Probability Density Function of M-ary FSK Signal in the Presence of Noise, Interference and Fading

Transcription

1 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc Probablty Dsty Fucto of -ary FSK Sgal th Prsc of Nos Itrfrc a Fag DRGN KRSTIĆ IHJLO STEFNOVIĆ NTŠ KPCINOVIĆ SRDJN JOVKOVIĆ DUŠN STEFNOVIĆ Faculty of Elctroc Egrg Uvrsty of Nš lsara vva 4 Nš SERBI ragaa@lfa..ac.yu bstract: - I ths ar th rcvr for th oulato of -FSK sgals th rsc of Gaussa os trsybol trfrc uls trfrc a fag s r. Th robablty sty fucto of -ary FSK sgal th rsc of os trfrc a fag s rv. Ths trfrcs ca srously gra th coucato syst rforacs. Ky-Wors: - -ary Frqucy Shft Kyg Gaussa Nos Itrsybol Itrfrc Iuls Itrfrc Raylgh Fag Naaga Fag Probablty Dsty Fucto Jot Probablty Dsty Fucto Itroucto I ths ar w r a syst for cohrt oulato of -ary FSK sgals th rsc of Gaussa os uls os a varabl sgal altu. Ths sturbacs ca srously gra th rforac of coucato systs []-[3]. I th ar [4] th rforac valuato of svral tys of FSK a CPFSK rcvrs was vstgat tal usg th of ot s tho. lso th rror robablty of th crosscorrlator rcvr for bary gtal frqucy oulato tcto s stu usg thortcal aalyss a coutr sulatos [5]. I [6] avrag bt-rror robablty rforac for otu vrsty cobg of ocohrt FSK ovr Raylgh chals s tr. Prforac aalyss of w-ba -ary FSK systs Raylgh fag chals s gv [7]. I orr to vw th fluc of Gaussa os trsybol trfrc a fag o th rforacs of -ary FSK syst w rv th robablty sty fucto of -ary FSK rcvr outut sgal th ot robablty sty fucto of outut sgal a ts rvatv a th ot robablty sty fucto of outut sgal at two t stats. Th bt rror robablty th sgal rror robablty a th outag robablty ca b tr by th robablty sty fucto of a outut sgal. lso th ot gratg fucto th cuulatv strbuto of a outut sgals a th ot a varac of outut sgals ca b rv by robablty sty fucto of th outut sgals. Th avrag lvl crossg rat a th avrag fa rvato of a outut sgal rocss ca b calculat by th ot robablty sty of a outut sgals a ts rvatv. xrsso for calculato of autocorrlato fucto ca b rv by th ot robablty sty fucto of outut sgal at two t stats. Us of th Wr-Hch thor gvs us th sctral owr sty fucto of outut -ary FSK sgal. Bcaus of that th rsults obta ths ar ar sgfcat. Bas o ths th rsults obta ths ar ar of grat sgfcac. Ths ar s orga as follows: frst scto s troucto. I sco scto th ol of -ary FSK syst s f. Sgal charactrstcs such as robablty sty fucto of outut sgal at o t stat ar gv thr scto. Sgal charactrstcs at two t stats ar rv forth scto. I th xt ffth scto th urcal rsults th cas a varous aratrs valus ar gv. Th last scto s th cocluso. ol of th -ary FSK Syst Th ol of a -ary FSK syst r ths ar s show Fg.. Ths syst has brachs. Each brach sts of baass fltr a corrlator. Corrlator s stg of ultlr a lowass fltr. ISSN Issu 5 Volu7 ay 8

2 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc Th sgal at th ut of th rcvr s gtal frqutly oulat sgal corrut by atv Gaussa os trfrc a varabl sgal altu. s t Fg.. Syst ol for cohrt oulato of -ary FSK sgal Trastt sgal for th hyothss H s: s t ω t whr ots th altu of th oulat sgal a ca hav Raylgh or Naaga- strbuto our ar. Gaussa os at th ut of th rcvr s gv wth: t t t x ω t y s ω t... whr x a y ar th coots of Gaussa os wth ro as a varacs. Th trfrc t ca b wrtt as: t ω t 3 If t s uls trfrc t has strbuto gv by: t c ω t... c s tat a has Posso s strbuto: ω ω ω t ω ω t! ω t ax 4 5 s tsty of uls rocss. Phass... hav ufor robablty sty fucto. Ths sgals ass frst through baass fltrs whos ctral frqucs ω ω... ω corrso to hyothss H H...H. ftr ultlyg wth sgal fro th local oscllator thy ass through lowass fltr. Th fltr cuts all sctral coots whch frqucs ar gratr tha th borr frqucy of th fltr. If ar th outut sgals of th brach of th rcvr th th -FSK rcvr outut sgal s: ax{... } 6 Th robablty sty of outut sgal s F 7 Th ot robablty sty fucto of th outut sgal a ts rvatv s: F 8 3 Sgal Charactrstcs at O T Istat I th cas of th hyothss H trastt sgal s: t ω t 9 s whl th outut brach sgals of th rcvr ar: x x 3 It s cssary to f th robablty sty fuctos of th brachs outut sgals a th cuulatv sty of ths sgals to obta robablty sty fucto of -ary FSK rcvr outut sgal. Th cotoal robablty sty fuctos for th sgals... th rsc of trsybol trfrc a Raylgh fag ar:. / / 3 ISSN Issu 5 Volu7 ay 8

3 / 3 4 By avragg to 4 w obta th robablty sty fuctos of th brachs outut sgals as: Th cuulatv strbutos of th sgals... ar: F 8 F 9 F Th robablty sty fucto of th -ary FSK rcvr outut sgal th cas of th hyothss H ca b obta fro: F Th cotoal robablty sty fuctos for th sgals... th rsc of uls trfrc a Naaga- fag ar: / c ots staar vato. / c 3 3 By avragg a 3 w obta th robablty sty fuctos of th brachs outut sgals: c! Γ 4 c! Γ 5 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc ISSN Issu 5 Volu7 ay 8

4 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc c 6! Γ Th cuulatv strbutos of th sgals... ar: F! r c Γ r 7 Th -ary FSK rcvr outut sgals at stat t ar... a at th stat t thy ar.... Th xs for th ut sgals ar: frst x s th ubr of th brach a th othr sgs t stat obsrv. For th outut sgal th x rrsts th t stat obsrv. Th hass of th trfrc ach brach ra tat at both t stats... t. ω ω ω t ω ω ω a F r c Fg.. Syst ol for cohrt oulato of -ary FSK sgal at two t stats! Γ r 8 Th ot robablty sty of th sgals a s: F r c! Γ r 9 Th robablty sty fucto of th -ary FSK rcvr outut sgal th cas of th hyothss H ca b obta fro 7: F 4 Sgal Charactrstcs at Two T Istats Th syst ol for cohrt oulato of - ary FSK sgal at two t stats r ths chatr s show Fg.. 3 whr ots th corrlato coffct of th os. Slarly th ot robablty sty of th sgals a s: ISSN Issu 5 Volu7 ay 8

5 3 3 Th ot robablty sty of th sgals a wh uls os a Naaga- fag ar rst at th syst ut s: c c c c Γ! 33 whr ots th corrlato coffct of th os. Slarly th ot robablty sty of th sgals a s: c c c c Γ! 34 c c c c Γ! 35 Th -ary FSK rcvr outut sgals at two t stats a ar qual: { }... ax 36 { }... ax 37 Th ot robablty sty of th outut sgals a s: l l F l 38 Th ot robablty sty of sgals a a thr rvatvs a s: WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc ISSN Issu 5 Volu7 ay 8

6 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc Slarly 39 F l l l 4 Slar stuato s wh Naaga- fag s rst at th syst ut. 5 Th Nurcal Rsults W ow r th ual brach FSK rcvr bcaus of ts asy ltato a vry goo rforacs. It s loy ay ractcal tlcoucato systs. Th robablty sty fucto ths cas has th for: F F 43 Fg. 3. shows th fluc of th trsybol trfrc o th robablty sty fucto at th outut of th ual brach FSK rcvr. a: 4 4 Th ot robablty sty of th sgals a a thr rvatvs a s: robablty sty fucto sgal vlo 3 4 Fg. 3. Probablty sty fucto of ual brach FSK rcvr outut sgal for ffrt valus of sgal-to-trfrc owr W ot that wth th crt of th sgal-totrfrc owr rato.. wth crasg of th trsybol trfrc altu th curv has lowr axu a b wr. lso w ca s that wth cras of aratr axu cras a traslatory shft alog sgal vlo axs. ISSN Issu 5 Volu7 ay 8

7 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc Fg. 4. shows that slar trasforatos of curv ha wh sgal-to-os owr rato vars whl trfrc ra tat. I Fgs. 7. to 9. robablty sty fuctos vrsus outut sgal ar gv s o aratr..55 robablty sty fucto SNR6 SNR5 SNR sgal vlo Fg. 4. Th robablty sty fucto o th ual brach FSK rcvr outut sgal for ffrt valus of sgal-to-os owr rato robablty sty fucto c outut sgal c 4 c 4 c Fg. 6. Th robablty sty fuctos for th aratrs a Th robablty sty fuctos vrsus outut sgal wth a wthout trfrc a uls os for varous valus of fag svrty aratr a avrag owr ar show Fgs. 5 a 6 rsctvly. Th robablty sty fuctos vrsus outut sgal wh c s o staar vato ar gv Fgs.. to. robablty sty fucto c c 4 c 4 c outut sgal Fg. 5. Th robablty sty fuctos for th aratrs a robablty sty fucto outut sgal c c.5 c c 4 Fg. 7. Th robablty sty fuctos for th aratrs.5 a ISSN Issu 5 Volu7 ay 8

8 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc robablty sty fucto outut sgal c c.5 c c 4 Fg. 8. Th robablty sty fuctos for th aratrs a robablty sty fucto outut sgal c c.5 c c 4 Fg.. Th robablty sty fuctos for th aratrs.5 a robablty sty fucto outut sgal 4 c c.5 c c Fg. 9. Th robablty sty fuctos for th aratrs a robablty sty fucto outut sgal c c.5 c c 4 Fg.. Th robablty sty fuctos for th aratrs a ISSN Issu 5 Volu7 ay 8

9 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc robablty sty fucto outut sgal c c.5 c c Fg.. Th robablty sty fuctos for th aratrs a 6 Cocluso I ths ar th statstcal charactrstcs of th sgal at th outut of th rcvr for cohrt FSK oulato ar rv. Th ut sgal of th rcvr s gtal frqutly oulat sgal corrut by atv Gaussa os trsybol trfrc a varabl sgal altu. Th trfrc aars ach rcvr brach. I ths ar th robablty sty fucto of a -ary FSK rcvr outut sgal th ot robablty sty fucto of outut sgal a ts rvatv a th ot robablty sty fucto of outut sgal of two t stats ar rv. Th ot robablty sty fucto of th outut sgal at two ts stats s ortat for th cas wh th os s corrlat. Th bt rror robablty a th outag robablty ca b tr by th robablty sty fucto of a outut sgal. Th lvl crossg rat a avrag fa urato of a outut sgal rocss ca b calculat by th ot robablty sty fucto of a outut sgal a ts rvatv. xrsso for calculato autocorrlato fucto of th outut sgal ca b obta by a ot robablty sty fucto of a outut sgals at two t stats. lso by ths fucto a llhoo fucto of a - FSK syst wh th cso s o by two sals ca b calculat. ay of th wrlss coucato systs us so for of vrsty cobg to ruc ultuath fag aar th chal. og th slr vrsty cobg schs th two ost oular ar slcto cobg SC a swtch a stay cobg SSC. I th ar [6] lou a So vlo aaly a ot a sl for of ual-brach swtch a stay cobg SSC. Our futur ars wll b vlo to vrsty cobg a rucg fag th vrsty systs. Rfrcs: [] T. Oosh a K. Kuch Cohrt Otcal Fbr Coucatos. Toyo Jaa: KTW Sctfc 988. [] J.. Sor Otcal Fbr Coucatos. Nw Yor: Prtc Hall 99. [3] J. G. Proas Dgtal Coucatos. Nw Yor cgraw-hll 989. [4] Hao -Jog a B. Sth Wcr Prforac valuato of FSK a CPFSK otcal coucato systs: a stabl a accurat tho J. Lghtwav Tchol. Vol. 3 No. 8 ug [5] K.. Farrll a P. J. cla Prforac of th cross-corrlator rcvr for bary gtal frqucy oulato IEEE Tras. Cou. Vol. 45 No 5 ay 997. [6] arv. K So a oha-sl lou vrag Btt-Error Probablty Prforac for Otu Dvrsty Cobg of Nocohrt FSK Ovr Raylgh chals IEEE Tras. o Cou. vol. 5 No rl 3. [7] Yo Kyoo Jog Kwag Bo L Prforac aalyss of w-ba -ary FSK systs Raylgh fag chals IEEE Tras. O Cou. Vol.48 Issu Dc [8] H. Huyh. Lcours Iulsv Nos Nocohrt -ary Dgtal Systs IEEE Tras. o Cou. vol. 3 Issu Fb 975. [9].. Blaco Dvrsty rcvr rforac Naaga fag Proc. 983 IEEE Southastr Cof. Orlao [] G. Luatla: Th statstcal tlcoucato thory a forato thory Gravsa ga Bogra 98 srba. [] P. Stavroulas: Itrfrc alyss a Ructo for Wrlss Systs Loo rtch hous 3. []. D. Whal Dtcto sgals os Bll Tlho Laborators Whay Nw ISSN Issu 5 Volu7 ay 8

10 WSES TRNSCTIONS o COUNICTION Dragaa Krstc halo Stfaovc Natasa Kaacovc Sra Jovovc Dusa Stfaovc Jrsy cac Prss Nw Yor Sa Fracsco Loo 97. [3].. bu-dayya a N.C. Baulu Outag robablts of vrsty cllular systs wth cochal trfrc Naaga fag IEEE Tras. Vh.Tchol. vol Nov. 99. [4] chl Daou Yacoub Clauo Rafal Cuha otro a Slva a Jos Eso Vargas Bautsta Sco-orr statstcs for vrsty-cobg tchqus Naaga-fag chals IEEE Tras. Vh. Tchol. vol Nov.. ISSN Issu 5 Volu7 ay 8