CYCLOSAIONARIY-BASED BLIND CLASSIFICAION OF ANALOG AND DIGIAL MODULAIONS Oavia A. Dobe Ali Abdi 2 Yehekel Ba-Ne 2 Wei Su 3 Fauly of Eng. Applied Siene Memoial Univeiy of Newfoundl S. John NL AB 3X5 Canada 2 CCSPR Dep. of ECE New Jeey Iniue of ehnology Newak NJ 72 USA 3 RDECOM Fo Monmouh NJ 773 USA dobe@eng.mun.a abdi@adm.nji.edu bane@yegal.nji.edu wei.u@u.amy.mil Aba - he poblem of blind analog digial modulaion laifiaion i akled in hi pape. A yloaionaiy-baed laifie i popoed whih doe no euie eimaion of aie phae feueny offe ignal noie powe iming eovey a pepoeing ak. Numeial imulaion ae pefomed o validae heoeial developmen. I. INRODUCION Blind modulaion laifiaion (MC i a majo ak of an inelligen eeive wih boh miliay ommeial appliaion []. A modulaion laifie eenially involve wo ep i.e. ignal pepoeing modulaion eogniion. Pepoeing ak may inlude eimaion of aie phae feueny offe ignal noie powe iming eovey e. Depending on he laifiaion algoihm employed in he eond ep diffeen pepoeing ak an be euied. Claifiaion algoihm whih ely le on pepoeing ae ough. Reeah on MC ha been aied ou fo a lea a deade []-[9]. Many algoihm popoed in he lieaue adde he digial modulaion laifiaion poblem []-[5]. A uvey of uh laifiaion ehniue i peened in []. Nonehele laifiaion of analog digial modulaion i of inee a ome exiing yem ill ue analog ommuniaion ehniue. Algoihm fo he eogniion of analog digial modulaion ae alo popoed in he lieaue [5]-[8]. Howeve he pefomane of mo of hee algoihm i peened unde he aumpion of pefe pepoeing i an ubanially degade wih model mimahe uh a aie phae feueny offe. In hi pape we popoe a novel algoihm whih exploi ignal yloaionaiy fo analog digial modulaion laifiaion. he algoihm doe no euie eimaion of aie phae feueny offe ignal noie powe iming eovey in he pepoeing ep i appliable o he following pool of modulaion: ampliude modulaion (AM double ideb (DSB ingle ideb (SSB M-ay phae hif keying (M-PSK M-ay uadaue ampliude modulaion (M-QAM. he e of he pape i oganized a follow. he ignal model oeponding aiial haaeizaion ae peened in Seion II he popoed yloaionaiy-baed laifiaion algoihm i inodued in Seion III imulaion eul ae given in Seion IV onluion ae dawn in Seion V. empoal yloaionaiy paamee ae defined in Appendix A eul fo he fi- / o eond-ode yloaionaiy of he ignal of inee ae deived in Appendix B. II. SIGNAL MODEL AND CORRESPONDING SAISICAL CHARACERIAION A. Signal Model Le he eeived baeb wavefom ( be he um jθ j2π f ( = e e ( + w( ( whee θ i he aie phae f i he aie feueny offe ( = I ( + jq ( i he anmied ignal wih modulaion i w ( = wi( + jwq( i zeo-mean omplex Gauian noie of powe N wih indexe I Q ing fo in-phae uadaue epeively j =. he in-phae uadaue noie omponen wi ( wq ( ae zeo-mean unoelaed Gauian poee unoelaed wih (. he anmied ignal i given by ( = A( +µ Ax( fo i=am ( = Ax( fo i=dsb ( = A( x( ± jx (( fo i=ssb ( = A k px( k ε fo i=m-psk k M-QAM whee A i he ignal ampliude µ A i he modulaion index x( i he zeo-mean eal-valued b-limied modulaing ignal ( x( i he Hilbe anfom of x( px ( i he anmi pule hape i he ymbol peiod ε epeen a iming eo wih ε< k = k I + jk Q i he ymbol anmied wihin he k h peiod dawn fom he ignal onellaion i i=m-psk M-QAM M i he numbe of poin in he ignal onellaion []. he daa ymbol { k } ae aumed o be zeo-mean independen idenially diibued ( om vaiable wih { i ki } ( { i kq } unoelaed. A he eeive-ide nomalizaion of he ignal i aied ou wih epe o he eeived ignal powe o emove any ale fao fom daa hen ampling i pefomed a a ampling ae f (no aliaing ou []. he nomalized diee-ime ignal i ( k: = ( / S + N wih S a he ignal powe. = kf
B. Cyloaionaiy of Signal of Inee Reul deived in Appendix B fo he fi- /o eond-ode yli empoal umulan funion (CCF of he ignal of inee ae peened in he euel wih emphai on hei appliaion o modulaion laifiaion. he fi-ode/ zeo-onjugae CCF of AM DSB SSB M-PSK M-QAM nomalized diee-ime eeived ignal a yle feueny (CF β ae a follow jθ /2 (AM ( β = Ae ( S + N wih β= f f (2 ( i ( β = any β i = DSB SSB M-PSK M-QAM. (3 One an hu onlude ha he eeived AM ignal exhibi fiode yloaionaiy wheea all ohe ignal of inee do no. hi popey of he AM ignal i exploied fo i eogniion. he eond-ode/ zeo-onjugae CCF of DSB SSB M-PSK M-QAM M 4 BPSK nomalized diee-ime eeived ignal a CF β delay τ ae a follow (DSB 2 2 j 2 f f j θ π τ x (; βτ = A ( S + N e e m ( τ f (4 wih 2 f f β= mx( τ f a he eond-ode/ zeoonjugae momen of xk ( = x ( = kf ( i (; β τ = any β i =SSB M-PSK M-QAM M 4(5 2 j2πγfε (BPSK (; βτ = A ( S + N f e (6 j2θ j2π fτf j2πγk e e px ( k px ( k+τ e k = wih 2 f f β=γ+ γ= f k/ k inege 2 p X ( k = p X ( kf. = hu fo i=dsb BPSK (; βτ a CF β= 2 f f 2 ff + f k/ 2 epeively wheea fo i=ssb M-PSK M-QAM M 4 (; β τ = any β. hi popey i exploied hee o diinguih beween DSB BPSK SSB M-PSK M-QAM M 4. he eond-ode/ one-onjugae CCF of DSB SSB M-PSK M-QAM nomalized diee-ime eeived ignal a CF β delay τ ae a follow 2 j 2 f f π τ x (DSB (; βτ = 2 A ( S + N e m ( τf (7 + w( β; τ wih β= mx( τ f a he eond-ode/ one-onjugae momen of x( k (; w βτ a he of eond-ode/ one-onjugae yli umulan of wk ( = w ( / S+ N a CF β delay τ = kf 2 j2π fτf (SSB (; βτ =2 A ( S + N e (8 ( mx( τf m jm ( x( τf + w( β; τ wih β = m( (. x a he Hilbe anfom of m (. x 2 j2πγfε j2π fτf (; βτ = A ( S + N f e e (9 * j2πγk px ( k px ( k+τ e + w( β; τ k = wih β=γ= f k/ k inege 2 i=m-psk M-QAM M 2 * a he omplex onjugae. hu fo i = DSB SSB (; β τ a β = only wheea fo i=m-psk M-QAM M 2 (; β τ a β= f k/ k inege 2. hi popey i exploied o diinguih beween DSB BPSK a well a beween SSB M-PSK M-QAM M 4. III. PROPOSED CYCLOSAIONARIY-BASED CLASSIFIER he binay deiion ee algoihm popoed fo blind eogniion of analog digial modulaion i depied in Fig.. A eah node of he ee ignal yloaionaiy i exploied o make a deiion on he modulaion foma of he eeived ignal. A. Binay Deiion-ee Claifiaion Algoihm A Node of he ee diiminaion beween AM DSB SSB M-PSK M-QAM M 2 i pefomed by exploiing he peene of a CF in he fi-ode/ zeo-onjugae CCF of he AM ignal. he eogniion poe i blindly pefomed oniing of he following wo ep: Sep I: he magniude of he fi-ode/ zeo-onjugae CCF i eimaed fom he finie lengh eeived daa euene a idae CF β' ove he [ //2 ange. Fo an AM ignal a peak ou in hi magniude a CF β= f f i value deeae wih a eduion in he ignal o noie aio (SNR defined a SNR:= S/ N (ee (2. hu below a eain SNR he value of hi peak beome ompaable wih he aiially inignifian pike whih appea in he eimaed CCF magniude a idae CF β' β due o he finie lengh of daa euene 3 (example ae povided fo illuaion in Seion IV. If a peak i deeed in he eimaed magniude of he fi-ode/ zeo-onjugae CCF Sep II i applied o he idae CF oeponding o hi peak. Ohewie i i deided ha he modulaion foma i no AM. Sep II: he yloaionaiy e developed in [2] i ued o e he idae CF oeponding o he peak deeed in Sep I. Definiion of he empoal yloaionaiy paamee i given in Appendix A. 2 Noe ha he inege k ake value in a denumeable e whih depend on he yloaionaiy ode ignal bwidh [2]. 3 Appaenly wih a finie lengh daa euene neihe he fi-ode eimaed CCF of AM a idae β β ' no of DSB SSB M-PSK M-QAM M 2 a any idae CF β ' ae idenially zeo. Noe ha eul of Appendix B need o be undeood a aympoial value whih ae obained by aveaging ove an infinie-ime ineval.
Deail on hi e ae given in Seion III. B. If he eed idae CF β ' i deided o be indeed a CF β he ignal i idenified a AM; ohewie a belonging o he DSB SSB M-PSK M-QAM ignal la. A Node 2 diiminaion beween DSB BPSK SSB M-PSK M-QAM M 4 i pefomed by exploiing he peene of a CF in he eond-ode/ zeo-onjugae CCF of he DSB BPSK ignal fo zeo delay ( τ=. he laifiaion poe i imila o ha fo Node. A Node 3 Node 4 diiminaion beween DSB BPSK SSB M-PSK M-QAM M 4 i epeively pefomed by exploiing he peene of non-zeo CF in he eond-ode/ one-onjugae CCF of M-PSK M-QAM ignal M 2 fo zeo-delay ( τ=. he laifiaion poe i imila o ha fo Node wih Sep I applied o he ixh-ode/ hee-onjugae yli empoal momen funion (CMF Sep II o he eond-ode/ one-onjugae CCF. he eondode/ one-onjugae CCF magniude i no ued in Sep I a i value a non-zeo CF β=± f / i low even a high SNR peak oeponding o hee CF anno be diinguihed fom he aiially inignifian pike whih appea due o eimaion baed on a finie lengh daa euene. Inead he wo ymmei peak ae idenified fom he ixh-ode/ hee-onjugae CMF magniude 4. he idae CF whih oepond o he highe peak ou of he wo i hen eed in Sep II. I i o be noed ha he laifiaion poe doe no euie knowledge of he aie phae feueny offe ignal noie powe iming eovey. In addiion i i o be noed ha laifiaion beyond Node 4 an be pefomed by applying ohe algoihm popoed in lieaue uh a []-[3] fo linea digial modulaion [7] fo SSB ignal. B. Cyloaionaiy e A menioned in Seion III. A he yloaionaiy e developed in [2] i ued o hek he peene of a CF a eah node l l =...4 of he popoed laifiaion algoihm. he e i fomulaed a a wo hypohei-eing poblem i.e. H : he eed idae CF β ' i no a CF H : he eed idae CF β ' i indeed a CF. he e oni of hee ep i applied hee fo zeo delay 5 a follow. Sep : he nh-ode/ -onjugae CCF 5 i eimaed fom K ample a he idae CF β ' eleed in Sep I fo zeo delay i.e. ˆ ( β'; 5. hen he veo 4 By uing he yli umulan-o-momen fomula along wih (2-(22 (25-(26 one an eaily pove ha he magniude of he ixh-ode/ hee-onjugae CMF a β=± f / i geae han ha of he eondode/ one-onjugae CCF a he ame CF. 5 Fo Node n = = fo Node 2 n = 2 = fo Node 3 4 n = 2 =. ˆ : [Re{ ˆ ( '; } Im{ ˆ ( '; }] ( ( = K β K n n β 5 ( i fomed wih Re{.} Im{.} a he eal imaginay pa epeively. Sep 2: he aii ˆ = K Σ n n n ˆ ˆ 5 ( i ompued fo he eed idae CF β '. Hee - denoe maix anpoe invee epeively ˆ Σ i n an eimae of he maix Re {( Q + Q / 2} Im {( Q Q / 2} Σ = 5 (2 n Im {( Q + Q /2} Re {( Q Q /2} whee ( K ( K Q : = lim Cum[ ˆ ( β'; ˆ ( β'; ] n K ( K ( K* Q = ˆ β ˆ β K : lim Cum[ ( '; ( '; ] wih Cum[.] a he umulan opeao 6. he ovaiane Q Q ae given epeively by 7 8 [2] K = k n K = +ξ ξ= j2π2 β' k j2 πβ' ξ Q lim K Cum[ L ( k; L ( k ; ] e e K * = =- n K k= ξ +ξ j2 π( β' ξ Q lim K Cum[ L ( k; L ( k ; ] e whee L ((* ( ( k n i u i ; : = ( k i he n h-ode/ u = -onjugae lag podu of ( k fo zeo delay-veo wih ( u u =... n a a poible onjugaion o ha he oal numbe of onjugaion i. Sep 3: he aii 5 eimaed a eed idae CF β ' fo zeo delay i ompaed again a hehold Γ ( l ( l l =...4 fo deiion-making. If 5 one deide ha he eed idae CF i indeed a CF fo zeo delay; ohewie i i no delaed a CF. he hehold i e fo a given pobabiliy of ( l ( K ( l fale alam PF = P{ n H} 5 l =...4 by aking ino aoun ha 5 ha an aympoi χ 2 diibuion wih wo degee of feedom unde H [2]. 5 IV. SIMULAION RESULS A. Simulaion Seup Fo he geneaion of analog ignal he modulaing ignal x( i obained by low-pa fileing a euene of zeo-mean Gauian om numbe wih uni vaiane. he analog ignal 6 Fo he umulan definiion ee e.g. [3] Ch.2. 7 Noe ha hee eualiie ae alway valid if n = wheea only fo a zeo-mean poe if n = 2. 8 Fo he ovaiane eimao ee e.g. [2] e. (48. 5
ae aled o ha he ignal powe i. Fo he AM ignal he modulaion index µ A i omly hoen beween. Fo he linea digial modulaion la we imulae BPSK QPSK 8-PSK 6-QAM 64-QAM wih uni vaiane onellaion. he ignal powe i alo e o he pule hape px ( i oo-aied oine wih.25 oll-off fao []. he ignal bwidh i B = 3 KHz. A he eeive-ide a low-pa file i ued o eliminae he ou-of-b noie he eeived ignal i ampled a a ae f = 48 KHz. he obevaion ineval available 4 a he eeive i eond whih yield K = 4.8 ample. he SNR i defined a he ignal powe o he noie powe a he oupu of eeive file. All eeived ignal ae affeed by a phae θ unifomly diibued ove [ π π a aie feueny offe f f =.. In addiion linea digial ignal ae affeed by a iming eo ε wih ε =.8. B. Eimaed CCF Magniude Saii Ued fo Deiion- Making hehold Seing he magniude of he fi-ode/ zeo-onjugae CCF of AM BPSK ignal ae given fo illuaion in Fig. 2 a b a -db -2dB SNR epeively. Fo AM one an noie he peak in ˆ (AM ( β ' a β=β= ' f. f = a well a he eduion of i value wih a deeae in SNR. On he ohe h non-zeo pike due o eimaion baed on a finie lengh daa ( euene ae o be noed in ˆ K ( i ( β ' i=am DSB. Baed on ˆ ( β'; 5 he aii 5 i alulaed a eah node l l =...4 aoding o (. A Kaie window of lengh 6 paamee wa ued o ompue he ovaiane eimae in (2 8. A eah node l he aii i ompaed again a hehold Γ ( l ( (2 l =...4. Hee Γ =Γ =8.42 (3 (4 Γ =Γ =3.86. hee value oepond o a 4 pobabiliy of fale alam of 3 epeively [4] ae e baed on he eimaed value of he aii ued fo deiion-making. C. Claifiaion Pefomane o evaluae he pefomane of he popoed laifiaion algoihm we define he aveage pobabiliy of oe ( l 2 ( li li laifiaion a eah node l l =... 4 a P = 2 P li = ( i i wih P l l a he aveage pobabiliy o hooe he banh l i l i = 2 when indeed he banh l i i he oe hoie. he ( i i pobabiliy l l ( N li mod ( l ( i i li P i fuhe ompued a P / i N = mod wih ( l N i mod a he numbe of poible modulaion fo he l i h banh ( i P l i a he pobabiliy o hooe he l i h banh when he ( li i h ignal i anmied i =... Nmod. Fo example wih ( ( ( l = (Node l = (AM banh N mod = P = P. ( i he P l i i eimaed baed on 3 Mone Calo ial. he aveage pobabiliy of oe laifiaion a eah node ( l of he popoed binay deiion ee algoihm P l =...4 i ploed veu SNR in Fig. 3. A Node ( l = a pobabiliy of oe laifiaion of one i ahieved a SNR a low a -23dB o diiminae AM fom DSB SSB BPSK QPSK 8-PSK 6-QAM 64-QAM. A Node 2 ( l = 2 uh a pefomane i aained a SNR a low a -8dB o diiminae DSB BPSK fom SSB QPSK 8-PSK 6-QAM 64-QAM. A Node 3 4 ( l = 3 4 5dB SNR i euied o diiminae wih pobabiliy one beween DSB BPSK SSB QPSK 8-PSK 6-QAM 64-QAM epeively. Noe ha he aii whih i uenly ued a Node 3 4 an be applied o diinguih beween analog digial modulaion a Node i.e. o diiminae AM DSB SSB fom M-PSK M-QAM. Howeve highe SNR would be euied o ahieve he ame laifiaion pefomane a Node hu he pefomane of he whole laifie would be limied. I i alo o be noed ha eul peviouly peened ae ahieved fo eond obevaion ineval. Appaenly if a lage obevaion ineval i available a he eeive-ide he eimae will be moe auae whih in un will eul in a bee laifiaion pefomane. V. CONCLUSION A yloaionaiy-baed binay deiion ee algoihm ha been popoed fo analog digial modulaion laifiaion. he algoihm povide a good laifiaion pefomane in addiive Gauian noie doe no euie eimaion of aie phae feueny offe ignal noie powe iming eovey in he pepoeing ep. REFERENCES [] O. A. Dobe A. Abdi Y. Ba-Ne W. Su A uvey of auomai modulaion laifiaion ehniue: Claial appoahe new end o be publihed in IEE Po. Commun. 26. [2] O. A. Dobe Y. Ba-Ne W. Su Highe-ode yli umulan fo high ode modulaion laifiaion in Po. IEEE MILCOM 23 pp. 2-7. [3] C. M. Spoone Claifiaion of ohannel ommuniaion ignal uing yli umulan in Po. ASILOMAR 995 pp. 53-536. [4] O. A. Dobe A. Abdi Y. Ba-Ne W. Su Seleion ombining fo modulaion eogniion in fading hannel in Po. IEEE MILCOM 25 pp.-7. [5] E. E. Azzouz A. K. Ni Auomai Modulaion Reogniion of Communiaion Signal. Kluwe 996. [6] Y.. Chan L. G. Gadboi P. Yanouni Idenifiaion of he modulaion ype of a ignal Signal Poe. vol. 6 pp. 49-54 989. [7] O. A. Dobe A. Abdi Y. Ba-Ne W. Su he laifiaion of join analog digial modulaion in Po. IEEE MILCOM 25 pp.-6. [8] Y. O. Al-Jalili Idenifiaion algoihm of uppe ideb lowe ideb SSB ignal Signal Poe. vol. 42 pp. 27-23 995.
[9] W. A. Gadne Cyloaionaiy in Communiaion Signal Poeing. New Yok: IEEE Pe 994. [] A. B. Calon P. B. Cilly J. C. Ruledge Communiaion Syem 4h ed. MGaw Hill 22. [] A. Napoliano Cyli highe-ode aii: inpu/oupu elaion fo diee- oninuou-ime MIMO linea almo-peiodially imevaian yem Signal Poe. vol. 42 pp. 47-66 995. [2] A. V. Dawade G. B. Giannaki Saiial e fo peene of yloaionaiy IEEE an. Signal Poe. vol. 42 pp. 2355-2369 994. [3] C. L. Nikia A. P. Peopulu Highe-Ode Spea Analyi: A Nonlinea Signal Poeing Famewok. Penie Hall 993 [4] M. Abamowiz I. A. Segun Hbook of Mahemaial Funion. New Yok: Dove Publiaion 972. [5] R. C. Cabo A noe on he appliaion of he Hilbe anfom o ime delay eimaion IEEE an. Aou. Speeh Signal Poeing vol. 29 pp. 67-69 98. APPENDIX A: EMPORAL CYCLOSAIONARIY PARAMEERS In he faion-of-ime (FO pobabiliy famewok ignal ae modeled a ime-eie ahe han ealizaion of ohai poee [9] []. hu wih he FO pobabiliy appoah aiial paamee ae defined hough infinie-ime aveage ahe han enemble aveage. hi appoah i ubeuenly ued o peen empoal yloaionaiy paamee. Le ( be a oninuou-ime omplex-valued ime-eie L % (; τ % be he n h-ode/ -onjugae lag podu of ( defined a L n % ((* ( (; i u i : = ( + u= u (3 whee τ % = [% τ = % τ2... % τ n] i he delay-veo. he ime-eie ( i aid o be n h-ode yloaionay fo a given onjugaion onfiguaion ( -onjugae if m% % I L% % e d I /2 j2πα% ( α % ; τ : = lim ( ; τ I I /2 j2πα : =< L% % ( ; e (4 exi i non-zeo fo ome delay veo a denumeable e of eal α% wih α % [9]. E. (4 define he CMF wih α% a CF. I i o be noed ha CMF aie fom a onideaion of he finie-engh addiive ine-wave omponen in he lag podu. he um of all uh omponen i given by he empoal momen funion (MF [9] { α} j 2πα m% ( ; τ % :=E % [ L% ( ; τ % ] = m% ( α% ; τ % e % (5 ( i ( i m ( i n n α κ % % m whee κ % = { α% : m% ( α% ; } { α% E } [.] i he expeaion opeao in he FO pobabiliy famewok whih eplae he uual opeao E[.] in he ohai famewok. hi aually pefom a ine-wave exaion opeaion i.e. i exa all addiive ine-wave omponen exien in i agumen. he n h-ode/ -onjugae empoal umulan funion (CF of ( an be expeed in em of he n h- lowe-ode MF by uing he momen-o-umulan fomula [9] % ( ; : = Cum[ (... ( +τ % ( + ] ( i ( i(* ((* i n ((* i n n n = m ( i {... } % z= z nz z ( (! ( ; (6 whee {... } i a paiion of = { 2... n} wih z z =... a non-empy dijoin ube of o ha hei eunion i i he numbe of he ube in a paiion ( n z i a delay veo whoe omponen ae elemen of { } n u u= wih indie peified by z n z i he numbe of elemen in he ube z fom whih z oepond o onjugae em. Noe ha z= nz = n z= z =. Fuhemoe % ( i (; un ou o be an almo-peiodi funion of ime ha an be wien a [9] % % % % % e πβ (7 j 2 % ( i (; τ = ( i ( ; β κ % β τ % whee % % % I % % e d (8 I /2 j2πβ% ( i (; β τ = lim ( i (; I τ I /2 i he n h-ode/ -onjugae CCF a CF β % κ % = { β% : % ( β% ; }. By uing (5 (6 (7 he n h-ode/ -onjugae CCF of ( a a CF β % an be expeed a [9] % % % ( i (; β τ = ( (! {... } m ( i =β % α% z= z α% % z nz z ( ; (9 whee α % = [ α%... α% ] i a e of CF i a -dimenional one veo. E. (9 i efeed o a he yli momen-oumulan fomula. A imila expeion an be wien o expe he CMF a a funion of CCF. hi i efeed o a he yli umulan-o-momen fomula i given by [9] m% ( ( α i % ; = ( i ( β ; {... } % % % =α z= z % z nz (2 β z whee β % = [ β%... β% ] i a e of CF. Fo he diee-ime ignal ( ( k i = ( obained by = kf ampling he oninuou-ime ignal ( a a ampling ae f he n h-ode/ -onjugae CCF he e of CF ae epeively given a (unde he aumpion of no aliaing [] % f f (2 (; β τ = ( β ; τ κ = { β [ /2;/2 : β = β% / f ( i ( β; τ } (22 whee τ = f wih omponen τ u = uf u =... n. Eimao of he yloaionaiy paamee defined in he FO famewok ae obained by onideing finie-ime aveage onvege aympoially o he ue value whih ae he infinie-ime aveage of he ame uaniie [9].
APPENDIX B: DERIVAIONS OF (2-(9 Subeuenly fi- / o eond-ode empoal yloaionaiy paamee of he ignal of inee ae deived. Wih a zeo-mean puely aionay modulaing ignal x( 9 by uing (3 (4 wih he ignal given in ( aking ino aoun he aiial popeie of he noie one an eaily how ha he fi-ode/ zeo-onjugae CMF of he eeived ignal ( i = AM DSB SSB ae a follow (AM j2πα% m% (AM ( α% :=< ( e jθ j j2 f 2 j Ae f e θ π e A( Ax( e πα % α= % (23 =< + µ = ohewie m% ( i ( α % = any α% i=dsb SSB. (24 A he fi-ode CMF CCF ae idenial (23 (24 hold alo fo he fi-ode/ zeo-onjugae CCF i.e. % ( β % = m% ( α% β % =α% i=am DSB SSB. he n h-ode/ -onjugae CCF of linea digial modulaion i [2]-[3] n n j2πγε % jθ( n2 j2 π f ( u uτu % = (; β % = A e e e n (25 n (* u j2πγ% p ( +τ % e d+ % ( β% ; u= X u w n whee β=γ+ % % ( n2 f (26 wih γ=k % / k inege i he n h-ode/ -onjugae n umulan fo he ignal onellaion i i=m-psk M-QAM ( u i he opional minu ign aoiaed wih he opional onjugaion (* u u =... n. Noe ha % w(; β% n in (25 i non-zeo only fo n 2 =... n β=. Fo value of he n h-ode / -onjugae umulan fo diffeen linea digial modulaion ( n =...8 =... n ee e.g. [2]-[3]. n Odd-ode umulan (n odd eual zeo fo ymmei ignal onellaion [2]. hu wih = he fi-ode/ zeo-onjugae CCF of M-PSK M-QAM ignal beome zeo i.e. % ( β % = any β % i=m-psk M-QAM. Seond-ode yloaionaiy i ubeuenly inveigaed fo DSB SSB M-PSK M-QAM ignal. One an eaily how ha he eond-ode/ zeo-onjugae CMF of he eeived ignal ( i=dsb SSB ae a follow (DSB (DSB j2πα% m% (DSB ( ατ % ;% :=< ( ( +τ % e = 2 j 2θ j2π f Ae e m% x( α % = 2 f (27 = ohewie m% ( ατ % ;% (SSB (SSB j2 :=< ( ( +τ e πα % % = any α%. (28 (SSB 9 Baed on hi aumpion one an eaily how ha < ( x ( = j2πα% j2πα% < xx ( ( + τ % e =< ( xx ((( + τ % e = any α %. he ideniie m% x = m% x( ( m% xx =m% xx( ( ( [] ae ued o deive (28 wih m% x : =< x(( x +τ % m% x : =< x(( x + τ % ( ( ( m% xx : =< x(( x +τ % ( ( m% xx : =< x(( x + τ % ( (. A he fi-ode CMF eual zeo fo i = DSB SSB a well a fo he noie w ( he yli momen-o-umulan fomula yield he eualiy of eond-ode/ zeo-onjugae CCF CFM i.e. % (; β% τ % = m% ( α% ; β % =α% i = DSB SSB. By applying (25 wih n = 2 = = [ ] = fo i=bpsk = fo i=m-psk M-QAM M 4 [2] he eond-ode/ zeo-onjugae CCF of M-PSK M-QAM ignal ae found a 2 j2πγε % j2θ j2π f A e e e j2πγ% % ( βτ %; % = px ( px ( e d i BPSK +τ % = (29 i = M-PSK M-QAM M 4 whee β % =γ+ % 2 f wih γ=k % / k inege. Similaly one an how ha he eond-ode/ one-onjugae CMF of he DSB SSB ignal ae given epeively a (DSB ( DSB * j2πα% m% ( ατ % ;% := ( ( ( + e ( DSB 2 j2π f 2 Ae mx( τ + mw( α% ; τ α % = = % % % % (3 ohewie m% ( ατ % ;% := ( ( ( + e (SSB (SSB * j2πα% (SSB 2 j2π f 2 Ae ( m% x( jm% x( + m% w( α% ; α % = = m ( (3 ohewie * whee m% x : =< x( x ( + τ % m( % ( x i he Hilbe anfom of m% ( x. he ideniy m% xx = mx( τ ( (% % [5] wih * m% xx : =< x( x ( + τ % ( ( i ued in deiving (3. Noe ha m% x = m% x( m% xx = m% xx( ( ( a x( i a ealvalued ignal. By uing he yli momen-o-umulan fomula one an eaily how ha he eond-ode/ one-onjugae CCF CMF of ( i= DSB SSB ae epeively eual. In ohe wod % (; β% τ % = m% ( α% ; β % =α% i= DSB SSB. Fo i=m-psk M-QAM by applying (25 wih n = 2 = = [2] i beome aighfowad ha 2 j2πγε % j2π f % (; βτ % % = A e e (32 * j2πγ% px ( px ( +τ % e d+ % w( β% ; wih β=γ=k % % / k inege. Wih he eul peviouly deived fo oninuou-ime ignal by applying (2 (22 aking ino aoun ignal nomalizaion one an eaily obain (2-(9 fo he nomalized diee-ime ignal.
AM DSB SSB BPSK QPSK 8-PSK 6-QAM 64-QAM DSB SSB BPSK QPSK 8-PSK 6-QAM 64-QAM No ( ( Ye DSB No (2 Ye AM (2 DSB BPSK SSB QPSK 8-PSK 6-QAM 64-QAM No (3 (4 Ye No ( (3 (4 Ye K β' β' BPSK Fig.. Popoed yloaionaiy-baed binay deiion ee laifiaion algoihm. SSB QPSK 8-PSK 6-QAM 64-QAM.35.3 ^(K (AM (β'.3.25.2.5 ^(K (BPSK (β'.25.2.5...5.5 -.5 -.4 -.3 -.2 -...2.3.4.5 Cidae yle feueny β'.2 a -.5 -.4 -.3 -.2 -...2.3.4.5 Cidae yle feueny β'.4 ^(K (AM (β'..8.6.4 ^(K (BPSK (β'.35.3.25.2.5..2 -.5 -.4 -.3 -.2 -...2.3.4.5 Cidae yle feueny β' Fig. 2. he eimaed fi-ode/ zeo-onjugae CCF magniude of AM BPSK ignal veu idae CF wih 4 K = 4.8 ample a SNR=-dB b SNR=-2dB. b.5 -.5 -.4 -.3 -.2 -...2.3.4.5 Cidae yle feueny β'.95 l = l =2 l =3 l =4.9 P ( l.85.8.75.7-3 -25-2 -5 - -5 5 5 2 SNR (db Fig. 3. he aveage pobabiliy of oe laifiaion veu SNR a eah node l l =... 4.