd te atce wt a le ozeo autote of te -t ouce. We ae ta foced to clude te o-optal D zz f atce wt cotbuto of te -t ad eveal ote ouce. I ode to ze te devatat fluece of cote [73] we ave to eue te ube of ouce cotbut to te elected D zz f atce a all a poble. We ca tll ely o cteo fucto 4. ad o te fact tat te tace of a atx equal to te u of t daoal eleet: j z tace D z f d d. 4.4 jj j e electo of te optal D zz f atce u te cteo 4. decbed oe detal [7]. 36

55 4..5 Noe fluece Mz te teval [aa] o wc te We-Vlle dtbuto calculated we aulled te te avea of eaueet ad ece lot te avea of te zeoea oe. A a eult we educed te obute of te peeted appoac ad expoed t to te fluece of oe [ ]. e te avea ca be patally eplaced by avea of te D zz f atce wt cotbuto fo a le ot ecealy te ae ouce: wee [ f D f ] U D f U D zz j ωω j Dωω j f 4.5 j j j D ω ω f Bω ω B 4.6 deote te fluece of oe ω. Recall te oe odelled a Gaua wte ad zeo-ea Subecto 3.4: I ω N σ 4.7 wee tad fo N L K vecto wt all eleet equal to zeo wle te ould be ead a»dtbuted accod to«. e te oe fluece 4.5 ca be expeed a: l D ωω q f σ BB. 4.8 e oe vaace σ ca be etated fo te coelato atx R x te exact pocedue explaed Secto 5. wle te wte atx B kow. allow u to ply ubtact te fluece of oe fo 4.5. e fluece of oe ca fute be educed by avea te ecotucted ouce H U Bx. 4.9 A tated 3.7 te peeted decopoto appoac wll ecotuct LK- delayed epetto of eac ouce wc ca be aled te ad aveaed aftewad. Nevetele te oe wll alo fluece te cteo 4. wc uaatee te uquee of te ecotucted atx U. We ca teefoe expect a dop of pefoace te evoet wt to oe ad eavly ovelapped ouce. 37

56 4. W-lce ad e-ode cuulat e teoy of e-ode tattc HOS a bee well etabled alo te feld of MIMO yte [ ]. It a bee ow tat te yte epoe ca be decovolved weeve te yte put ca be odelled a a depedet detcally dtbuted..d. zeo-ea ado wte oe [ 66 67]. I addto to te yte detfcato HOS ae fequetly appled to te feld of ada ad oa al ae poce eoloy ad alo peec aaly. Altou uffe fo a vey coputatoal coplexty HOS-baed etod ave ay ce popete. ey ca be uccefully appled we te ube of ouce ly exceed te ube of eaueet wle te upepoto of pule epoe do ot de a toou decopoto. Moeove by be etve to all Gaua ado pocee te cuulat of ode e ta two ae alo eetly etat to te Gaua oe. I te lat decade te ue of HOS a expaded to te feld of bld ouce epaato [ ]. Suppo te ouce depedet o-gaua ado pocee te cetal lt teoe aue te dtbuto of te u covee towad te Gaua dtbuto. Becaue all e-ode cuulat of te Gaua poce equal zeo te HOS ca be ued a a eaue of te oauaty of te xed ouce ad ece eve a a bac cteo ouce decoupl. dea a bee ued ueou bld ouce epaato appoace. e copeeve ovevew ca be foud [4 84]. I t capte we wll tudy oe of ay poble oute towad te HOS-baed MIMO yte detfcato te o called w-lce. e etod baed o Balett-Blle- Roeblatt equato [] ad eable ecotucto of te yte epoe. e fact tat te pule ouce caot be odelled a..d. wte ado oe doe ot coupt te cuulat a lo a tey ae coputed at te la tat do ot exceed te u te-pule dtace. A aleady explaed deal wt boedcal al te u te-pule dtace exceed te let of te odelled yte epoe.e. te let of MUAP Secto 3... I te ubequet ubecto we ft befly outle te defto of e-ode cuulat. We expla te teoy bed te w-lce ad exted t to te feld of te MIMO yte. We alo dcu t applcato to te covolutve xtue of pule ouce ad deotate t capablty to et te Gaua oe. We fute ow tat w-lce ca be ued to copute oly a coae etato of yte epoe ad decbe te Newto-Gau etod [6] fo te optato. We coclude te capte wt te dcuo about te fluece of te oe ad of ootooal ouce. 38

57 4.. He-ode tattc Gve a et of ado vaable X... X M te jot cuulat of ode defed by cu X k... X k M M j l Φ ω... ω k k ω... M ω M M ω ωm 4.3 wee k km ad Φ ω... ω M deote te jot caactetc fucto of te vaable X... X M : j ω X ω X M Φ ω... ω E[ e ]. 4.3 M Fo a tatoay dcete te ado poce X te cuulat of ode defed a a cuulat of te ado vaable X X... X : C ω M X X... cu X X... X. 4.3 e cuulat defto 4.3 ly teoetcal. I ode to calculate te exact value of cuulat oe ould kow te dtbuto of te obeved ado vaable. I pactce we ofte ely o te fact tat te -t ode cuulat ca be expeed wt te oet of te -t ad lowe ode. Suppo X zeo-ea te td ode cuulat fo exaple ca be wtte a: C 3 X cu X X X l X X X Suppoe ow te put ouce ou MIMO yte..d. wte ad zeo-ea ado pocee. Accod to te Balett-Blle-Roeblatt equato [] te cuulat of te eaueet x ca be wtte a: C x... x... cu x N j γ x j... x M L j l j l j l M l 4.34 wee γ j deote te cetal value te -t ode cuulat atx of ouce S j : γ C j j j 39

58 4 Let Hl deote te atx wc cope all yte epoe: l l l l l l l l l l MN M M N N H Accod to 3. we ca wte l l L l ω H x Defe ow te -t atx of te td ode cuulat of eaueet wee M coepod to te fxed eaueet: M M M M M M x x x x x x x x x x x x x x x x x x x x x x x x x x x C C C C C C C C C C U 4.34 ad 4.38 te C atx ca be ewtte a [66] L l N l l l H Γ H C 4.39 wee Γ...N deote te cuulat atx of ouce wt a le ozeo eleet 3 γ at daoal poto : 3 γ Γ 4.4 C a fou-deoal tuctue wc cota td-ode auto-cuulat of eaueet x ad all co-cuulat wt t eaueet. Accod to 4.38 ad 4.39 te cuulat ca be calculated two dffeet way. ey ca be etated out of te yte output 4.38 o deteed fo te odel If te two atx equato ae ade equal at fxed la a yte of o-lea equato appea woe oluto lead to ukow yte epoe j l ad put kewe 3 γ. If te yte epoe ae of let L te total ube of ukow equal

59 NML N 4.4 wee M tad fo te ube of eaueet N fo te ube of ouce ad L fo te let of yte epoe. e ft te 4.4 tad fo te ukow aple epoe j l ad te ecod fo te ukow kewee γ 3 te put ae uppoed tepoally ad patally ucoelated. O te ote ad we eed at leat tat ay o-lea equato a we ave ukow. e ube of poble equato deped o dffeet value obtaable C. L L I eeal te td-ode auto-cuulat atx C x x x eeate dffeet value te co-cuulat atx baed o dffeet eaueet L 3L C x x x cotbute wle te co-cuulat atx baed o 3 j dffeet eaueet cotbute 3 L L dffeet value. We deal wt a MIMO yte wt N put ad M output te total ube of dffeet td-ode cuulat atce C a follow: auto-cuulat atx M- cocuulat atce baed o eaueet ad co-cuulat M M atce baed o 3 eaueet. u te total ube of dffeet cuulat value C yeld: L L L3L M M M 3L L L [3M M ] [ M 7 3M 3] L [ M 3M ]. 4.4 e yte of olea equato ca be eolved ad toouly decopoed weeve 4.4 copute e ta 4.4. I pte of t teoetcal eult all te eo accopay te cuulat etato de ucceful cloed-fo decopoto pactce. It a bee ow oweve tat a oluto ay be foud tou o-lea optato [6]. e coepod eac fo te lobal optu eed a talato pot wc be te optato te vcty of te lobal oluto. I ote wod to fd te yte cael epoe.e. te MUAP ou cae a cuulat-baed optato ut be wt a ood appoxato of tee epoe a poble. 4

60 4 4.. Coae etato of yte epoe: w-lce Ao vaou appoace av bee popoed fo MIMO yte detfcato we wat to ely upo toe wt a lttle a po foato eeded a poble. Oe of te o called w-lce etod [55]. Bede te u ube of yte cael to be odelled te oly addtoal foato eeded t appoac about te value of te tat aple of yte-cael epoe a we wll ee te equel. MIMO w-lce etod baed o a peupto tat uc a atx of wet MM M M w w w w W 4.43 ext tat te follow elatop ca be deved: l l L l G H W C 4.44 wee l l l l l l N δ δ δ W H Γ G 4.45 ad l δ deote te Dac pule. Fo 4.44 t follow MN M M N N L l l l G H W C Hece te caual pat of atx C ca be ued to detfy te yte epoe. Recall te yte epoe ae caual Subecto 3.4: l j l<...m j N. 4.47

61 eefoe fo te at-caual pat of te atx C Eq yeld: C W L H l G l l M M N N MN 4.48 we ad L < C W H l G l l 4.49 we <. Let te dce exted ove te let of te yte epoe L L. Fute let C a jk Cx C x j xk x x j xk L L L C x x j xk C x x j xk L L L 4.5 deote te at-caual ad C c jk C C x x x x j j x x k k L L L C C x x x x j j x k x L k L L 4.5 te caual pat of ete cuulat atx C. Accod to 4.48 ad 4.49 we ca wte: x x j xk a C a C M C C a M a MM a W a M a N a MN 4.5 wee a j j j j [... L ] [...] deote te at-caual pat of te yte epoe l wle j j 43

62 w W w M w w M MM 4.53 wt w [ w L w L... w L ] 4.54 j j j tad fo te ete atx of wet. I te ae way te caual pat of 4.46 ca be ewtte a: j c C c C M C C c M c MM c W c M c N c MN 4.55 wee c j j j [ L... ] tad fo te caual pat of te yte epoe j l. e eult 4.5 ad 4.55 ae depedet of te fxed eaueet. Hece te dffeet ow of te left-ot atx 4.5 ad 4.55 ca be cotucted out of te cuulat wt te dffeet fxed eaueet. Cobe ow te at-caual ad caual pat of all auto-cuulat ad all co-cuulat baed o two dffeet eaueet to te atce S a ad S c epectvely: S a a C a C a C MM C C C a a a MM C C C a M a M a MMM 4.56 S c c C c C c C MM C C C c c c MM C C C c M c M c MMM 4.57 Accod to 4.5 ad 4.55 te atx of wet W ca be coputed a: W S E E E # a S # a D

63 wee E [...] ad [...] tad fo te vecto of let L ad # deote te Mooe-Peoe atx peudovee. e t-ad de of 4.58 uaatee te uquee of te wet W oweve t foce te cael epoe j l to be by f j ad by otewe. Fally te yte epoe ca be calculated a: c c c M c c c M c N c c N MN S c W 4.59 wee c j j j [ L... ] tad fo te caual pat of te yte epoe j l. Matce S c ad S a cope oly auto-cuulat ad co-cuulat baed o te pa of dffeet eaueet. A aleady explaed te dffeet ow of atce S c ad S a ca be foed by calculat te cuulat wt dffeet fxed eaueet. By elect all poble value of 4.56 ad 4.57 we ave ply ceaed te coputatoal tablty a all dffeet cuulat wee ued to etate te yte epoe Nolea optato of coae etate e coae appoxato of te yte-cael epoe ve by 4.59 ca fute be poved by o-lea optato a popoed [6]. e coe of te etod Newto-Gau teato wc coplete o-lea ato of cot fucto V θˆ : [ cˆ θ cˆ ] Σ [ cˆ θ cˆ ] V ˆ θ 4.6 wee te c ad ĉ vecto ae obtaed by plac te tapoed ow of te atx C C C C C MM C C C C MM M M C MMM 4.6 a C jk colu oe bellow te ote. C jk c 4.6 deote te ete cuulat Cjk atx of eaueet j ad k. e vecto ĉ cotucted wt te cuulat C jk etated accod to 4.38 fo te yte output tee ae codeed te efeece 45

64 value to be ftted te optato pocedue wle c θˆ coepod to te cuulat coputed fo 4.39 at eac tep of teato te value ae be optzed. e vecto of paaete [ L L... L γ ] θˆ MN MN 3... γ cobe te value of te yte epoe ad put kewee etated at a ceta tep of teato wle Σ tad fo te ayptotcally oalzed covaace atx of eo [6 68]: Σ l [ E{ˆ c cˆ} cˆ θ cˆ θ ] 3 N wee E deote ateatcal expectato ad te let of MIMO output. I pactce te atx Σ eplaced by t etate [ {ˆ ˆ } ˆ ˆ] P c c c θ c θ p Σˆ N 4.64 P p P p P wee dce p P dcate tat cuulat ĉ p P ae calculated accod to 4.33 oly ove te aple fo p to p ad P a potve abtay lae P P tee but uc alle ta. Mzato of te cot fucto 4.6 caed out by teat te Newto-Gau update foula utl te coveece eaced [6]: ˆ ˆ cˆ θ ˆ cˆ θ cˆ θ ˆ θ θ δ Σ Σ ˆ c cˆ θ θˆ ˆ 4.65 ˆ θ θ cˆ θ wee deote te Jacoba atx of c θˆ ad Σ ˆ te atx vee of θˆ pojecto of Σ oto te pace of potve e-defte atce. Σ ˆ calculated by ett all te eatve eevalue of atx Σ to zeo wle te potve eevalue λ ae eplaced by te ecpocal value [68 5]. e tep ze δ deteed λ by u a le eac alot at eac teato e.. te etod of becto cubc tepolato Newto etod etc. wle te devatve fo te Jacoba atx cˆ θ ae defed Appedx 4A. θˆ 46

65 Iput kewee γ caot be optzed by Newto-Gau teato. e eao tat te devatve of cuulat wt epect to te kewee ca be expeed a a lea cobato of devatve wt epect to yte epoe: cˆ θ γ j 3γ j M q j cˆ θ j Iclud te ukow kewee γ to te optato teato caue te Jacoba cˆ θ atx to becoe ak-defcet ad pevet te extece of te vee atx θˆ cˆ θ ˆ ˆ c θ Σ ˆ ˆ 4.6. Itead te ukow γ ould be etated fo te θ θ cuulat 4.38 ad 4.39: γ S jk θˆ cˆ jk 4.67 γ cobe te kewee of all put. Matx θ ˆ wee vecto [ γ ] γ N cot of te auto- ad co-cuulat of yte epoe: S jk S jk L l j l k l l ˆ θ L l j l L k l l L N l jn l kn l l 4.68 L N l jn l L kn l l ˆjk wle te vecto c cobe L cuulat of eaueet: cˆ jk [ cu x xj xk... cu x xj L xk ] 4.69 wt abtay eaueet j ad k j k [... M]. Altou te put kewee [ γ ] γ γ N ca be etated at te optoal value of dce j k ad la t advable to epeat te calculato fo eveal dffeet et of value ad to aveae te eult aftewad. 47

66 Ifluece of o-gaua oe ad ootooal ouce e w-lce ad Newto-Gau optzato ae baed o te Balett-Blle- Roeblatt equato. e latte uppoe all te co-cuulat of dffeet ouce equal to zeo.e. t aue te ouce ae depedet. We te ouce pule ovelap te co-cuulat ca ot be elected ay oe ad te Balett-Blle- Roeblatt equato fo 4.34 yeld: N j N j L l L l x x j j l l l l l l l C x x x cu C wee M l l C jm j deote co- cuulat of ouce j jm.... A ceae of te ube of ovelapped pule ceae te value of co-cuulat ad coequetly te dffeece betwee te cuulat 4.38 ad A a eult te w-lce ad Newto-Gau wll ty to atc Eq ad 4.7 wc wll lead to a wo etato of te yte epoe. A la effect ca be obeved te peece of o-gaua oe M ]... [ ω ω ω. Suppo ω ae depedet of ouce te Balett- Blle-Roeblatt equato yeld: N j L l j j j x x j C l l l x x cu C γ ω ω ω ω ω ω 4.7 wee C ω ω deote te cuulat of oe.

67 49 Appedx 4A: Devatve of cuulat Let N q kq L l jq q q jk l l l c γ 4A. deote te eeal co- cuulat value at te abtay la ad ad obeve all poble devatve c q jk j k. We dtu tee dffeet cae: a devatve of te auto-cuulat.e. jk: ]; [ γ c q q q q q q q q 4A. b devatve of co-cuulat baed o two dffeet eaueet.e j k jk o j k: j k [ ] γ c q q q q q q k 4A.3 [ ] γ c q q q kq k 4A.4 k j [ ] γ c jq q jq q q q j 4A.5 [ ] γ c q q q jq j 4A.6

68 5 jk [ ] γ c jq jq q q jj 4A.7 [ ] γ c jq q jq q q jq jj ; 4A.8 c devatve of co-cuulat baed o tee dffeet eaueet.e. j k: [ ] γ c kq jq q q jk 4A.9 [ ] γ c kq q q jq jk 4A. [ ] γ c jq q q kq jk. 4A.

69 5. Ivee coelato baed etod I pevou capte two ovel appoace to bld ouce epaato of covolutve xtue wee toduced te etod of te-fequecy dtbuto ad w-lce. Ft we owed ow jot daoalzato of SFD atce ca be utlzed to etate te x atx ad pule ouce up to a cala facto. We detfed te devatat fluece of cote ad explaed ow tey ca be avoded. e two a dawback of te peeted etod tued up to be t ablty to cofot te eavy oe ad te eed to fd at leat oe o-ovelapped pule of eac ouce. Wle te ft poble ca patally be olved by avea te SFD atce te ecod poble uc oe eou. Keep te ube of ouce all ad ouce otooal te tak of elect te ot-ovelapped pule tval. I te cae of cloe-to-otooal ouce ad al-to-oe ato te caddate pule ca be elected u te cteo 4.. But by cea te ube of ouce te pobablty of fd te oovelapped pule dop datcally ad te effcecy of te peeted appoac deceae. Moeove be etve to oe te cteo 4. alo caue a dop of pefoace oy evoet. O te ote ad be baed o HOS w-lce ad Newto-Gau optato ae eetly etat to te Gaua oe. Ufotuately tey ca oly be ued to ecotuct te yte epoe wle te foato about te ouce lot te decopoto poce. e obvou dawback of all HOS-baed decopoto tecque alo te coputato coplexty. e teoetcally expected beavou of te cuulat calculated out of te eaueet ca oly be obeved f te poce al ae at leat eveal aple lo. A a eult HOS-baed etod ae bot te ad eoy dead. e etod toduced te pevou capte ae aote coo dawback. ey ae baed o te aupto of oe eaueet ta ouce. Altou vey coo BSS t aupto caot alway be etabled pactce. Uually we eve do t kow te exact ube of ouce ad eve f we dd t oete ad to eue eou eaueet. I te cae of uface EMG al te axu ze of te aay of electode lted by te ze of te vetated ucle. Reduc te teelectode dtace te dffeece ao te detected MUAP d ad te x atx becoe ak-defcet. O te ote ad tee ca ealy be eveal te of MU actve te detecto volue. I t capte we dcu ow te decopoto ca be exteded to te udedeteed cae.e. we te ube of ouce exceed te ube of eaueet. 5

70 Real t uc eae to etate te bay ouce ta te x atx eeal we ft toduce a copletely ew decopoto cocept. ouout t devato we oe all te foato cotaed te x atx ad focu tctly o te popete of pule ouce. I ote wod we ty to cacel te tefe effect of te x atx. A a eult we oly aae to ecotuct te etate of ouce wle te x atx copletely lot. We fute ow t ot a ltato at all becaue te yte epoe ca alway be etated by avea te eaueet te vcty of te ecotucted pule. e capte dvded fou ecto. I Secto 5. we ft code te ouce tctly otooal. We alo aue oe eaueet ta ouce ad deve a ovel decopoto appoac te o called»vee coelato IC baed etod«. e decopoto of o-otooal ouce toduced Secto 5. wle Secto 5.3 evaluate te fluece of oe. e lat ecto dcue te pact of te ube of eaueet ad exted te ovel appoac to te cae of ude-deteed MIMO yte.e. te yte wt oe ouce ta eaueet. 5. Decopoto of otooal ouce Suppoe te ube of ouce N alle ta te ube of eaueet M ad exted te vecto of oy eaueet by te delayed epetto Secto Fute uppoe te exteded ouce otooal ad deote by R te coelato atx: H R l da[... N K L N K L ] 5. wee da[] tad fo te daoal atx wt t paaete o te daoal * j j l j j te autocoelato fucto of ouce j ad * j deote te coplex cojuate value of j. Let y x ω deote te vecto of oy eaueet ad ecall te x atx H a KM N L K ectaula atx wc coepod to te ovedeteed yte of equato. Accod to 3. tee ae at leat KM N L K eevalue of R x equal to zeo. Coequetly te oe vaace ˆ σ ca be etated by avea te KM N L K allet eevalue of R y. Subtact ˆ σ fo R y we obta te etato of coelato atx of eaueet: R x R y ˆ σ I HR H. 5. 5

71 53 Suppo te x atx H of full colu ak: K L N a H 5.3 ad ultply te vecto of te exteded eaueet wt Mooe-Peoe peudovee of x R we toduce te o called actvty dex # # # # o o o o o Id K L N j j j j j ω ω ω ω ω x R H H R H H H H HR H x R x 5.4 wee # deote te Mooe-Peoe atx peudovee ad o ω eplace te pact of oe. I 5.4 we alo ued te equalty # R R wc eeally vald oly fo o-ula atce. Nelect te fluece of oe te actvty dex Id could be tout of a a dcato of a lobal ouce actvty. It dffe fo zeo oly at te te tat wee at leat oe ouce actve. Sce te ouce ae otooal tee ca be at ot oe actve ouce evey te oet. We a aple dex foud wc te j-t ouce actve te ete pule ta of te j-t ouce ca be ecotucted a: # o o o v j j j j ω ω ω x R x R x 5.5 au tat j. Accod to 5.5 te pule ta of dffeet ouce ca be ecotucted by ply coo dffeet te oet. Kow te pule ouce te yte epoe ca be ecotucted by avea coepod eaueet te vcty of te pule [46]: Ψ F k k j j l x l F l ˆ 5.6 wee k tad fo te te oet coepod to te k-t ecotucted pule of te -t ouce F deote te ube of ecotucted pule of te -t ouce wle l Ψ a abtay wdow fucto. pocedue cooly efeed to a a pke teed ld wdow avea tecque [46].

72 5. Decopoto of o-otooal ouce Suppoe ow cloe-to-otooal ouce.e. ouce wt all but fcat ube of ovelapped pule. ple te coelato atx R a doat daoal but ot tctly daoal. e fact tat tee ca be eveal ouce actve te ae te oet a a cucal pact o te actvty dex. Suppoe tee ae Q ouce actve te ae te oet ad deote te by a et of dce G { j... j } wee G {... K L N}. Nelect te Q fluece of oe te value of te actvty dex te oet yeld: Id R v j j Q v j j 3 j j v j j j 3 j Q j v j j v j j v j j Q Q j j j Q j j v j j j 5.7 v wee j deote te j-t eleet of atx R. Accod to 5.7 te actvty dex at a ve te oet cope te cotbuto of all te ouce tat ae actve tat vey oet. e o-daoal atx R caue te actvty dex alo v cota all poble cobato j j of actve ouce. e ae tue fo te ecotucted pule ta of ouce. U 5.5 te cae of te ovelapp ouce we ca eeally ecotuct oly te upepoto of all te ouce: v Q N L K Q R G q j q v j q q v j q 5.8 wee v deote te -t ow of atx R wle j uppoed equal to. Suppoe ow te atx R daoal doat. e a pove Appedx 5A t vee a a upeo daoal wle all te off-daoal eleet ae uc alle ta te daoal oe. ple te upepoto v G a to cotbuto oly fo te ouce tat ae cotaed et. e cotbuto fo all ote ouce ae uc alle apltude ad ca be eoved by a ple teold opeato. Eq. 5.8 ow deve to: G q v j jj j j j j. 5.9 G v j G j G v 54

73 Accod to 5.9 te ete pule ta of te j-t ouce ca tll be ecotucted povd we ave foud ay te oet wt a cotbuto pule fo oly te j- t ouce. But uc oet ae vey ad to fd. We ca patally ely o te actvty dex a t value ceae wt te ube of ultaeouly actve ouce. Howeve te evee ot eeally tue. A value of te actvty dex ca ete be te coequece of a be ube of ultaeouly actve ouce o ca dcate te actvty of oly oe ouce wt a low f ate. Naely accod to 5.7 te cotbuto fo dffeet ouce ae ultpled by facto v wc ae a ow Appedx 5A popotoal to te vee of te coepod f ate. Coequetly te actvty dex ut be ued wt pecauto ad oe foal pocedue fo epaato of te ouce upepoto eeded. j 5.. Sepaato of te upepoed ouce A explaed Secto 3. te f of a pule ouce ca be odelled by a ado poce. Eac aple te pule ta codeed a ado vaable wt two poble outcoe pule o pule. It pobablty dety fucto deped o te outcoe of te eveal pevou aple ad ece vey dffcult to defe. I te equel we wll ate opeate wt codtoal pobablte. We wll aue tee at leat oe ouce actve te obeved te oet ad defe te pobablty tat two o oe ouce ae actve te ae oet. I ote wod we wll ty to awe te queto of ow pobable t tat te k-t pule of j-t ouce ovelap wt ay ote pule fo ay ote ouce. Suppoe ow te ovelap of two pule fo dffeet ouce a ado evet. It pobablty popotoal to t elatve fequecy of occuece ad at leat te cae of cloe-to-otooal pule ouce elatvely low uually well below.. ay be ee oe eadly by ealz tat te elatve fequecy of ovelap betwee te -t ad j-t ouce popotoal to te j-t eleet of te atx R. e axu ube of ultaeouly teed pule uually uc alle fo te ube of all exteded ouce. I te cae of boedcal al te efactoy peod pevet te delayed epetto of te ae ouce to ovelap. Coequetly te ube of ultaeouly teed pule caot exceed te ube of oal ouce N av N ovelapped pule would ecealy ply tat at leat two epetto of te ae ouce ovelap. Code a oe eeal cae ad aue te ouce ca be aaed to R ubet eac cop Q ouce weea te ouce fo te ae ubet caot ovelap wt eac ote. Defe te ovelap of two ubet a te ovelap of ay two ouce fo dffeet ubet ad aue all ouce wt te ae ubet ae equally pobable to ovelap wt ouce fo te ote ubet. Fo te ake of plcty alo uppoe all poble ovelap of all poble ubet ae depedet equally pobable ado evet ad deote te pobablty by p Appedx 5B. 55

74 We ae ow poto to dcu te epaato of upepoed ouce. U 5.8 ecotuct te upepoto of te pule ta of ouce actve a ve te oet : v j jj 5. G j G wee G { j... j } deote te et of all ouce tat fed. Wtout lo Q of eealty we ca aue te et G ot epty. Radoly elect te pule v ad deote te oet of t occuece wt. Fute deote by G G { j... j } te et of ouce tat wee actve oet. Becaue te Q oet wa coe fo te upepoto v tee at leat oe ouce v G tat actve bot oet ad. ple te teecto at leat oe ouce: G G cota G G {} 5. wee {} deote te epty et. e codtoal pobablty tat ouce f tee at leat oe ouce eac et Appedx 5B: G ad R p G G cad G > cad G > P cad G G cope G ca be etated a 5. Q wee cad G G deote te cadal ube of teecto G G. Accod to 5. te pobablty P cad G G cad G > cad G > deceae apdly wt te ube of expected coo ouce ad alo wt te ube of ouce eac ubet Q. Coequetly te et G ad G wll ot pobably dffe fo eac ote Appedx 5B. e ae cocluo ay apply fo upepoto ad v j jj 5.3 G j G v 56

75 v. 5.4 G G v e upepoto v ad v ovelap all te pule tat belo to te ouce cotaed G G G G. ey wll alo ae toe pule wc ae ovelapped fo te ouce fo G G by te pule of ouce fo G G wee M M tad fo te et dffeece M wtout M. Suppoe te upepoto v G ad v G ae a pule at te oet. ee ae two poble copleetay explaato: Aupto 5.: At leat oe ouce fo G G fed at : G G :. 5.5 ; Aupto 5.: ee o ouce G G wt a pule at. Howeve at leat two dffeet ouce fed at ; te ft cotaed G G : G G te ecod ; G k; k G G G : :. k j; j G G : j 5.6 Let te upepoto v ad v ae aote pule at te oet 3 G G ad aue tat te oet fulfl te Aupto 5. wle at 3 Aupto 5. fulflled. e accod to Aupto 5. te teecto G G G cota at leat oe ouce uc tat: ; G G : G G G {} 5.7 wle Aupto 5. uaatee ; G G : G G G {} 3 3 j 3 3 G : {}. k 3 G G 3 j ; j G G : G G {} k; k G 5.8 U te cocluo fo Appedx 5B we quckly ealze te pobablty of av oe ta oe ouce ay teecto G G ; j becoe elble a oo a j Q te ube of ouce eac ubet exceed te ube of ubet R. I la 57

76 fao te teecto G G G ad 3 G G G ae wt 3 potablty epty et ad te ube of pule te poduct v v v v ca be etated a Appedx 5C Cae : G G G G 3 3 R I pq Fp 5.9 Q wee fo te ake of plcty we aued te total ube of pule fo all te ouce equal F Appedx 5C. Aue ow bot te oet ad 3 fulfl Aupto 5. ad let tee be at leat oe ouce actve all fou te oet: ad 3. e te teecto G G G G cope at leat oe ouce wle cocluo Appedx 5B 3 uaatee te pobablty of av oe ta oe ouce G G G G 3 elble. Aaloouly to te cae above we ca etate te ube of pule te poduct v v v v a Appedx 5C Cae : G G G G 3 3 p I pq F Q Accod to 5.9 ad 5. te upepoed ouce ca be epaated by ply copa te pule v v v ad v v v. G G G G G 3 G 3 G U 5.8 we ft ecotuct te upepoto v all te oet wc coepod to te pule te poduct v G v G. Fo all poble pa j wee j we ceck te ube of pule te poduct v G G G v v v j G 5. ad cobe all te oet ad j fo wc te ube of pule te poduct 5. exceed a pedefed teold F/ a coo et A. I a ate ulkely cae we te te oet ad j ae oe ta oe coo ouce we et eveal et A. Recall te pobablty of av oe ta oe ouce actve eveal dffeet te oet covee to zeo wt te ube of dffeet te oet Appedx 5B. Hece te et A wll cota te te oet coepod to te pule of oe le ay te -t ouce. By avea te vecto of exteded eaueet ove all te oet cop te et A we et: j A x x cad A 5. A 58

77 wee cad A tad fo te cadal ube of et A. Vecto x A wll ave a to cotbuto fo te -t ouce wle te cotbuto of all ote ouce wll ted to zeo a te ube of eleet A wll e ove all lt: l x A He cad A 5.3 wee e tad fo te -t ut o vecto wt a le ozeo eleet at te -t poto. U te vecto x A 5.8 we ecotuct te -t ouce v A x A v R # x o x o ω A. ω A e R o ω A Noe fluece U te atx calculu te oe fluece o ω 5.5 ca be expeed a: o ω # # # x ω H R ω R ω R H ω. 5.5 Aue ow te pocedue fo epaato of upepoed ouce wc wa deved te pevou ecto ucceeded to fd a et of te oet A {...} coepod to te pule of te -t ouce. Let aveae te exteded vecto of oy eaueet ove all te oet A : A y y cad A 5.6 A ad ecall te oe odelled a a wte zeo-ea Gaua ado poce. U te Cae-Rao lowe lt [9] t eay to ow te vaace of te aveae te peece of te wte Gaua oe lted by wee σ tad fo te oe vaace. It tval to ee σ σ y A 5.7 cad A l cad A σ y A

78 6 Accod to 5.8 te fluece of te oe vecto ω 5.5 wll ted to zeo we te vecto A y wll be calculated ove a lae eou et of te oet A. Coequetly 5.5 uet # A o A v v A ω R x y y 5.9 ad te oe fluece A o ω ead: # # # A A cad A A cad A A o ω H R ω R ω R H ω x ω. 5.3 Wle te ft two facto 5.3 covee to zeo te avea vecto A y adly cae te lat te # A ω H R. It pact ca be educed by avea te ecotucto of dffeet epetto of te ae ouce. ee yet aote vey effcet way towad a bette al-to-oe qualty of te eult. We peet t te ext ubecto Noe te pace of ouce e oe fluece ca alo be obeved te pace of ouce. Fo te ake of plcty uppoe te quae x atx H of full ak ad defe. ω x H x H ω 5.3 ak to accout te tadad o equalty H x te fluece of oe te pace of ouce ca be etated a [68 5]: t x ω x ω H H x H ω H ω H ω κ 5.3 wee H H κ tad fo te codto ube of atx H. O te ote ad te equalty x H ple x ω x H H ω H ω ω κ 5.33

79 ece ω κ x ω ω κ x Accod to 5.34 te fluece of oe deped toly o te codto ube κ H H. If H well-codtoed a all oe te pace of eaueet ca ot poduce lae eo te ouce pace but f H ll-codtoed eve a all oe ca eult lae coupto of te ouce pace. A tee ae ay dffeet o tee ae alo ay dffeet atx codto ube. We wll focu oly o te codto ube deved fo te ecod o. It defed a te ato betwee te laet ad te allet ula value of atx H. e ula value of atx H dcate ow uc dtoto ca occu ude te tafoato by H. e laet ula value of atx H eveal te uppe lt to wc te ut vecto ca be aplfed by atx H wle te allet ula value defe t lowe lt. e actual deee of dtoto deped alo o te oetato of te vecto ete te tafoato wt epect to te t ula vecto of atx H. Suppoe te oe vecto ω aled wt te t ula vecto wc coepod to te laet ula value ad te ouce vecto aled wt te t ula vecto wc coepod to te allet ula value. e te elatve pact of oe wll be aplfed by facto κ. Deote by D te daoal atx of ula value of atx H ad by U ad V te atce cop te left ad t ula vecto of atx H epectvely U ad V ae by defto utay: H UDV UU U U I 5.36 VV V V I Fo te ake of plcty uppoe te coelato atx of ouce detty atx ad expe te coelato atx of eaueet R x R x a: R equal to te HR H HH UDV VDU UD U Becaue te atx factoze t to R x yetc ad potve edefte we ca alway uquely 6

80 wee eevalue of atx te follow elato: ad R U x L ΛU L 5.39 UL deote te utay atx of eevecto ad Λ te daoal atx wt te R x. e uquee of te eedecopoto [5] uaatee U U L 5.4 Λ D. 5.4 Accod to 5.4 alte te eevalue of te atx R x cae te ula value of te x atx H. Suppoe ow te eevalue Λ oted a deced ode fo te laet to te allet eevalue ad et te k allet eevalue to zeo: I Λˆ KM k KM k KM k k Λ DJkD k KM k 5.4 k k IKM k KM k KM k k wee J k ad I KM k KM k tad fo te detty atx of k KM k k k ze KM k KM k wle k k a atx wt all eleet equal to zeo. Cotuct te ew coelato atx ad ewte 5.4 to KM Rˆ x : ˆ U ˆ 5.43 R x LΛU L x Rˆ # x x VDU UD J k D U k H UDJ DU H UDV VJ k V otooal pojecto atx wc covee to detty atx a k covee to zeo [68 5]: l VJ kv I k It fluece wll be aeed te ext ecto wee te decopoto of oe eaueet ta ouce wll be tuded. Fo ow t uffce to ay tat cutt te 6

81 eevalue of te atx R x toduce te effect la to ootooal ouce but fcatly pove te codto ube of te x atx H. e optal deee of cutt te value of paaete k deped o te oal codto ube of te x atx H ad o te al-to-oe ato: te toe te oe te e te optal deee of cutt. 5.4 Decopoto of ude-deteed yte I ecto 5. wee te IC etod wa deved we aued te ube of ouce exceed te ube of eaueet. But te ube of ouce uually ad to etate. Suppo te ube of ouce ukow we caot coectly etate te exteo facto K ad we k te udedeteed yte. O te ote ad te ube of ouce ca alway exceed te ube of eaueet. Suppoe te ube of ouce N eate ta te ube of eaueet M M<N o equally te exteo facto K too all to atfy te elato KM > N L K. e KM N L K x atx H ow ectaula ad a oe colu ta ow. Wtout lo of eealty we ca uppoe te x atx H of full ow ak: It decopoto to ula value yeld: a H KM H UDV 5.47 wee U ad V deote utay atce of ze KM KM ad N L K N L K epectvely. D daoal atx wt te ula value of atx H o t daoal. Suppo te ouce ae otooal ad atx 5.5 tafo to R equal te detty atx 5.4 ad Id y R H # x y HH # H o VD DD # ω DV o ω o ω 5.48 ad v # j x ω y R y o

82 I KM KM O # KM N L K KM wee VD DD DV V V O N L K KM KM O N L K KM N L K KM otooal pojecto atx wc poject te exteded ouce vecto to te pace of eaueet. It popete ae decbed Appedx 5D. Wat ca ow be ad about te actvty dex? Becaue te atx potve edefte Appedx 5D te actvty dex tll o-eatve evey poble te oet. But te tepetato of t value tu out to be tcky. I pevou ecto we aw tat av oe eaueet ta ouce te value of te actvty dex ceae wt te ube of ultaeouly actve ouce. ay ot ecealy be tue ay oe te cae of ude-deteed MIMO yte. Let tee be two ouce ad j actve te te oet. e value of te actvty dex te te oet yeld: Id t t t 5.5 j j jj j j wee t j deote te j-t eleet of te atx. Hav tee ouce j ad k actve te oet te value of te actvty dex yeld: Id t j t j j t j k jj t k k k j t kk k t jk. 5.5 Suppo ow te pule of all te ouce equal to ad aue t k / < t ad t <. Copa 5.5 ad 5.5 kk jk t > t > t we ca alway jj kk t t t t t t < t t t t < t t t 5.5 jj kk j k jk jj j jk jj j we ee ow te value of te actvty dex ca deceae altou te ube of actve ouce ceae. Altou pefectly poble t ot a vey lkely cae becaue ay dffeet codto ave to be et. e dex actvty ca teefoe tll be ued a a dcato of lobal actvty of ouce. e expeetal eult ad te teoy Appedx 5D ow te atx wll ave at leat a pat of t daoal eleet upeo to all off-daoal eleet a lo a KM<NLK-<KM. e ouce coepod to te doat daoal eleet of te atx tll obey te teoy fo te ove-deteed cae: v G tj j t G N L K j e pocedue deved Subecto 5. ad 5.3 ca ece eadly be appled alo te cae of ude-deteed MIMO yte. e oly dffeece betwee te ove- G 64

83 deteed ad te ude-deteed cae tu up to be te deee to wc te x atx H cacelled. Wle H copletely lot te ove-deteed cae t tll patally peet te fo of atx te ude-deteed cae. We H cloe to te quae atx cloe to te detty atx. R te ato betwee te ube of ouce ad te ube of eaueet adually loo t daoal fo ad te pact of te x atx ceae Appedx 5D. Nevetele keep te ato KM : N L K below at leat a pat of all ouce wll be copletely ecotucted. cocluo wll be fute cofed by copeeve et of expeet wc ae decbed te ext capte. 65

84 Appedx 5A: Ivee of daoally doat atx Suppoe te ouce coelato atx R of ze N N daoally doat.e. te atx wc te atude of eac daoal ety exceed te u of te atude of te off-daoal ete te coepod ow: wee j > j 5A. j j deote te j-t eleet of atx R. Suppo te ouce cloe-tootooal we alo ave j j : j < ε; ε << 5A. wee ε uually uc alle ta.. e atx atce: R ca alway be wtte a a u of te daoal ad o-daoal R D N 5A.3 wee D da R N N 5A.4 deote te daoal ad N R da R N N N N N N 5A.5 o-daoal pat of te atx expeed a [5]: R. U te Neua ee te vee of R ca be 66

85 67 [ ] D N D D D N D I N D I D N D R k k 5A.6 povd te abolute value of eac eevalue of te atx N N N N N N N N N N D 5A.7 alle ta. Accod to te Geco teoe te eevalue of N D ae cotaed te uo of te Geco ccle defed by [5]: : < N j j j N j j j k k λ 5A.8 wee k λ tad fo k-t eevalue of te atx N D ad te elatop fo 5A. wa ued. U 5A.8 t eay to ow N D k k l. 5A.9 u te ft-ode appoxato of R yeld: N N N N N N N N N N N N N N N N N D N D D R. 5A. U equalty 5A. we obta

86 68 N N N N N N ε ε ε ε ε ε D N D D R 5A. ad N N N N N N N N ε ε ε ε ε ε D N D D R 5A. wc we te equalty << ε take to accout pove te vee of daoally doat atx R a a upeo daoal. Accod to copeeve uecal ulato la cocluo apply alo to te atce wc ae ot tctly daoally doat te ee of 5A. but tll fulfl te codto 5A.. Moeove eac ulated atx te ato betwee te daoal ad off-daoal eleet wa yet poved by te atx vee pu te upeoty of te daoal eleet eve e. e exact poof of te obeved peoeo eace beyod te cope of t detato.

87 Appedx 5B: Pobablty of ovelapped pule Let te pule ouce be oazed to R ubet eac cop Q ouce ad aue te ouce fo te ae ubet caot ovelap wt eac ote. Defe te ovelap of te -t ouce wt te -t ubet of ouce a a et of all poble ovelap of te -t ouce wt ay ouce fo te -t ubet. Fute aue tat ovelap of all poble ouce wt all poble ubet ae depedet equally pobable ado evet ad deote te pobablty by p. Fo te ake of plcty alo aue tat ay ouce fo a ve ubet equally pobable to ovelap wt ay ouce fo ay ote ubet: j k ; j k : pj k p Q 5B. wee p j k deote te pobablty tat a le pule of te j-t ouce ovelapped wt ay pule of te k-t ouce weea we aue te j-t ad k-t ouce do ot belo to te ae ubet. Let be te te oet wc at leat oe ouce wa actve ad deote te actve ouce by dex j. e pobablty of av R ouce actve ve te oet equal te pobablty tat te j-t ouce ovelap wt - out of R- poble ubet of ouce: P cad G R R cad G > p p 5B. wee te et G cope te ouce tat wee actve te oet cad t cadal ube wle deote te ube of cobato of eleet take eleet at a te. e ube of all poble cobato of out of RQ ultaeouly actve ouce G R R C Q Q 5B.3 ad te pobablty of av elected ouce actve ve te oet yeld: P cad G C R Q cad G >. 5B.4 69

88 7 Fally exclud elected ouce fo te et of all poble ouce te ube of all poble cobato of ovelapped ouce ead: R Q Q Q R C. 5B.5 Obeve ow two te oet ad wt te pule fo at leat oe ot ecealy te ae ouce. Deote by te ube of ouce tat wee actve oet ad by te ube of ouce tat wee actve oet. U te deotato > G G cad G P cad p ad > G G cad G P cad p 5B.6 te pobablty of av coo ouce actve bot te oet ca be obtaed a: p p C C G cad G cad G G P cad p G R R G R Q R Q G G > > 5B.7 wee te fact tat at leat ouce ad to fe bot te oet ad : ad take to codeato. U 5B.5 we obta: C C p p p R Q R Q G R R G G G.!!!!!!!!!! G R R G Q Q Q R R R R p p 5B.8 Ow to!!!!!!!!! R R R R 5B.9

89 5B.8 yled: p G G R R R!! R R! Q p G p G! R! R!!! R! R! R! R! R R! R! R!! Q R!! R!! 5B. p p R p p R. Becaue R te pobablty pg ca be etated a follow: G p G G Fally u R! R! R R! Q R R R p R p p R p R. 5B. R R R R R R p p p p p p 5B. we et p G G R! p R p! R! Q Q. 5B.3 e etato 5B.3 epeet te uppe boud wc eve attaed ad accuate oly fo lae Q. Nevetele we ca ee tat te pobablty of av coo ouce two te oet dop apdly wt te ube of coo ouce ad alo wt te cadal ube of ubet Q F. 5B. 5B. ad 5B.3. I te cae of EMG al te ube of ubet coepod to te ube of actve MU wle Q epeet te ube of te delayed epetto of eac evato pule ta. ak to accout tat Q uually eate ta wle te pobablty p adly exceed.5 we ca coclude te pobablty of av two coo ouce actve two dffeet te oet elbly all. fact ca eadly be ee alo fo F. 5B.. 7

90 Pobablty uppe boud Q p.4.5 Fue 5B.: e uppe boud 5B.3 of pobablty pg at ad R G depedece of te ube of ouce eac ubet Q ad te pobablty of ovelapped ubet p. Pobablty uppe boud Q p Fue 5B.: e uppe boud 5B.3 of pobablty pg at ad R G depedece of te ube of ouce eac ubet Q ad te pobablty of ovelapped ubet p. 7

91 Pobablty uppe boud Q p Fue 5B.3: e uppe boud 5B.3 of pobablty pg at 3 ad R G depedece of te ube of ouce eac ubet Q ad te pobablty of ovelapped ubet p. e pobablty of av coo ouce actve tee o oe te oet ae uc alle ta pg ad ca tu be elected at leat we Q p b all eou. G 73

92 Appedx 5C: Aveae ube of pule te poduct of te upepoed pule ta of ouce I t appedx we ue te aupto ad cocluo fo Appedx 5B to etate te aveae ube of pule a poduct of te upepoed pule ta of cloe-tootooal ouce. Suppoe fou te oet ad 3 wt at leat oe ot ecealy te ae actve ouce ad ue 5.5 to ecotuct te coepod upepoto of ouce pule ta: v j j j 3. 5C. G j G v Deote by G j... j } te et of ouce tat wee actve te oet { 3. Accod to Appedx 5B te pobablty tat te et et. G dffe fo ote Fo te ake of plcty aue te ube of pule eac ouce equal F ad obeve te aveae ube of pule te eleet-we poduct v v G G j j. e poduct v v G wll cetaly cota te pule of all te ouce fo te teecto G j G G j ube of pule tat te ouce fo v v G ca be etated a: G j j j. Code te utual ovelapp te aveae G G j cotbute to te poduct j F f j 5C. wee follow te coveto fo Appedx 5B j te cadal ube of G G j f p Q 5C.3 deote te coecto facto toduced by te fact tat te pule of te ouce p G G j ovelap ad p Q tad fo te pobablty tat te pule of two abtay Q ouce ovelap Appedx 5B. 74

93 e poduct v v G wll alo cota all toe pule of ay ouce fo j G j j G G wc adoly ovelap wt te pule of ay ouce fo M M deote te et dffeece a: G G wee j M wtout M. e ube ca be etated F p Q f j j f j j j 5C.4 wee deote te cadal ube of G ad f p Q 5C.5 a eealed coecto facto 5C.3 wc accuulate te fluece of all poble ovelap ao te ouce fo ubet G G ad G G epectvely ad exclude te fluece of ouce fo G G j j j. ak to accout te fact tat ot all ouce ca ovelap Appedx 5B te total aveae ube of pule te poduct v v G yeld: G j j I pq j j F [ j f j pq f j j f j j j ]. 5C.6 We ca follow te ae oute to etate te aveae ube of pule te poduct v v v v. Fo te claty eao defe ft te follow G G G G 3 utually djuctve et: 3 M M M M G 3 G G G j k G G G M3 j G G M 3 j j k M j k G M j k M j k M j k k 3 k 3 5C.7 wee j k {3 } ad j k j k. By defto te uo of all te M et equal te uo of all te G et: G M M j j M j k M3 j k 5C.8 75

94 76 ad all poble teecto of te M et ae epty. Follow te coveto fo Appedx 5B deote by cadm te cadal ube of te et M ad etate te aveae ube of pule te poduct 3 3 v v v v G G G G a: ]. [ f f f f p f f f p f f f p f f f p f f f p f f f p f f f p f f p f f p f f p f f p f f p f f p f f p f F p I Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q 5C.9 Suppo te pobablty Q p p Q all Appedx 5B ad elect all te facto wc ae ultpled by te ecod o te e powe of Q p 5C.9 yeld: ]. [ f f p f f p f f p f f p f f p f f p f f p f F p I Q Q Q Q Q Q Q Q 5C. Accod to 5C. te aveae ube of pule te poduct 3 3 v v v v G G G G deped aly o te ube of ouce te

95 et G G G ad G 3 ad te dvo te coepod M et. I te equel we ae o to tudy jut two of ay poble cae. Cae : Suppoe te et M cota a le ouce wle et M 3 M 3 M 3 M ae epty: ad Fute aue ad ad 3 ae all equal to o le ta. e 5C. yeld: I pq Q f F p [ f f 3f ]. 5C. Recall tat ad tat o aveae te ube of ouce te et G 3 ca be etated a 3 Rp. Calculate te value of all fucto f ad elect te facto wc ae ultpled by te ecod o e powe of p Q to et te fal etate R I p R Q F Fp. 5C. Q Cae : Suppoe te et M 3 cota a le ouce wle et M M 3 M 3 ad M 3 ae epty. Fute aue lke fao to Cae tat 3 3 ad 3 ae equal to o le ta. e 5C. plfe to I p F[ f 3p f f ]. 5C.3 Q Q Nelect all te facto wc ae ultpled by te ecod o e powe of et te fal etate p Q we p I p R Q F F 3. 5C.4 Q 77

96 Appedx 5D: Popete of otooal pojecto atx Defe te atx a # H HH H 5D. wee # deote te Mooe-Peoe atx peudovee [68 5] ad H abtay x atx of ze KM N L K wt KM<NLK-. Wtout lo of eealty we ca uppoe te x atx H of full ow ak. U te ula value decopoto: te atx ca be expeed a: VD H UDV 5D. I KM KM O # KM N L K KM DD DV V V O N L K KM KM O 5D.3 N L K KM N L K KM wee V N L K N L K utay atx ad D KM N L K daoal atx wt te ula value of atx H o t daoal. It eay to ow tat: wc pove Popety. 5D.4 Popety : e atx otooal pojecto app te N L K - deoal pace of ouce oto KM-deoal pace of eaueet. ak to codeato te coo popete of te otooal pojecto atce we ave [68 5]: ad N L K tace t KM 5D.5 KM N L K t v v 5D.6 j j j KM j 78

97 wee t tad fo te -t daoal eleet of atx v j deote te j-t eleet of te utay atx V wle tace. te atx tace te u of all daoal eleet. A a otooal pojecto te atx a ay ueful popete. I te equel oly toe wc ae cucal fo te IC etod wll be outled. Popety : by defto potve edefte [5]: ; R N L K : 5D.7 ad det 5D.8 wee deote ay pcple ubatx of atx.e. te atx obtaed fo by delet te ae et of colu ad ow. Apply 5D.8 to te abtay pcpal ubatx of ze we obta: t det t j tj t t t jj jj t j. 5D.9 Moeove u 5D.7 ad ett [......] te vecto woe -t eleet equal j-t eleet equal - wle all te ote eleet equal zeo we et t t t. 5D. jj j Ca te of te j-t eleet vecto.e. yeld: [...] 5D.7 t t t. 5D. jj j By e te equalte 5D.6 5D.9 5D. ad 5D. we et t j t t jj ax t t jj 5D. ad j t j t t t t t jj jj t t jj. 5D.3 79

98 Fally fo 5D. ad 5D.3 t follow: Popety 3: e abolute value of te off-daoal eleet t j lted to te teval t t jj t j t t jj. 5D.4 Popety 4: Accod to 5D.5 ad 5D.6 tee ae at leat KM daoal eleet t dffeet fo zeo. Aue ow te atx a j-t daoal eleet equal to. Accod to 5D.4 we ave t jj t k jk t kj 5D.5 ad t t. 5D.6 jk kj Fally we ave wc pove Popety 5. kj k k t jj t jktkj t jk k : t. 5D.7 Popety 5: e otooal pojecto atx wt te j-t daoal eleet equal to ut ave all te eleet of te j-t ow ad j-t colu all. e lat popety we ae o to outle baed o a le foal poof. Fo 5D. t follow: l KM N L K I N L K N L K 5D.8 Obeve ow te -t colu of te atx V. Deote t by v ad code tee dffeet cae:. e ajoty of te eey of v cocetated te ft KM eleet. te deal cae a otooalty of te atx V uaatee te daoal eleet t wll be cloe to wle all off-daoal eleet t j ad t j wll be cloe to. 8

99 . e eey of v ufoly dtbuted ao all eleet. ot pobable cae ad ple te value of te daoal eleet t wll lowely N L K deceae we te ato ceae wle te off-daoal eleet KM t j ad t j wll lowly ceae. 3. e ajoty of te eey of v cocetated te lat KM N L K eleet. te wot cae a 5D.6 uaatee te daoal eleet t wll be cloe to zeo weea accod to Popete 3 ad 4 te off-daoal eleet t j ad t j wll be uc alle ta at leat eveal daoal eleet. e lat cocluo ca be fute jutfed by otc tat te otooalty of te atx V uaatee te ube of te colu v cocotat wt te cae 3 lted. Naely be otooal te atx V peeve te o ad te ale betwee abtay vecto. Suppo tat all t colu coepod to cae 3 t would ecealy ply te atx V la to te pojecto atx wc ap N L K -deoal pace oto N L K KM -deoal ubpace. dectly cotadct te peevato of ale betwee te abtay vecto [68 5]. N L K Popety 6: Suppo te ato all eou te atx wll ave at KM leat a pat of t daoal eleet upeo to all off-daoal eleet. e uecal ulato eveal te doat daoal eleet ae at leat eveal te e ta te coepod off-daoal eleet a lo a te ube of ouce N L K doe ot exceed te ube of eaueet KM by facto. 8

100 8

101 6. Suface EMG decopoto eult I pevou capte we deved tee ovel decopoto appoace. We explaed te teoy bed te fluece of oe ad of o-otooal ouce ad te lat capte we evaluated te pact of te ato betwee te ube of ouce ad ube of eaueet. We fute etated te pobablty of ovelapped ouce ad tated vey eeal cocluo ead te effcecy of te toduced etod. Suppo eeal pule ouce te deved etod deped ete o te type of te pule ouce o o te atue of te x poce. ey ca ece be ued fo epaato of bot te covolutve ad ultplcatve xtue. e oly aupto wc tey follow te exaple of boedcal al coce te al let of te te-pule teval wc ut be loe ta te coepod yte epoe. aupto wa fute odeated te cae of te IC etod wee we uppoed te ouce ae dvded to R ubet eac cop Q ouce. I t way te decopoto ca ealy be exteded to clude oe eeal cae. I te cae of boedcal al we take RN ad Q L K [77] wle o te ote ad we allow te epetto of te ae ouce to ovelap we take R N L K ad Q. Follow te coo coveto fo te teoy of BSS te x atx wa fute uppoed cotat te. I t capte te pefoace of te developed decopoto appoace wll be teted o te ytetc ad eal SEMG al. A aleady explaed te Capte 3 te MU ca be teated a a pule ouce. e efactoy peod ad axal poble f fequecy uaatee te delayed epetto of te ae evato pule ta caot ovelap wle te evato ta of dffeet MU do ot ovelap fcatly at leat at low ucle cotacto level. O te ote ad te aupto of te tatoay x atx poe oe cucal etcto o te uface EMG eaueet. e latte ut be take teady cotolled codto du oetc ucle cotacto ad wtout ay obevable effect of te fatue. Wle te decopoto of te EMG al to te MU f patte a poved to be a vey potat clcal ue tee yet aote oe teccal eao fo tet te developed decopoto appoace o te uface EMG. Naely te pyolocal popete of MU put oe evee ltato o bot te evato pule ta ad o te ape of MUAP. Evaluat te eult o te eal al tee ltato ca be eadly ued te vefcato ad valdato pocedue. ue cucal a te oal evato ta ae uually ukow. 83

102 e capte dvded to two lae ecto. Secto 6. evaluate te decopoto o te ytetc uface EMG al wle Secto 6. tude te decopoto o te eal uface EMG al. Bot ecto ae fute dvded to tee ubecto eac povd te eult of dffeet decopoto appoac. e eult wll be dcued ad utually copaed te ext capte. 6. Reult o ytetc uface EMG al Decopo te eal uface EMG al te oal evato pule ta ad eve te ube of actve MU ae ukow. We ca patally ely o te dect eaue wc wll be decbed oe detal te ext ecto. Nevetele te dffeet decopoto pocedue ould ft be evaluated o te ytetc al. way te pact of te ube of actve MU te f fequecy dept te ucle tue etc. a well a te fluece of oe ca be eadly evaluated. Moeove te expeece aed o te ytetc al ca pove cucal we evaluat te pefoace o te eal al. Howeve to be able to eao fo te te eult te ytetc al ut eeble te eal oe a uc a poble. o eue te epeetatvee all te ytetc EMG al wee eeated u te advaced uface EMG ulato bult by te wold ecoed expet fo te boeee laboatoy LISN Laboatoo d Ieea del Stea Neuoucolae at Poltecco d oo Italy. e ulato allow ulato of ult-cael patally flteed uface EMG al eeated du volutay cotacto by te actvty of a lae ube of oto ut [56]. All te a featue of te uface EMG al ae odelled clud te fatue te eeato ad extcto peoea of te acto potetal at te ed-plate ad tedo eo te ze ad ape of te ecod electode ad te clato of te fbe wt epect to te detecto yte. e volue coducto decbed a a aotopc layeed edu wt ucle fat ad k laye. Moto ut ae placed adoly te detecto volue ad ae actve at electable f ate. e detecto yte ae placed ete alo wt o taveal to te fbe decto. e fluece of two facto wa evaluated du ou ulato: a te ube of actve MU ad b te al-to-oe ato SNR. e ube of actve MU wa et to 5 ad epectvely wle SNR aed fo db to db tep of 5 db. e ote potat SEMG paaete wee et a follow: Sk wa ulated a tck otopc laye ad ubcutaeou fatty tue a 3 tck otopc laye. Actve MU coted of a ado ube of fbe ufoly dtbuted betwee 5 ad 3 wt te ccula MU tetoe of fbe/. e dept of actve MU te aotopc ucle laye vaed ufoly fo 3 to wle addtoal ado ft ufoly dtbuted betwee - ad fo te cete of te electode aay te decto taveal to te ucle fbe wa appled to eac MU. 84

103 MU f ate wa oally dtbuted aoud te ea of 5 Hz wt tadad devato of 4 Hz. e IPI vaablty wa odelled zeo-ea Gaua vaable wt te vaace equal to % of te IPI ea. Coducto velocty wa aued oally dtbuted wt te ea of 4 / ad tadad devato of /. e evato zoe wee aued placed te ddle betwee te tedo ad te e-fbe let wa et to 7. e pead of evato zoe wa lted to 5. e aveae let of ulated MUAP wa etated to 5. 5 aay detecto yte wt ow ad 5 electode pe ow wa ceted ove te dtal alf of ucle fbe colu aled wt te decto of fbe. Rectaula by electode wt te te-electode dtace of 5 wee ulated. Meaueet wee uppoed lotudal le-dffeetal. Sytetc SEMG al duato of 3 wee apled at 4 Hz. ulato wee pefoed fo eac ube of actve MU 5 ad. I eac of te ulato u te dept of te actve MU te f ate ube of fbe ft te decto taveal to te ucle fbe ad coducto velocty wee adoly elected. I addto al fo eac ulato u wee coupted by addtve oe 5 ealato of oe fo eac SNR eult 5 tet al fo eac ube of actve MU. 6.. Metod baed o te-fequecy dtbuto We ft teted te etod baed o te-fequecy dtbuto. Eploy all 5 le-dffeetal eaueet of ytetc uface EMG ad av te aveae let of MUAP etated to 5 aple te vecto of eaueet wa exteded by facto K3 te cae of 5 actve MU ad by facto K7 te cae of actve MU. Accod to 3. te ube of exteded ouce wa etated to 35 ad 3 epectvely. I ode to eue te ube of eaueet exceed te ube of ouce te al wt actve MU would ave to be exteded by te facto K7. A a eult te et of SFD D zz f atce ete te jot-daoalzato would cope at leat 8 atce of ze 8 8. Hav t coputatoal coplexty popotoal to te cube of te atx ze [74] te jot-daoalzato would eque eoou aout of poce te to coplete. Hece te etod baed o tefequecy dtbuto wa teted oly o te al wt 5 ad actve MU. deco fute jutfed by te eult wc a we wll ee pove te effcecy of te etod deceae wt te ube of actve MU. e etod baed o te-fequecy dtbuto aue te ube of ouce actve MU kow. We t ot te cae te ube of ouce ca be etated fo te ak of coelato atx of eaueet R x. Note alo te ube of ouce ca be ltly oveetated. A a eult te etod ecotuct oe exta ouce but 85

104 tey all ave apltude cloe to zeo. Nevetele tet te etod o te ytetc al we uppoed te ube of ouce wa kow. e exact decopoto pocedue ued ca be uazed te follow x tep:. U te ee-decopoto of te coelato atx of exteded eaueet ecotuct te wte atx B ad etate te oe vaace σ. e exact pocedue decbed [3 7].. Wte te exteded vecto of oy eaueet ad accod to 4.5. calculate te atce of peudo We-Vlle dtbuto D f. 3. Apply te cteo 4. to all D zz f atce ad jo te atce wc coepod to te le-autote of te ae ay te -t ouce to te et A. If te ube of et A eace te etated ube of ouce o to tep 4. Otewe ue 4. ad 4.4 to fd te atce wc coepod to te ouce ad cota a lttle cotbuto fo ote ouce a poble. 4. Aveae te atce eac et A ad ue 4.8 to eove te fluece of oe. Deote te eult atce by M. 5. Jot-daoalze all atce M to etate te utay atx U ad ue 4.6 to ecotuct te evato pule ta. 6. Mutually copae te obtaed evato ta ad clafy all te epetto of te ae evato ta to te ae oup. Al te all te delayed epetto of te ae evato ta ad calculate te aveae. Evaluat te eult oly te ecotucted evato ta cota at ot 3 % placed pule wee take to codeato. All ote ecotucto wee dcaded. e eult ae uazed able 6. ad exeplfed F. 6. ad 6.. e eult able 6. ow te decopoto effcecy deceae wt te SNR ad te ube of actve MU. At SNR of db te aveae ube of te ecotucted MU yeld 3.8 ad 7.3 te cae of 5 ad MU epectvely. Decea SNR to db te aveae ube of MU dop to.6 ad.7 epectvely. e dop of pefoace te cae of actve MU ca patally be explaed by ealz te etod baed o F dtbuto toly deped o te ube of o-ovelapped pule. e latte deceae apdly wt te ube of actve MU. U te eult of Appedx 5B ad ett R5 5 actve MU Q 7 ad p. 5 te pobablty of o-ovelapped pule yeld.4. O te ote ad ett R actve MU ad Q 3 te pobablty of o-ovelapped pule dop to.6. Icea te ube of actve MU by facto te ube of atce coepod to te le-autote deceae by facto 4. e ube of o-deal SDF atce ete te jot daoalzato ceae ad te pefoace dop Subecto zz 86

105 5 SNR [db] e [] Fue 6.: Oal ytetc evato pule ta black ad pule ta ecotucted by te etod baed o F dtbuto ey te cae of 5 actve MU ulato MU 5 ad at SNR 5 5 ad db. e pefoace alo dop wt oe. We ca patally cacel t fluece by avea te SFD atce ad apply 4.8. Howeve te oe alo fluece te electo of SFD atce wc a aleady explaed cucal fo te etato of te x atx. Sall pobablty of ovelapped pule te cae of 5 actve MU copeate te dop pefoace of cteo 4.. I te cae of actve MU te ube of ovelapped pule ceae ad te effect of oe becoe evdet. e F-baed etod ue te ecotucted x atx to etate te pule ouce. Accod to 3.6 te x atx cope K delayed epetto of eac yte epoe. A a eult te MUAP ca be ecotucted by avea te coepod ow of te etated x atx. Howeve accod to BSS deteacy 3. te ecotucted ouce ad coepod colu of te ecotucted x atx ca be abtaly peuted wt epect to te oal ouce. Hece befoe te avea te colu of te x atx ut be eaaed to atc te oal ode. Note te oal ode of ouce ad atx colu aleady deteed tep 6 of te decopoto pocedue decbed above. Wle te ecotucted ouce ca be aveaed ove K L delayed etato of eac ouce tee ae oly K delayed ecotucto of eac MUAP. I ou tet we ued K3 te cae of 5 actve 87

106 MU ad K7 te cae of actve MU. Avea ove oly 7 dffeet value ot vey effcet ad we ca eadly expect te ecotucted MUAP wll fcatly dffe fo te oal oe. e eult ae decbed able 6. wle te o of te MUAP dffeece wc ued a a eaue of ft wll be defed te ext ubecto. 5 SNR [db] e [] Fue 6.: Oal ytetc evato pule ta black ad te pule ta ecotucted by te etod baed o F dtbuto ey te cae of actve MU ulato 3 MU 8 ad at SNR 5 5 ad db. e eult able 6. wll be fute dcued te ext capte wee tey wll be copaed to te eult of ote two etod. Fo te te be t uffce to ealze te evato pule ta wee ecotucted uc oe accuately ta te coepod MUAP. 88

107 able 6.. e ube of ecotucted evato pule ta ea ± tadad devato te pecetae of ecotucted pule wt epect to te ube of pule te oal ytetc pule ta tue potve tattc ad te pecetae of placed pule wt epect to te ube of ecotucted pule fale potve tattc. e eult ae clafed accod to te ube of actve MU ad te ulated SNR. Nube of actve MU 5 SNR [db] Aveae ube of ecotucted evato pule ta Poto of ecotucted pule [%] Poto of placed pule [%] 3.8 ±.9 9. ±. 5.9 ± ± ± ± 7..8 ± ± 6..7 ± ± ± ± 8..6 ± ± ± ±. 89. ± ± ± ± ± ± ± ±. 5. ± ± ± ± ± 4..5 ± 9.6 able 6.. e ube of MU ea ± tadad devato ecotucted by te etod baed o F dtbuto ad te aveae o of te dffeece 6. betwee te MUAP ecotucted fo te x atx ad te efeece value. e eult wee obtaed o te ytetc al te cae of 5 ad actve MU epectvely. Nube of actve MU 5 SNR [db] Aveae ube of ecotucted MU Aveae o d [%] 3.8 ± ± ± ± 9..8 ± ± ± ± ± ± ±. 68. ± ±.8 7. ± ±. 7.4 ± ± ±.8.7 ± ± 3. j 89

108 6.. Metod baed o e-ode tattc e effcecy of all poble decopoto etod wc ae baed o e-ode tattc deceae weeve: te oe ot Gaua te ouce ae ot depedet ad detcally dtbuted..d. te poceed al ae too ot to povde te accuate etato of cuulat I te cae of ytetc EMG al te oe wa ulated wte ad Gaua ad we ca eadly expect te w-lce wll pove te etace to t. Hav te al lted to let of 37 aple we ca alo aue 4.34 wll elatvely accuately etate te value of cuulat. e a ouce of eo wll teefoe be te ootooal ouce. e latte ae te cae of ytetc uface EMG al tll depedet but exbt te ovelapp pule wc cotbute te o-zeo value to te co-cuulat of ouce. A a eult te Balett-Blle-Roeblatt equato fal to odel te yte accuately ad te Newto-Gau optato covee to te wo oluto. Howeve w-lce do ot uppoe te ouce exteded by te delayed epetto wc uaatee te ube of ovelapped pule uc alle ta te cae of ote two etod te IC etod ad F-baed etod. eefoe we ca aue te decopoto etod baed o te HOS wll tll aae to ecotuct ood etate of te tue MUAP. A ce popety of HOS baed decopoto etod te ablty to ecotuct te yte epoe alo we te ube of ouce by fa exceed te ube of eaueet. Gve te ube of ouce ad te let of te coepod yte epoe te equed ube of eaueet ca be calculated by 4.4. Sett te ube of ouce to ad te let of yte epoe to L5 we ee tat all te yte epoe ca be ecotucted fo a le eaueet. e deved etato ly teoetcal. I pactce vaou calculato eo de te decopoto ad te ube of ouce ecotucted fo a le eaueet dop datcally. HOS-baed decopoto tecque peeted t detato cope two tep. Wle te Newto-Gau optato allow oe ouce ta eaueet te w-lce aue te ube of ouce doe ot exceed te ube of eaueet. All te popete of HOS ae ot favouable. Oe of te ot obvou dawback te coputatoal coplexty. I te cae of w-lce te coputatoal coplexty ceae wt te quae of te let of yte epoe ad alo wt te quae of te etated ube of ouce. Coequetly te decopoto etod wa teted oly o te al wt 5 actve MU. e coputatoal coplexty of Newto-Gau optato alo epoble fo ou deco to tet te etod oly o 5 out of 5 avalable uface EMG eaueet. I all tet o te ytetc al oly te eaueet coepod to te elatve ulated uptake poto

109 36 ad 37 wee decopoed wee pa j deote te -t ow ad j-t colu. All te ote eaueet wee ply dcaded. We w-lce ae ued to coaely etate te MUAP we ft eed to kow te let. Sett te apl fequecy to khz te MUAP let ay be expected at aoud 5 aple. ee yet te codto fo 4.59 to be et: j j j I te cae of uface EMG al t codto elatvely dffcult to atta. e cloet poble appoxato aceved we te uface EMG eaueet ae fted te. ak te tavell of AP to codeato t ple a lotudal placeet of electode ad alo jutfe ou deco to take 5 eaueet fo te cetal colu of electode. O te ote ad oe ut be awae tat eac decopoed MUAP o to be oalzed by t tat aple et to. Hece te apltude of MUAP wll be lot te decopoto poce. e fact tat we caot etely et te codto fo 4.59 doe ot de te decopoto. A aleady explaed te w-lce ae ued to oly coaely etate te ape of MUAP wle te fal fo aed tou te Newto-Gau optato. Apply ou HOS-baed etod to te ytetc EMG al wt 5 actve MU te decopoto poce ca be uazed te follow 4 tep:. ake eaueet of uface EMG wc coepod to te lotudal poto of electode. I ou cae te eaueet fo te cetal colu of electode te eaueet coepod to uptake poto ad 37 wee poceed.. Calculate te td-ode cuulat of tee eaueet oly up to te la ad wc coepod to te etated MUAP let L5. 3. Ue 4.57 ad 4.58 to etate te caual ad at-caual cuulat atce S a ad S c ad calculate te coae etate of MUAP accod to 4.59 ad Opte te coae etato by te Newto-Gau optato. I ode to educe te coputatoal coplexty a uc a poble aly becaue of ue deo of atx Σ t we to elect a appopate ubet of te cuulat value atx C 4.6 uc tat te codto ve by 4.4 ad 4.43 ae jut fulflled. I ou expeet we codeed oly te ft quadat of cuulat C kl.e. L 4 L 4. e ze of tep δ 4.66 wa et by te etod of becto weea te vecto of ouce cuulat γ [ γ γ ] K wa accod to 4.68 aveaed ove all poble cobato of dce k l ad. 9

110 I all ou expeet te optato wa teated weeve all te poble - o of pa-we dffeece ao te MUAP ecotucted tee ucceve optato tep calculated below 5 % wt epect to te -o of te ecotucted MUAP. O aveae te optato coveed 3 optato tep. e ecotucted MUAP ae patally depcted F. 6.3 ad 6.4 wee oly two of ay ecotucted MUAP ae llutated depedece of SNR. Quck vual pecto of te eult cof te expected etace to te Gaua oe. Altou te ape of MUAP cae wt SNR te oe fluece elatvely all. I ode to quatfy te decopoto effcecy te ecotucted MUAP wee alo uecally copaed aat te efeece ape. I all ou expeet te -o of dffeece betwee te ecotucted ad te efeece MUAP defed 6. wa ued a a eaue. Sce te value of te MUAP dffeece deped alo of te o o te MUAP te copaed MUAP 6. wee ft oalzed. Expe te dffeece efe to te -o of te efeece MUAP ad tak to codeato tat te o of a oalzed vecto equal to te toduced copao cteo take t fal fo: d j j j ˆ ˆ j j j j j ˆ ˆ j wee we wte j [ j... j L ] ad ˆ j [ j... j L ] to deote te efeece ad te ecotucted MUAP j. j j 6. e eult wee aveaed ove all te ulato u wt te ae SNR ad ae uazed able 6.3. Oly te MUAP woe o dffeece 6. wee below te teold of 3 % wee clafed a detfed. e obute to te Gaua oe wa alo cofed by te uec eult. e ea -o dffeece 6. yeld. % at SNR db ad ceae oly fo 3.7 % at SNR db. able 6.3. e ube of MUAP ea ± tadad devato ecotucted by w-lce te cae of 5 actve MU ad te copao to te efeece ape te depedece o te SNR. e appled o of dffeece d j defed by 6.. e ube of actve MU 5 SNR [db] Aveae ube of ecotucted MU Aveae o dffeece d j [%] 3.6 ±.8. ± ±.6. ±. 3. ±.5.3 ±. 5.7 ±.8 3. ±.5. ± ±.9 9

111 .6.4 Oal MUAP SNR db SNR db SNR db MUAP apltude e [] a.5 Oal MUAP SNR db SNR db SNR db MUAP apltude e [] b Fue 6.3: e MUAP ecotucted by w-lce fo te ytetc uface EMG al of let 37 aple te cae of 5 actve MU ad at SNR db lt ey daed le SNR db dak ey daed le SNR db black dotted le: a ulato cotbuto of te ecod MU to te eaueet 34 b ulato cotbuto of te ecod MU to te eaueet 35. Refeece value ae depcted wt black old le. 93

112 .5 Oal MUAP SNR db SNR db SNR db MUAP apltude e [] a.4. Oal MUAP SNR db SNR db SNR db MUAP apltude e [] b Fue 6.4: e MUAP ecotucted by w-lce fo te ytetc uface EMG al of let 37 aple te cae of 5 actve MU ad at SNR db lt ey daed le SNR db dak ey daed le SNR db black dotted le: a ulato cotbuto of te ft MU to te eaueet 35 b ulato cotbuto of te fout MU to te eaueet 37. Refeece value ae depcted wt black old le. 94

113 6..3 Ivee coelato baed etod I Capte 5 we deved a copletely ovel decopoto appoac te o called vee coelato IC baed etod. We deotated t effcecy te cae of otooal ouce ad aeed te fluece of oe ad o-otooal ouce. We alo aued te IC etod ca be ued fo te decopoto of ude-deteed MIMO yte. Fally we explaed ow te pefoace dop wt te ato betwee te ube of eaueet ad te ube of ouce. We cocluded tat eve we te ube of ouce exceed te ube of eaueet by facto te etod wll tll be able to copletely ecotuct at leat a pat of te ouce. I t ecto we wll tet t pefoace o te ytetc uface EMG al. We wll ae te pact of oe ad ceck te ypote tat te MU wt low f ate ae oe lkely to be decopoed. Fally we wll evaluate te ecotucto of MUAP by te pke teed ld wdow avea tecque. o vefy ow te etod effcecy deped o te ato betwee te ube of ouce ad te ube of eaueet we fxed te ube of eaueet. e latte wee eadle to te ube of actve MU alway exteded by te facto K. I ote wod we added 9 delayed epetto to eac eaueet wc ceaed te ube of eaueet to 5. e ube of te exteded ouce deped o te ube of actve MU ad eace 7 34 ad 68 fo 5 ad actve MU epectvely. I te lat cae te ube of ouce exceed te ube of eaueet by.36. e appled decopoto pocedue ca be uazed te follow 8 tep:. Calculate te coelato atx of exteded eaueet etate te oe vaace ˆ σ ad ue 5.4 to calculate te actvty dex. U te etated oe vaace detee te oe teold: ˆ σ ˆ σ Rx t 6. wee x R deote -o of atx R x. Set te value wc te actvty dex doe ot exceed te oe teold to zeo. We R x ll-codtoed ue te pocedue decbed 5.3 to cut te eevalue of R x ad to pove t etvty to oe. e optal deee of cutt ca be deteed fo te eevalue of atx R x. e e te deee of cutt te lowe te etvty to oe. Epcally at leat 8 % of te eey of oal atx R x ut be peeved afte cutt ode to pevet te lo of foato o te x atx. 95

114 . Calculate te eda value of te actvty dex ad deote by a abtay te oet te aple of te actvty dex wc te eda value eaced. 3. Ue 5.8 to ecotuct a upepoto of evato pule ta v G ad adoly cooe oe of t pule exceed te oe teold t. Deote by σˆ te te oet of te elected pule ad ecotuct te upepoto v G. 4. I te poduct v G v G elect all te pule exceed te oe teold t ad deote te ube by Z. Deote by σˆ 3..Z te te oet wc te -t elected pule appea ad ecotuct te upepoto v. Multply eleet-we eac upepoto G v G by v v. G G 5. Mutually copae all te poduct v v v ad clafy te G G G accod to te ube of ovelapped pule. Put all te te oet ad j fo wc te poduct v v v ad v G G G G v v ovelap at leat J pule to te ae et A. e G G j j teold J ca be coputed a a poduct of te al let ad te lowet poble f fequecy appoxately 6 Hz te cae of EMG al. 6. Fo all te et A woe cadal ube exceed te teold J/3 calculate te aveae vecto of exteded eaueet: x A cad A x A ad ue 5.9 to ecotuct a le evato pule ta. Set to zeo te actvty dex value all te te oet wc coepod to te pule te ecotucted evato pule ta. 7. Copae te obtaed evato ta to all pevou ecotucted pule equece ad clafy t ete a a ew pule equece o aleady detected pule equece. 8. Repeat tep to 7 utl all te value of te actvty dex ae et to zeo. I ou expeet te cutt of te eevalue atx R x Secto 4.3 wa lted to 5 % 5 % of eevalue wee et to zeo. Altou te elected deee of cutt ay ee te elated eevalue epeeted oly % of te oal atx 96

115 eey o aveae. Follow te advce tep 5 te teold J wa etated to 6 38 wee 3 tad fo te let of te ytetc al ecod ad te 6 te lowet expected MU f ate Hz. e decopoto eult ae uazed able 6.4. Soe of te ecotucted evato pule ta ae fute llutated F ad 6.7. able 6.4: e ube of ecotucted MU' evato pule ta ea ± tadad devato te pecetae of ecotucted pule wt epect to te ube of pule te oal ytetc pule ta tue potve tattc ad te pecetae of placed pule wt epect to te ube of ecotucted pule fale potve tattc depedece of te ube of actve MU ad SNR. Nube of actve MU 5 SNR [db] Aveae ube of ecotucted evato pule ta Pecetae of accuately ecotucted pule Pecetae of placed pule 4.7 ±.6 99 ±.6. ± ±.6 99 ±.9. ±. 3.9 ±.7 97 ±.3. ± ±.6 95 ± 3.8. ±.. ±.9 93 ± 4.. ± ±.3 99 ±.8. ± ±.6 99 ±.. ±. 6.4 ±.9 97 ±.6. ± ±.7 94 ± 4.7. ±..6 ±.4 9 ± ± 4..4 ±.4 99 ±.7. ± ±.7 98 ±.3. ± ±.3 96 ± 3.5. ± ±.3 93 ± ± 5..3 ±.9 9 ± ± 5. Evaluat te ecotucted evato pule ta oly te ta cota at ot 3 % of placed pule wee codeed detfed. All ote ecotucto wee ply dcaded. I te cae of 5 ad actve MU te ove-deteed cae ad SNR of db alot all te ulated MU wee copletely ecotucted. Sulat actve MU te ude-deteed cae te poto of ecotucted MU dopped to 5 %. Icea te oe fluece bot ove-deteed cae 5 ad actve MU uffeed appoxately te ae dop pefoace. At SNR db te etod ecotucted appoxately 8 % of te MU wc wee detfed at SNR of db wle at SNR of db te poto of te ecoed MU dopped to 35 %. e dop pefoace wa oe obvou te cae of actve MU te ude-deteed cae. At SNR of ad db oly 55 % ad 5 % of te MU detfed at SNR db wee 97

116 ecoed epectvely. Readle te ube of actve MU ad te SNR te ecotucted evato pule ta exbted alot pefect atc wt te efeece value wle te ube of placed pule wa elble. e teoy pedct te IC etod ould ecotuct te MU wt lowe f fequece oe uccefully. I te decopoto poce te MU wt f fequece ae ultpled by te alle daoal eleet of te vee coelato atx of ouce Appedx 5A ad ae le lkely to exceed te oe teold. Follow te ae cocluo te MU wt low f fequece ave e cotbuto te ecotucted pule ta ad ae oe lkely to upa te oe level. o tet t ypote we obeved te f ate of te ecotucted MU. e f ate of te ulated MU wee oally dtbuted aoud te ea of 5 Hz wle te tadad devato wa et to 4 Hz. e lowet ulated f ate wee aoud Hz weea te et oe outeaced 9 Hz. e decopoto effcecy alo deped o ote paaete of MU aly o te dept of MU te ucle tue ad o te ube of ucle fbe cop eac MU. Bot paaete ae toly coelated becaue tey bot affect te apltude of a MUAP a detected o te k uface. Supefcal MU wt a ube of ucle fbe poduce toe MUAP ad ae expected to et te fluece of oe wt oe ucce. eefoe evaluat te ecotucted MU te fluece of all tee paaete te f ate dept te ucle tue ad te ube of fbe ut be take to codeato. Due to te claty eao we cobed te te follow opolocal dex: I M M Ap f A f f 6.3 wee Ap tad fo te aveae peak-to-peak apltude of a MUAP a detected by te -t electode ad f f deote te aveae MU f ate. e e te value of te opolocal dex te toe te cotbuto of te MU to te SEMG ecod. e decopoto eult ae peeted able 6.5. e eult able 6.5 daee wt te teoy. No fcat pact of te f ate wa otced. e ecotucted MU clealy ow te coelato wt te dept of MU te ucle tue ad wt te ube of ucle fbe. A expected te etod ot uccefully ecoe te upefcal MU wt a ube of ucle fbe wle all te ote ae teated a a backoud oe. Icea te oe te aveae dept of te ecotucted MU deceae wle te ube of te fbe ceae. Recall all te paaete wee coe adoly eac ulato u depedetly fo eac ote. eefoe t ly ulkely fo te MU wt low f ate oly to be deepe te ucle tue o to be copoed out of alle ube of fbe. fact alo cofed by te aveae f ate of te detfed ad ed MU able 6.5 colu 7 ad 8. e aveae f ate bot oup yeld 5.5 Hz o aveae 98

117 weea te tadad devato alway aoud 3.5 Hz. o coclude te eult clealy deotate te fluece of te MU f ate elble. able 6.5: Value of paaete ea ± tadad devato of te detfed ad ed MU: te MU dept te ucle tue te ube of fbe ad te aveae f ate veu te ube of actve MU ad SNR. e oalzed value of te opolocal dex ae depcted te t-ot colu. Reult wee obtaed o te ytetc SEMG al. Nube of actve MU 5 SNR [db] Aveae dept te ucle tue [] Aveae ube of fbe Aveae f fequecy [Hz] Aveae value of dex I f A ±. ±.6 ±53 ± 65 ± 3.5 ± 3.9 ±.4 ± ±. ±.5 ± 45 ± 4 ± 3. ± 3. ±. ± ±. ±.4 ± 4 ± 34 ± 3. ± 3.4 ±. ± ±.8 ±.4 ± 4 ± 53 ± 3. ± 3.4 ±. ± ±.4 ±.4 ± 4 ± 43 ± 3.3 ± 3.6 ±. ± ±. ±. ± 47 ± 5 ±.8 ± 3.8 ±. ± ±. ±.9 ± 45 ± 5 ±.9 ±.9 ±. ± ±. ±.8 ± 46 ± 46 ±.7 ±.8 ±.9 ± ±.9 ±.9 ± 48 ± 47 ±.8 ±.8 ±.6 ± ±.8 ±. ± 36 ± 49 ±.8 ±.9 ±.7 ± ±.6 ±.9 ± 4 ± 5 ± 3. ±.5 ±.8 ± ±.7 ±. ± 4 ± 5 ±.7 ± 3.6 ±.7 ± ±.5 ±. ± 48 ± 49 ± 3. ±.9 ±.8 ± ±.5 ±. ± 5 ± 49 ± 3. ±.7 ±.3 ± ±.3 ±. ± 4 ± 5 ±.8 ±.7 ±.4 ±.7 99

118 5 SNR [db] e [] Fue 6.5: Oal ytetc evato pule ta black ad te pule ta ecotucted by te IC etod ey te cae of 5 actve MU ulato MU 3 ad at SNR 5 5 ad db. 5 SNR [db] e [] Fue 6.6: Oal ytetc evato pule ta black ad te pule ta ecotucted by te IC etod ey te cae of actve MU ulato 5 MU 6 ad at SNR 5 5 ad db.

119 5 SNR [db] e [] Fue 6.7: Oal ytetc evato pule ta black ad te pule ta ecotucted by te IC etod ey te cae of actve MU ulato MU 6 ad at SNR 5 5 ad db. Fally let u evaluate te eult of te pke teed ld wdow avea tecque decbed Secto 5.. Recotuct te MUAP fo te ytetc al te ectaula wdow Ψ l of let 5 aple wa ceted to te poto of evey detfed pule of te coepod MU. Avea 5 eaueet 5 dffeet MUAP of eac MU wee ecotucted. U te o of dffeece 6. te ecotucted MUAP wee uecally copaed to te efeece value. e eult ae peeted able 6.6 wle a few exaple of te ecotucted MUAP ae depcted F ad 6..

120 .4. MUAP apltude Oal MUAP SNR db SNR db SNR db 5 5 e [] Fue 6.8: e MUAP cotbuto of MU 3 to te eaueet 35 ecotucted by te pke teed ld wdow avea tecque 39 aveae fo te ytetc uface EMG al of let 37 aple te cae of 5 actve MU ulato ad at SNR db lt ey daed le SNR db dak ey daed le SNR db black dotted le. Refeece value ae depcted black old..4. Oal MUAP SNR db SNR db SNR db MUAP apltude e [] Fue 6.9: e MUAP cotbuto of MU 6 to te eaueet 35 ecotucted by te pke teed ld wdow avea tecque 386 aveae fo te ytetc uface EMG al of let 37 aple te cae of actve MU ulato 5 ad at SNR db lt ey daed le SNR db dak ey daed le SNR db black dotted le. Refeece value ae depcted black old.

121 .6.4 Oal MUAP SNR db SNR db SNR db MUAP apltude e [] Fue 6.: e MUAP cotbuto of MU 6 to te eaueet 35 ecotucted by te pke teed ld wdow avea tecque 48 aveae fo te ytetc uface EMG al of let 37 aple te cae of actve MU ulato ad at SNR db lt ey daed le SNR db dak ey daed le SNR db black dotted le. Refeece value ae depcted black old. able 6.6: e ube of ecotucted MU ea ± tadad devato ad a copao of te ecotucted MUAP to te efeece value te -o of te MUAP dffeece d j defed 6. veu te ube of actve MU ad SNR. Nube of actve MU 5 SNR [db] Aveae ube of ecotucted MU Aveae o d j [%] 4.7 ± ± ± ± ± ± ±.6 7. ± 4.8. ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±.7 4. ± ±.3.7 ± ±.3.7 ± ±.9.3 ± 8. 3

122 6. Reult o eal uface EMG al e expeet wee coducted wt al fo te doat bcep bac ecoded Laboatoo d Ieea del Stea Neuoucolae LISN Poltecco d oo Italy. Ne ealty ale ubject of ae 6.8 ±. yea et 79 ± 7 c ad wet of 7. ± 8.3 k patcpated ou tudy. All ubject ave te foed coet. Suface EMG al wee detected by a aay of 3 5 electode wtout te fou coe electode of ze ad of te-electode dtace of 5 F Poxal 5 LDD y ax [] Dtal x ax [] Fue 6.: Aay of electode ey ccle ued fo detecto of eal uface EMG al ad t elatve poto wt epect to te vetated ucle. e y ax aled wt te ucle fbe wle t o led up wt te etated locato of te doat evato zoe. Poxal ad dtal alf of te ucle fbe ae alo dcated. e coodate xy below te coepod electode deote te elatve poto te aay wle LDD exeplfe ow tee dffeet electode cope te lotudal double dffeetal patal flte ouce: LISN Poltecco d oo Italy. e expeetal potocol wa deed by wold-ecozed boeee expet Pof. Robeto Melett D. Dao Faa ad Maco Gazzo ad coted of:. e doat a of te ubject wa placed to te oetc bace at F ee fve-ecod cotacto at axu volutay cotacto MVC foce wee pefoed epaated by ute. U te toque eo F. 6. te axu cotacto foce wa eaued ad aveaed ove all tee eaueet. Aftewad a 5-ute et wa ve to te ubject. 4

123 3. e k wa ltly abaded wt abave pate ad oteed to pove te electode-k cotact. 4. e locato of te evato zoe te doat bcep bac of te ubject wa deteed fo te tavell AP detected du volutay low foce ucle cotacto by te lea aay of 6 electode of ze ad te te-electode dtace of. Aftewad te lea aay of electode wa eoved ad te k eoteed. 5. e aay of 6 electode F. 6. wa placed ove te dtal alf of doat bcep bac wt t td electode ow ceted ove te etated evato zoe ad colu aled wt te ucle fbe. 6. Suface EMG wee ecoded du te 3 lo oetc volutay ucle cotacto utaed at 5 % ad % of MVC. Afte eac cotacto te ubject elaxed fo 5 ute. e cotacto foce wa eaued by te toque eo ad dplayed o te ocllocope to povde te vual feedback to te ubject. e detected al wee aplfed a et to by a 64-cael EMG aplfe LISN; Pa Boedcal & Spot evo Italy bad-pa flteed -3 db badwdt Hz 5 Hz ad apled at 5 Hz by -bt A/D covete. Lotudal doubledffeetal ecod tecque wa ued wt te adjacet electode pa alo te colu te electode aay F. 6. eult 5 SEMG ecod. e oe ad oveet atefact wee vually cotolled ad educed by apply wate to te k uface. Befoe ay fute poce all te eaueet wee dtally flteed to uppe te powe-le tefeece. I te cae of eal SEMG al of coue tee ae o efeece evato ta avalable. Hece te pefoace ut be evaluated by ote dect eaue. e oe ued t tudy wee: te eulaty of te ecotucted f patte te atue of IPI vaablty ad te ape of ecotucted MUAP. Suppo ot-te low level cotacto te MU evato ta ae well kow to follow eula elatvely lowly ca patte. IPI eulate ae ofte odelled a ealzato of a Gaua ado vaable woe axu value oal codto ould ot exceed te % lt of te ea IPI [56 6 9]. Ay lae eulaty te ee of dcuo above teefoe ut be take a a ealy wa of pobly wo decopoto. Slaly te ecotucted evato pule ca be evaluated wt epect to te IPI dtbuto. Seveal tattcal tet Jaque-Bea Kolooov-Sov Lllefo ca be appled to tet te pobablty tat te obtaed IPI ealato follow Gaua dtbuto o called p-value te ypote tet teoloy. Poble o-tatoate of te MU f ate ca be copeated by odell te ea IPI a a fucto of te. 5

124 Fue 6.: e oetc bace ued to keep te elbow ale cotat at deee du te acquto of uface EMG al. oque eo ae located at te jot of te dly coected pat of oetc bace ouce: LISN Poltecco d oo Italy. e lat cteo baed o te MUAP ape ad te ot tutve oe. Doubledffeetal ecod tecque ple elatvely f ltato to te ape of MUAP. Vaablty oly expected te peak-to-peak value te let of MUAP ad o ape detal aly te MUAP tal eo. Futeoe patally adjo electode ae expected to detect vey la MUAP ape. Fally by exa te MUAP ape dffeet cael te elatve poto of te detected MU wt epect to te aay of electode ad alo te tavell of AP ould be taceable. Decopo te eal al by tee dffeet decopoto tecque te eult ca alo be utually copaed. Fo te claty eao t te appopate to ft povde te eult of te ot effcet decopoto etod. Aftewad te ea decopoto tecque ca be evaluated wt epect to te upeo etod. e eult o te ytetc al poved te IC etod by fa outpefo te etod baed o F dtbuto a well a te HOS-baed etod. e la eult ca be expected alo o te eal al. eefoe we wll evee te ode of ubecto wt epect to te ubecto Secto 6.. We wll ft evaluate te eult of te IC etod. e etod baed o F dtbuto wll be evaluated Subecto 6.. wle te eult of te HOS-baed decopoto etod wll be decbed Subecto Evaluat te eult of te lat two etod we wll ely upo te eult of te IC etod. ak to accout all te decbed dect eaue te ube of uccefully ecotucted MU ta wll be codeed te ot fcat pefoace cteo. 6