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2 The Paame haaess o euey Resose uos o Nolea Sysems. J. Jg Z. Q. ag S. A. Bllgs ad G. R. Tomlso Deame o Auoma ool ad Sysems Egeeg The Uvesy o Sheeld Ma See Sheeld S JD U Reseah Reo No. 9 Augus 006

3 The Paame haaess o euey Resose uos o Nolea Sysems g-ja Jg Z-Qag ag Sehe A. Bllgs ad Geoey R. Tomlso Deame o Auoma ool ad Sysems Egeeg Uvesy o Sheeld Deame o Maheal Egeeg Uvesy o Sheeld Ma See Sheeld S JD U.. {.J.Jg Z.ag S.Bllgs & g.omlso}@sheeld.a.u Absa: The haaess o he euey esose uos o olea sysems a be evealed ad aalyzed hough he aalyss o he aame haaess o hese uos. To aheve hese obeves a ew oeao s deed ad seveal udameal ad moa esuls abou he aame haaess o he euey esose uos o olea sysems ae develoed. These heoeal esuls ovde a sga ad ovel sgh o he euey doma haaess o olea sysems ad umve a lage amou o omlaed egal ad symbol alulaos whh have evously bee eued o eom olea sysem euey doma aalyss. Seveal ew esuls o he aalyss ad syhess o olea sysems ae also develoed. Examles ae luded o llusae oeal alaos o he ew esuls. Ioduos Nolea sysems ae a moe omlex ha lea sysems ad a exhb hamos omlex e-modulaos ad eve haos Peaso 994. I ode o udesad ad uavel hese omlaed heomea may auhos have suded he aalyss o olea sysems boh he me doma ad he euey doma Gaham ad MRue 96 Sasy 999 hua ad Ng 979 Rugh 98. The sudes o olea sysems he euey doma ae based o he oe o he geealzed euey esose uos GRs Geoge 959 whh ae deed as he muldmesoal oue asomaos o he eel uos he Volea sees. May o-aame algohms have bee deved o esmae he GRs o uow o-lea sysems om u ouu daa Blla 958 m ad Powes 988 Beda 990 Nam ad Powes 994. Peyo-Joes ad Bllgs 989 deved a eusve algohm o omue he GRs o dsee me olea sysems desbed by NAR Nolea AuoRegessve model wh eogeous u models. A smla esul was develoed Bllgs ad Peyo-Joes 990 o ouous me olea sysems desbed by ego-deeal euaos. Swa ad Bllgs 00 exeded hese esuls o he ase o MIMO olea sysems. The devao o he GRs o olea sysems wh mea level o D ems was suded Zhag e al Based o hese esuls some moa haaess o he euey esose uos o olea sysems wee develoed Yue e al I ag ad Bllgs 996 ad ag ad Bllgs 997 he ouu euey esose uo ad he oesodg

4 haaess o olea sysems wee suded esevely. The boud haaess o he euey esose uos ad eegy ase haaess have also bee suded ad dsussed Zhag ad Bllgs 996 Bllgs ad ag 996 ag ad Bllgs 005. Alhough sga esuls have bee aheved may oblems ema usolved egadg he haaess o he GRs ad he sysem ouu euey esose uo ludg how he euey esose uos ae lueed by he aamees o he udelyg sysem ad he oeo o omlex o-lea behavous. The GRs ae aually a seuee o mulvaable uos deed a hgh dmesoal euey sae. The evaluao o he values o he GRs hghe ha ouh o h ode a beome had due o he lage amou o algeba o symbol maulaos ha ae volved Yue e al Moeove exsg eusve algohms o he omuao o he GRs do o exlly ad smly eveal he aalyal elaosh bewee he me doma sysem model aamees ad he sysem euey esose uos a lea ad saghowad mae. These hb he aal alao o he exsg heoeal esuls o a ea exe. Theeoe he develome o ew mehods o umve he omuaoal omlexy ad o moe lealy eveal he haaess o he euey esose uos s o gea moae o he aalyss ad syhess o olea sysems. Pevous esuls Peyo-Joes ad Bllgs 989 show ha o he NAR model he eusve omuao oedue o he GRs volves he obuo o dee model aamees o he sysem olea haaess o dee odes. ee a aleave aoah o aalyse he haaess o he euey esose uos ad he ees o dee yes o oleaes o he sysem would volve sudy o he haaess o he sysem model aamees o he euey esose uos. Theeoe a ovel ad eeve oee exao oeao s deed hs ae. Usg hs ew oeao seveal udameal esuls elag o he aame haaess o he euey esose uos o olea sysems ae deved. The aame haaess o he GRs ad he sysem ouu seum a easly be aheved usg a eusve algohm whh uses he sysem me doma model aamees whou he eed o deeme dvdual GRs. The esuls eveal he exl elaosh bewee he sysem me doma model aamees ad he sysem euey esose uos ad ovde a sga ad ovel sgh o he euey doma haaess o olea sysems. Imoa haaess o he euey esose uos o olea sysems ae also evealed ad aalyzed based o he ew aame mehod. The ew esuls deved he ese sudy ovde a ew ad eeve aoah whh should be vey useul o he aalyss ad syhess o olea sysems he euey doma. euey esose uos o olea sysems osdeg he lass o olea sysems whh ae sable a he zeo eulbum o ad whh a be aoxmaed he eghbouhood o he eulbum o by he Volea sees

5 4 N d u h y τ τ τ τ whee N s he maxmum ode o he sees ad h τ τ s a eal valued uo o τ τ whh s eeed o as he h ode Volea eel. The euey doma u-ouu deso o he sysem a be obaed as ag ad Bllgs 996 N d U Y σ π whee d d h τ τ τ τ τ τ ex s ow as he h ode Geealzed euey Resose uogr. Whe he sysem u s a mul-oe uo desbed by u os 4 The sysem ouu euey esose uo a be desbed as ag ad Bllgs 996: N Y 5 whee { } ± ± else 0 e. The GRs o se ad smle olea models a be deved by usg he obg mehod Rugh 98. osde he olea sysems whh a be desbed by he Nolea AuoRegessve model wh eogeous u NAR m m M m m u y y y y 0 6 whee y m s he mh-ode ouu o he sysem ad m. Nolea sysem 6 eeses a wde lass o olea sysems ad ludes seveal well ow olea u-ouu models as seal ases he ad Bllgs 989. I hs model he aamees suh as 0. ad 0. eese he lea sysem aamees oesodg o he lea ems he model suh as y- o u- e ad all ohe aamees eese he olea sysem aamees. s eeed o as he olea degee o he olea aamee. A olea aamee oesods o a olea em he model o he h degee o he om u y e.g.

6 5 u y. Obvously whe he oesodg aamees ae assoaed wh he lea model ems. e M m m m M 0 7 whh ludes all he aamees o he model 6. o he NAR model 6 he ollowg eusve algohm was deved by Peyo- Joes ad Bllgs 989 o omue he GRs: 0 0 ex ex 8 ex 9 ex 0 whee 0 ex. Moeove 9 a also be ewe as ex whee x x Thee ae hee yes o oleaes model 6: ue u oleaes oesodg o he olea aamees 0. whh lead o he s em he euey esose uos euao 8; ue ouu oleaes oesodg o he olea aamees 0. whh lead o he las em o euao 8; ad uouu oss oleaes oesodg o he olea aamees. whh lead o he seod em 8. I should also be oed om he eusve algohm o he h-ode GR gve 8- ha he olea aamees ae seaable om he sysem omlex valued uos hus eah em he h-ode GR has he om ~ m ae eusve omuaos whee m deoes a mullao o some olea aamees 7 whh wll be deemed lae ad ~ deoes a aoae omlex valued uo ad s deede o he olea aamees. Ths was uhe show ag

7 e al 006. The seaable oey o he euey esose uos o olea sysem 6 ovdes he bass o he sudy hs ae. Paame haaess Noe ha he olea aamees o dee olea degees oesod o dee degees o oleaes he sysem model ad he eusve algohm o he GRs. ee he haaess o he euey esose uos ad he ee o dee aamees o he sysem olea behavou a be suded hough he haaess o he oesodg me doma olea aamees he euey esose uos. The ous o hs seo s o aalyse he aame haaess o he GRs ad ouu euey esose uos o olea sysem 6 ad o sudy how hese euey esose uos ae deemed by he olea sysem me doma aamees. o hs uose a oweul oee exao oeao wll be deed ad he he aame haaess o he sysem GRs ad he ouu seum wll be vesgaed usg he ew oeao o eveal moa elaoshs bewee he euey esose uos ad he sysem olea aamees. o >.. oee exao oeao I ode o aalyze he aame haaess o he euey esose uos a useul oeao wll be deed as ollows. osde a sees whh a be we as whee he oees o ae eal o omlex umbes ad o ae eal o omlex valued uos. e [ ] [ ] T. Dee a oee Exao oeao : suh ha o ay he [ ] whee deoe all he omlex umbes ad s he -dmesoal omlex veo sae. Ths oeao has he ollowg oees whh also a as oeao ules: Redued veozed sum. [ ] whee eah eleme o belogs o bu o o.e.. Redued oee odu. ad edued hee meas ha hee ae o eeve omoes. Ivaa. a α α bu s o a oeed aamee; b 4 Uay. s o a uo o o. 6

8 Obvously whe hee s a uay hee s a osa em he oesodg sees whh has o elao wh he oees o. 5 Ivese. -. Moeove oe ha ha s he ode o he summao has o ee o he value o. Thus he oeao s also ommuave ad assoave hs sese. I should also be oed ha he oee exao s a oee oeed oeao. Tha s o all he oees volved a sees ae exaed ae alyg he oeao bu oly he oees o oe ae exaed. I wha ollows he oeao wll be used o sudy he aame haaess o he euey esose uos o sysem 6. o oveee le ad deoe mullao ad addo he sese o he edued oee odu ad veozed sum esevely o he sees. ude he odo *; whou ouso we smly as ad he oeaos ad smly as o ad esevely. Moeove dee he h degee aamee veo [ 4 4 ] whh ludes all he olea aamees m o he om. wh oleay degee 7. Noe ha a also be egaded as a se o he h degee olea aamees o he om. whh s a subse o 7. Reallg he seaable oey o he GRs oed a he ed o he las seo osde he h ode GR 8 as a sees ad he olea aamees 7 as he oees o he sees. Thus he oeao a be aled o he euey esose uos o eveal he deedee o hese uos o he olea aamees. I ode o llusae he use o he oeao he ollowg examle s gve. Examle. Aly he oeao o he GRs 8 u o he d ode. o 8 ex 0 Alyg he oeao o he s ode GR o he olea model aamees eallg he oees o he yelds o s ow om 8- ha 0 ex * * 7

9 ex ex ex ex ex ex Alyg he oeao o he d ode GR o he olea model aamees yelds ex ex ex ex o a be show om 8- ha 0 0 ex ex Alyg he oeao o he d ode GR o he olea model aamees yelds ex ex om

10 Subsug he above wo euaos ad o 4 yelds o 0 0 o o om euaos ad 5 a be eadly see ha how he olea aamees o olea degee ad ae a ole he omoso o he d ad d ode GRs. I euao dee yes ad degees o olea aamees mae deede obuos o he d GR. oweve euao 5 hee ae oss mullao ems bewee dee yes o olea aamees hus hee may be some seal olea behavou oesodg o hese ems he euey esose uo. Noe ha he s ou ems euao 5 ae olea aamees o olea degee ad he es all ome om he seod olea degee o he model aamees wh oss mullaos. Thus euao 5 lealy eveals whh ad how he dee olea aamees ae a ole he geeao o he d ode GR. The examle above demosaes ha he olea model aamees a sees o a olyomal a be eevely exaed by he oeao ad hus he haaess ha dee o-eeve aamees geeae a sees o a olyomal a be evealed by egleg he oesodg mulled uos. Theeoe he oeao ovdes a useul ool o he aalyss o he ees o he olea model aamees o he GRs ad he sysem ouu seum.. Paame haaess o he GRs I hs seo he aame haaess o he GRs ae deved ad aalyzed. The esuls ae summazed by he ollowg oosos. Pooso. The aame haaess o he h-ode GR a be obaed as ollows: 9

11 PROO O PROPOSITION : The esul a be obaed by dely alyg he oeao o euaos 8 ad. Alyg he oeao o euao 8 yelds 7 ex ex a Alyg he oeao o euao yelds ex 7b Subsug euao 7b o euao 7a he ooso ollows mmedaely. Ths omlees he oo. Pooso ovdes a exl exesso o he aame haaess o he hode GR whh eveals lealy whh ye o olea model aamees 7 ae luded he desos o he GR ad how he GRs ae deemed by hese olea model aamees. The esul daes ha he om o he seaable olea aamees he GR eeseaos a ow be desbed lealy based o euao 6. Noe ha hee ae may eeve ems euao 6 whh a also be see om he devao o 5. I ode o mae he aame haaess om Pooso moe omehesble ad o ovde lea sgh he ollowg esuls a be aled. Pooso. ludes he olea aamee 0 ad all he oeeve moomal uos o he olea aamees 7 o he om whee he subss sasy 0 ad. Tha s he se o all he subs

12 ombaos o he om oesodg o he olea aamee moomals o he om whh ae luded ae M 0 PROO O PROPOSITION : 0 s he s em euao 6. osde he las em o euao 6. Noe ha sasyg ludes all he ombaos o ad. Also oe ha se hee ae o olea aamees ad ay eeve ombaos x mae o obuo. ee a be see ha lude all he ossble o-eeve ombaos o sasyg should ad. Smlaly o. Eah o he subs ombaos oesods o a moomal o he volved olea aamees. Thus ludg he em 0 ludes all he ossble o-eeve moomal uos o he olea aamees o he om sasyg ad. Regadg he seod em o euao 6 he olea aamees aea he om hs ase ad he esuls ae smla o he above. ee he ooso ollows. I should be oed ha eeve moomals ae o osdeed Pooso. o sae he subs ombaos 0 ad 0 oesod o he olea aamee moomals o 0 ad 0 o esevely. Boh ae he same moomal hus oly oe s oued. ollowg Pooso o he seal ase: whe he ; oe ha 0 ad hus 0. Theeoe he bgges olea degee o olea aamees luded s oesodg o he olea aamees wh. Ths a be veed by Examle. I ode o uhe llusae he esul Pooso he ollowg examle s ovded. Examle. osde he d ode GR. The.

13 Whe 0 he volved olea aamees ae 0 0 ; Whe 4 whh has he ollowg o-eeve ombaos : he he volved olea aamee moomals ae: o 0 o o 0 0 o 0 0 o 0 Noe ha 0 hus he alulaos so a. The esul s osse wh Euao 5. Pooso.. PROO O PROPOSITION : Aodg o Pooso ludes all he moomals sasyg ad 0. Euao 7b a be ewe as ollowg he same dea he oo o Pooso he seod em o he gh o he eualy hs euao a be we as A Tha s all he ems A sasy ad 0.e. ad 0 oesodg o he subss o he olea aamee moomals. ee he ems A ae luded. The ooso s oved. Based o Pooso may eeve ems he exesso o he aame haaess o 6 a be aelled se he aame haaess o ae he same as hose o. The ollowg esul a be uhe obaed dely om Poosos ad. Pooso 4. Z whee Z >; Z whee Z > ad a leas oe >0 whe >. Noe ha a b deoes all he elemes a ae elemes b. om Pooso 4 s easy o deeme whh olea aamees ae luded a se GR o ay ode ad how a se olea aamee aeas dee

14 odes o he GRs. o sae osde a olea aamee. whh oesods o he olea em y 5 u. Aodg o Pooso 4 ollows Z-5. Tha s hs olea em has oly a deede obuo 5 h ode GR 5. ad has o ee o he GRs less ha he 5 h ode. Noe ha o a ovege Volea sees he magude o 5. may be vey small. ee he ee o he olea em 5 y u o he sysem behavou s sue o be vey small uless. s oely desged o he magude o he sysem u s vey lage. Based o Pooso he exesso o he aamee haaess o 6 a ow be smled as 8 0 osdeg he symmey o he las em o euao 8 oly hal o he sum s eough o lude all he ossble moomal ombaos exe he ew em 0. ee 8 a be uhe we as whee meas he ege a o.. Euao 9 s moe ose ha euao 6 ad s easy o eusvely deeme usg a omue ogam. Poosos -4 ad euao 9 demosae whh ad how he olea aamees aea he GRs. These esuls ovde o oly a lea sgh o he elaosh bewee he olea aamees o he sysem me doma model ad he sysem GRs bu hey also ovde a useul ool o aalysg he haaess o he GRs. Based o he esuls above some uhe esuls a be obaed eseally o some seal bu euely eoueed ases a o whh wll be suded Seo Paamee haaess o he ouu euey esose uos Based o he deo o he oeao ad he heoeal esuls aheved above hee exss a omlex valued uo veo wh aoae dmeso whh s a omlex uo o suh ha 0 whee s deed 9. Subsug 0 o euao yelds 9

15 4 N N d U d U Y σ π σ π e σ π d U gvg N Y a Euao a ovdes a exl exesso o he ouu euey esose uo o he NAR model 6 ude a geeal u whh s desbed as a olyomal om ems o he model olea aamees. A smla esul was also obaed ag e al 006 usg a dee mehod. The ese sudy odues hs exesso wh muh moe deal ad eveals he elaosh bewee he model aamees ad he ouu euey esose uo moe lealy. om euao a he aame haaess o he ouu seum ude a geeal u s obvously N Y b Smlaly whe he sysem 6 s sube o a mul-oe u he ouu euey esose uo a be obaed om 5 ad 0 as N N Y e ~ gvg N Y ~ Obvously he aame haaess o he ouu seum ude a mul-oe u ae he same as euao b. Euaos 0- gve he aamee haaess o he sysem ouu euey esose uos whh show a aalyal elaosh bewee he sysem model aamees ad he sysem euey esose uos ad ovde a moa sgh o he euey doma haaess o olea sysems. 4 Some uhe esuls Some haaess o he GRs ad he ouu seum a be evealed ad deved easly by aalyzg he aame haaess o he euey esose uos. Based o he heoeal esuls develoed he evous seo some uhe esuls ae ovded o a seal ase o he NAR model 6 he ollowg o demosae

16 oeal alaos o he heoeal esuls. Moe dealed sudes o he alao ssues wll be eseed ohe ublaos. o he NAR model 6 hee ae oly ue ouu oleaes he model he M y 0... y δ m 0 u m m 0 whee δ m. o may egeeg alaos hs model a be used o 0 else eese a olea eedba ool sysem ad oseuely has sgae he aalyss ad syhess o olea eedba ool sysems ae Jg e al The aamee haaess o he GRs Nog ha oly he olea aamees 0. o > ae ozeo hs ase he GRs o model a be we om 8 as 0 4 om euao 9 he aamee haaess o he h-ode GR s Euao 5 s us a seal ase o euao 9. I hs ase ag he olea aamees 0. o he Pooso 5 ollows. Pooso 5. 0 aeas m. om he mh ode whee m-. I 0 0 o he... 0 o all.... The vese o hs o does o hold. PROO O PROPOSITION 5: Aodg o Pooso o Pooso 4 he em 0 should be he GR m. whee m s omued as mm-. ee we have m -. om 5 ludes all he olea aamees 0 0 o ad o olea aamees 0 0 o > ae luded. Theeoe 0 0 o he 0 whh uhe ollows om 0 ha... 0 o all.... Noe ha 0 aeas. o om Pooso 5 whh mles 0 oly obues o he odd ode o he GRs. ee eve ea eve ode GR s zeo 0 a sll be ozeo. Thus he vese o he saeme above s o ue. Ths omlees he oo. om Pooso 5 he d ad d degee olea aamees ae all zeo 0 0 ad 0 0 he.0 ad.0. oweve eve 0 0 o > bu hee ae ozeo 5

17 ems 0 o 0 he h ode GR. may o be zeo beause 0 aeas. o ad 0 aeas. o om Pooso 5. Ths mles ha he olea aamees 0 ad 0 ae muh geae oles he GRs ha ohe olea aamees. Tha s he hghe he olea degee o a aula model ems he smalle he ee o he assoaed olea aamees o he sysem beomes se he hghe degee > olea aamees oly ae a ole he hghe ode GRs. Theeoe may ases 0 om o ae lely o om he bass o he GRs ad he sysem ouu seum. Moeove he esuls obaed above also demosae ha dee yes o oleay have dee ees o he sysem ouu seum ad olea ems o he same olea degee may have a smla ee o he ouu seum. 4.. A seal ase I hs subseo aohe ase o he NAR model s osdeed o demosae ha some haaess o he GRs a be suded by oly aalysg he aame haaess o he GRs whou he eed o omleely evaluae he GRs. Ths seal ase ludes he wo dee suaos o 0.0 o 0.0 o. Some sga olusos egadg he haaess o he GRs hese wo suaos a be eahed. Pooso 6. o he GRs o olea sysem I 0.0 ad 0. 0 o he.0 ad. 0. I 0. 0 ad 0.0 o he. 0 ad. 0. PROO O PROPOSITION 6: om he deo o he oeao ad he heoeal esuls above o ove.0 o 0 we eed oly o ove.0 o 0. Aodg o euao 5 we have 0 0 o. o. o o 0. o a 6b I 0.0 o he euao 6a ollows o. 50 o 4. 0 o. 7 whee s he lages odd umbe less ha o eual o. Noe ha - -4 ae all eve umbes. ee om he eusve alulaos o 7 a be show ha 0 o all Smlaly euao 6b ollows 0 0 o. 0. whee s he lages odd umbe less ha o eual o. Noe ha 0 0 hus 0. Ths oves he s oluso o he ooso. 6

18 I 0.0 o he euao 6a ollows 0 0 o. 40 o. 0 o. whee s he lages eve umbe less ha o eual o. Noe ha 0 0 hus 0. Smlaly euao 6b ollows o. 40 o whee s he lages eve umbe less ha o eual o. ee om he eusve alulaos o 8 a be show ha 0 o all Ths omlees he oo. Pooso 6 shows lealy ha some haaess o he GRs a be suded easly hough he aame haaess o he oesodg GRs ad hs ovdes a ew ad eeve aoah o he aalyss o olea sysems he euey doma. Noe ha eah GR dees he ouu seum o olea sysems ove a oesodg ouu euey age ag ad Bllgs 996. Thus.0 o some mles ha hee ae o ouu euey sea ove he euey age oesodg o.. Theeoe Pooso 6 mles ha 0. 0 ad 0.0 o he he ouu euey esose o he sysem s avalable ove a wde euey age ha he ase whee 0.0 ad 0. 0 o. Ths omme a be veed by he ollowg examle. Examle. osde a smle olea sysem desbed as y 0.y 0.5y 0.u ay by whh a be we he om wh a 0 b else 0 ad M. Thee ae oly ue ouu oleaes hs model. e u0s50. osde wo ases: 0.0 ad 0. 0.e. a0 ad b-0.005; 0. 0 ad 0.0.e. a ad b0. I he wo ases he sysem ouu seum a be obaed by alyg he T o he sysem me doma ouu. The esuls ae show gue ad. I a be see om he wo gues ha he ouu seum o he same u s ue dee. As exeed whe a ad b0 hee ae moe ouu euey omoes e.g. a euees ha he ase whee a0 ad b

19 8 Ouu euey Seum Y euey Rad/se gue. Sysem ouu seum whe a0 ad b Ouu euey Seum Y euey Rad/se gue. Sysem ouu seum whe a ad b0 8

20 4. Deemao o he ouu euey esose uo om euaos ad he ouu seum o olea sysem a uomly be ewe as N Y Y 9 Y ˆ I 9 a be omued aodg o 5 ad ˆ a be deemed usg a eeve daa aalyss based mehod oosed ag e al Theeoe he ouu euey esose uo o sysem a be obaed whou a lage amou o eusve alulaos o he GRs ad he oesodg omlaed egal ems o 5. uhe sudes wll vesgae emloyg he dealed omao abou he suue o euao 9 ovded by he aalyss o he aamee haaess o he sysem GRs o dely deeme a auae deso o he sysem ouu euey esose uo. 5 olusos Seveal udameal ad moa esuls elag o he aame haaess o he GRs ad he ouu euey esose uos o olea sysems desbed by a NAR model have bee deved hs ae. These esuls eevely eveal he elaosh bewee he sysem model aamees he me doma ad he sysem euey esose uos ad ovde o he s me a sga ad ovel sgh o he euey haaess o olea sysems. uhe heoeal esuls ad aal algohms o he aalyss ad syhess o olea sysems he euey doma a be develoed based o hese ew esuls. uhe eseah wll ous o he alao o he aame haaess o olea sysems deved he ese sudy o vesgae he aalyss ad desg o olea sysems he euey doma. The aalyss wll vesgae he ees o he sysem model aamees o he GRs ad he ouu seum o eveal how he model aamees ae he sysem behavo he euey doma. Desgs o omal values o some o he model aamees o aheve a desed sysem ouu euey esose wll also be suded lae ublaos. Aowledgeme The auhos gaeully aowledge he suo o he Egeeg ad Physal See Reseah oul U ad he EPSR-uhso Whamoa Doohy odg Posgaduae Awad o hs wo. Reeees J.S. Beda Nolea Sysem Aalyss ad Ideao om Radom Daa New Yo: Wley

21 S.A. Bllgs ad Z.Q. ag A boud o he magude haaess o olea ouu euey esose uos Ieaoal Joual o ool Pa Vol 65 No ad Pa Vol 65 No S.A. Bllgs ad J.. Peyo-Joes Mag olea ego-deeal euao o he euey doma Ieaoal Joual o ool Vol M.B. Blla Theoy o he aalyss o o-lea sysems Tehal Reo 45 MIT Reseah aboaoy o Eleos ambdge Mass Ma S. he ad S. A. Bllgs. Reeseao o o-lea sysems: he NARMA model. Ieaoal Joual o ool O. hua ad.y. Ng euey doma aalyss o olea sysems: geeal heoy IEE Joual o Eleo us ad Sysems D.A. Geoge ouous olea sysems Tehal Reo 55 MIT Reseah aboaoy o Eleos ambdge Mass. Jul D. Gaham ad D. MRue Aalyss o olea ool sysems. New Yo; odo : Wley 96..J. Jg Z.Q. ag ad S.A. Bllgs. euey Doma Aalyss Based Nolea eedba ool o Suessg Peod Dsubae. The 6h Wold ogess o Iellge ool ad Auomao Jue - ha 006.I. m ad E.J. Powes A dgal mehod o modellg uadaally olea sysems wh a geeal adom u IEEE Tasaos o Aous Seeh ad Sgal Poessg Z.Q. ag ad S. A. Bllgs. Ouu euey haaess o olea sysems. Ieaoal Joual o ool Vol Z.Q. ag ad S. A. Bllgs. Ouu euees o olea sysems. Ieaoal Joual o ool Vol. 67 No Z.Q. ag ad S.A. Bllgs Eegy ase oees o olea sysems he euey doma Ieaoal Joual o ool Vol Z.Q. ag S.A. Bllgs R. Yue ad J. Ouu euey esose uos o olea Volea sysems Reseah Reo No 90 Deame o ASE Uvesy o Sheeld 006. A ull veso has bee ovsoally aeed by Auomaa S.W. Nam ad E.J. Powes Alao o hghe-ode seal aalyss o ubally olea-sysem deao IEEE Tasaos o Sgal Poessg Jul R.. Peaso Dsee me dyam models Oxod Uvesy Pess994. J.. Peyo Joes ad S.A. Bllgs. Reusve algohm o omug he euey esose o a lass o olea deee euao models. Ieaoal Joual o ool Vol. 50 No W.J. Rugh Nolea Sysem Theoy: he Volea/Wee Aoah Balmoe Maylad U.S.A.: Johs os Uvesy Pess 98. S. Sasy Nolea sysem : aalyss sably ad ool. New Yo : Sge 999 A.. Swa ad S.A. Bllgs. Geealzed euey esose uo max o MIMO olea sysems. Ieaoal Joual o ool. Vol. 74. No R. Yue S. A. Bllgs ad Z.-Q. ag A vesgao o he haaess o olea euey esose uos. Pa : Udesadg he hghe dmesoal 0

22 euey saes. Ieaoal Joual o ool Vol. 78 No ; ad Pa New aalyss mehods based o symbol exasos ad gahal ehues Ieaoal Joual o ool Vol Zhag ad S.A. Bllgs Ga bouds o hghe ode olea ase uos Ieaoal Joual o ool Vol 64 No Zhag S.A. Bllgs ad Q.M. Zhu euey esose uos o olea aoal models. Ieaoal Joual o ool