RF Phase Modulation of Optical Signals and Optical/Electrical Signal Processing. Nikolaos I. Andrikogiannopoulos

Transcription

1 RF Phae Modulaion of Opical Signal and Opical/Elecrical Signal Proceing by Nikolao I. Andrikogiannopoulo Dipl., Naional Technical Univeriy of Ahen (004) Submied o he Deparmen of Elecrical Engineering and Compuer Science in parial fulfillmen of he requiremen for he degree of Maer of Science in Elecrical Engineering and Compuer Science a he MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 006 Maachue Iniue of Technology 006. All righ reerved. Auhor... Deparmen of Elecrical Engineering and Compuer Science May 5, 006 Cerified by... Vincen W. S. Chan Joan and Irwin M. Jacob Profeor Elecrical Engineering & Compuer Science and Aeronauic & Aronauic Direcor, Laboraory for Informaion & Deciion Syem Thei Supervior Acceped by... Arhur C. Smih Chairman, Deparmen Commiee on Graduae Suden

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3 RF Phae Modulaion of Opical Signal and Opical/Elecrical Signal Proceing by Nikolao I. Andrikogiannopoulo Submied o he Deparmen of Elecrical Engineering and Compuer Science on May 5, 006 in parial fulfillmen of he requiremen for he degree of Maer of Science in Elecrical Engineering and Compuer Science Abrac Analog RF phae modulaion of opical ignal ha been a opic of inere for many year, mainly focuing on Ineniy Modulaion Direc Deecion (IMDD). The virue of coheren deecion combined wih he advanage of Frequency Modulaion, however, have no been explored horoughly. By employing Frequency Modulaion Coheren Deecion (FMCD), he wide opical ranmiion bandwidh of opical fiber can be raded for higher ignal-o-noie performance. In hi hei, we derive he FM gain over AM modulaion he maximum achievable ignal-onoie raio (by preading he ignal pecrum) for pecific carrier-o-noie raio. We hen employ FMCD for a cheme of remoe anenna for which we ue opical componen and ubyem o perform ignal proceing uch a nulling of inerfering ignal. The performance of opical proceing on differen modulaion cheme are compared, and ome imporan concluion are repored relaing o he ue of convenional FMCD, FMCD wih opical dicriminaor (FMCD O-D), and IMDD. Specifically, he uperioriy of convenional FMCD i hown; and, on he oher hand, he inferioriy of FMCD O-D i hown (ame performance a IMDD) becaue of he ue of an O-D. Finally, he remoe anenna cheme i generalized for N anenna and N uer. Thei Supervior: Vincen W. S. Chan Tile: Joan and Irwin Jacob Profeor of Elecrical Engineering & Compuer Science, and Aeronauic & Aronauic Direcor, Laboraory for Informaion and Deciion Syem

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5 Acknowledgmen I would like o hank, fir and foremo, my advior, Profeor Vincen Chan for hi paience, uppor and guidance. I ha been inpiring working cloe o uch an experienced and alened reearch advior, liening o hi inighful uggeion and being augh how o evaluae he ignificance among differen reearch direcion. I would alo like o expre my graiude o Dr. Terrence McGary for hi helpful inigh during our lenghy dicuion and hi uppor hroughou my hei. Alo, I would like o hank Guy Weichenberg, Yonggang Wen, Jihwan Parick Choi and Lillian Dai for all heir help. I am alo graeful o MIT, he Pari Kanellaki Fellowhip, and DARPA for heir financial uppor. La, my deepe hank o my paren for all heir love, uppor and acrifice hroughou he year wihou which hi hei would never have come ino exience. 5

6 Conen Inroducion. Inroducion and Moivaion.... Prior Work Thei Ouline... 8 Modulaion of Microwave Signal ono Opical Carrier. Inroducion.... IMDD and CD.... Noie Mechanim in Opical Link....4 Ineniy Modulaion Direc Deecion (IMDD) Comparion beween Coheren and Direc-Deecion Dyem... 7 Frequency Modulaion Coheren Deecion (FMCD) 9. Inroducion Fundamenal of FMCD.... Convenional FM Receiver Seup Weak Noie Analyi...7 6

7 .. Probabiliy of Cycle Skipping for Convenional FM Complee SNR of FMCD FM Threhold Phenomenon Dependence of Rician Approach on <m ()> Opimum Angle Modulaion Syem Approximaion of SNR max, FMCD FMCD wih O-D Configuraion Analyi of O-D Comparion wih he Convenional FMCD Specific Cae of Sinuoidal Signal Comparion among Modulaion Scheme Dicuion Opical/Elecrical Proceing of Remoe Anenna Signal Inroducion Remoe Anenna Seup and Aligning FM Signal Convenional FMCD in a Simple Remoe Anenna Scheme Configuraion SNR of FMCD in hi Scheme...8 Special Cae: Single Signal Alernaive FMCD Seup Nulling Inerference wih IMDD Nulling Inerference wih FMCD O-D Iniial Obervaion on FMCD O-D Configuraion

8 4.5. Equal Delay Time Difference Comparing Nulling in Remoe Anenna uing FMCD, IMDD, FMCD O-D Generalizing FMCD for N Anenna and N Uer Inroducory Noe Adapive Beamforming in FMCD Cae Sudy : Davie Beamforming for FMCD Cae Sudy : Scaling Scheme Cae Sudy : Scaling Scheme Concluion of Generalizaion Concluion Obervaion and Concluion 5. Inroducion Propoed Modulaion Mehod Remoe Anenna Scheme and Technique Combining Modulaion wih Remoe Anenna Scheme Avenue for furher Reearch... Reference 5 8

9 Li of Figure Figure -: Remoe Anenna Scheme... 4 Figure -: IMDD ranmiion yem, [7]... Figure -: Opical Coheren Syem, [7]... Figure -: FM Bandwidh Expanion... Figure -: Convenional FM receiver... 7 Figure -: Phaor diagram, []... 4 Figure -4: SNR of FM a a funcion of he CNR and he modulaion index β for a convenional FM receiver Figure -5: SNR a a funcion of CNR for differen value of he modulaion index β Figure -6: SNR a a funcion of β for differen CNR value Figure -7: Maximum value of modulaion index β max a a funcion of CNR for convenional FM receiver wih m()uniformly diribued in [-,]... 5 Figure -8: β max for differen econd order characeriic of he unmodulaed ignal... 5 Figure -9: SNR for differen econd order characeriic of he unmodulaed ignal Figure -0: SNR for differen econd order characeriic of he unmodulaed ignal Figure -: Threhold Exenion uing a Feedback Receiver (FMFB) Figure -: Peak SNR difference beween rue peak SNR and he approximaion Figure -: Opical Dicriminaor Figure -4: Variaion of phoocurren a oupu of FM dicriminaor a a funcion of inerferomeer differenial delay (τ) or inpu opical frequency (ω)... 6 Figure -5: Opical Dicriminaor error olerance Figure -6: Performance comparion among ype of modulaion Figure 4-: Remoe Anenna Aligning FM ignal Figure 4-: Remoe Anenna Seup... 8 Figure 4-: Remoe anenna cheme wih ingle uer Figure 4-4: Two-Anenna Scheme

10 Figure 4-5: Gain of Scheme over Scheme... 9 Figure 4-6: Remoe Anenna Seup for IMDD... 9 Figure 4-7: Single Uer - Single Inerferer Scheme wih Opical Dicriminaor Figure 4-8: Opical Dicriminaor wih equal ime delay Figure 4-9: Performance Comparion of differen modulaion and cheme Figure 4-0: Davie Beamforming wih hree anenna and hree uer Figure 4-: N Anenna, N uer, Adapive Beamforming for Narrowband Signal a high frequencie... 7 Figure 4-: N Anenna, N uer, Adapive Beamforming for Broadband Signal a high frequencie

11 Li of Table Table -: IMDD-SNR Limi of pracical inere... 6 Table -: Sho noie limied CNR for η0.9, Β50MHz according o (.8)... 6 Table -: β max for differen CNR value for a uniformly diribued ignal in [-, ] of Hz bandwidh... 5 Table -: β max for differen CNR value for a uniformly diribued ignal of 50 MHz bandwidh... 5 Table -4: Performance of modulaion cheme... 7 Table 4-: SNR for differen kind of FMCD and IMDD... 04

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13 Chaper Inroducion. Inroducion and Moivaion Opical fiber ha been eablihed a he medium of choice in high-capaciy digial ranmiion yem. However, applicaion ha involve poin-o-poin rouing of analog ignal have alo benefied from he excellen propagaion characeriic of opical fiber. Analog ranmiion of microwave ignal over opic ha been demonraed for video-ignal diribuion, micro-cellular radio, and remoe microwave-anenna cheme [4]. The ue of analog modulaion for microwave ignal wihin fiber, known a Radio Frequency (RF) Opical can benefi much from he fac ha he whole received ignal pecrum can be modulaed opically. Thi replace he need for demodulaing he received ignal and convering i from analog o digial forma. The equipmen ued for hi laer funcion analog-o-digial (A/D) converer for broadband ignal i very expenive, and in ome cae canno mee he pecificaion for very broadband ignal. There are many kind of Analog Opical Modulaion and deecion cheme. The wo baic approache, which we will dicu in Chaper, are: ) Ineniy Modulaion Direc Deecion (IMDD) ) Coheren Deecion (CD), uing Analog Modulaion (AM), Frequency Modulaion (FM), or Phae Modulaion (PM)

14 In hi hei, we will focu on CD uing FM. The advanage of uing Frequency Modulaion Coheren Deecion (FMCD) are manifold. Compared o IMDD, FMCD can achieve a higher ignal-o-noie raio (SNR) by expanding he bandwidh of he ignal ino an exremely wideband FM opical bandwidh. An exenive analyical dicuion on he benefi of hi i provided in Secion.4. The echnique of noie uppreion in convenional FM receiver ha been known a weak noie uppreion and i reponible for he uperior performance of FM radio. However, hi rade-off beween bandwidh and SNR preen a cerain hrehold above which he noie induced ino he pecrum degrade he performance of he ignal ignificanly. Thi o-called FM hrehold i one of he opic ha will be dicued in deail in hi work. Having derived he neceary analyical ool for FMCD, we nex look ino imporan applicaion where he ue of coheren deecion could provide an advanage over IMDD. One of he mo widely udied applicaion for IMDD ha been for anenna remoing [4], [4], which i proceing ignal received by everal remoe anenna. IMDD ha been preferred becaue of i impliciy. Our goal i o employ FMCD for hi applicaion and o juify he added complexiy by proving he uperior performance of our propoed modulaion cheme. In hi conex, he econd par of he preen udy focue on he cae of remoe anenna conneced o a cenral aion hrough a fiber-opic nework. In hi cheme, we propoe ha mo of he ignal proceing be conduced in a cenralized way, allowing for furher proceing combining conrucively verion of he ignal, nulling inerfering ignal - and ue le proceing equipmen near he anenna a in Figure -. Figure -: Remoe Anenna Scheme 4

15 Cenralized proceing provide benefi over ypical decenralized proceing echnique. In our deploymen, differen verion of he received ignal (from every anenna) reach he cenral aion and hey can be combined o achieve beer SNR han wih ju one of he anenna. Wha i more, inerference beween uer can be uppreed, or even cancelled, due o he diveriy ha can be achieved hrough he many verion of he received ignal. Cenral proceing echnique alo benefi from analog opical ranmiion. The oolbox of opical proceing coni of delay fiber, hifer, opical wiche, combiner, and plier. All of hee equipmen are cheap, ineniive o modulaion bandwidh, and can be ued before digiizing he ignal and performing he uual ignal proceing ha i preenly done digially. Moreover, he opical proceing can be effecive a frequencie much higher han he limi of elecrical equipmen and can herefore be ued for handling wideband ignal. To ummarize, he ue of opical ignal for he ranmiion of microwave ignal in conjuncion wih opical-elecrical cenral proceing ha he poenial of ignificanly lowering he co of a remoe anenna applicaion uch a a mobile nework or he co of conrucion of a phaed array anenna and providing broadband, inexpenive, and exremely efficien ignal proceing compared o elecrical proceing. The ue of FM in hi echnique i novel, and provide ignifican RF gain over IMDD, and hence uperior SNR performance. 5

16 Our conribuion include:. Derivaion of an analyical expreion for he SNR of FMCD over all he range of CNR indicaing he hrehold phenomenon. Thi udy wa made poible by combining elemen from he lieraure [7], [9], [].. Performance comparion among FMCD wih convenional FM receiver, IMDD and FMCD O-D. The performance of he convenional FMCD i baed on our own reul for he maximum performance of FMCD which i when i operae exacly on he derived hrehold and alo FMCD O-D i enirely baed on he analyi ha i carried in hi hei.. Propoal of a imple remoe anenna rucure employing FMCD wih wo anenna and wo uer where opical proceing coni of adjuing wo delay line o a o null an inerfering ignal. 4. Comparion among he modulaion cheme in he conex of he imple anenna cheme 5. Propoal of a new imple anenna cheme employing FMCD wih eparae heerodyne deecion for every ignal. Thi cheme i hown able o cale for N anenna and N uer. Dicuion on he ue of he exiing lea-mean-quare algorihm (LMS) and he modificaion required for i o work under FMCD.. Prior Work Analog opical link have been he ubjec of coniderable aenion for a variey of applicaion, including remoing anenna and cable eleviion diribuion yem. Work on analog link ha concenraed almo excluively on direc-deecion (DD) yem uing ineniy modulaion (IMDD) mainly becaue of i impliciy []-[6]. To dae, analog link baed on coheren deecion have received relaively lile aenion []-[6] compared o IMDD due o he increaed complexiy of coheren deecion (CD). CD yem have everal poenial advanage over DD yem. Coheren yem can approach ho-noie-limied performance wih ufficien LO laer power. In addiion, coheren yem can deec he phae of he opical carrier. Thu, while DD yem are be uied for ampliude or ineniy modulaion (AM or IM), CD yem can ue AM, PM, or FM. In our work, we inveigae he performance of FMCD moivaed by he ucce of angle modulaion echnique in microwave ranmiion yem. 6

17 Prior work on opical FM ha concenraed on deriving he SNR of FMCD link [], [7] and on he variaion of hi SNR along he ranmiion diance []. In hi hei, we do no inveigae he diance limi of he FM SNR. Inead, by aking ino accoun he opical noie ource and combining i wih he analyi ued for ypical radio FM yem [9], we derive he SNR of FMCD very cloe o wha ha been done in [7] for opical FM. We proceed o find he FM hrehold a a funcion of he modulaion bandwidh. Mo of he previou work, even on ypical radio frequency FM [8] ha no been inereed in finding an analyic expreion for he hrehold phenomenon of FM due o he ochaic naure of he phae induced noie. The FM hrehold ha been udied very lile and only in he field of eimaion heory by Van Tree [], []. Our approach i baed on he analyical expreion for he probabiliy of anomaly derived by Rice for convenional elecrical FM yem [9]. Our reul on he FM hrehold are conien wih he reul of []. Moreover in [], he Rician analyi ha been hown o be cloe o experimenal reul. Our work preen a complee formula for he SNR over all he range of carrier-o-noie raio CNR by incorporaing properly he probabiliy of anomaly of Rice. The FMCD menioned o far can alo be called convenional FM dicriminaion becaue i mainly coni of a quare law deecor followed by an elecrical dicriminaor. I i called convenional becaue i ue a microwave FM dicriminaor operaing under he ame principle ha elecrical yem do. Before uing our reul on he FM hrehold for convenional FM receiver in a remoe anenna cheme, we characerize he performance of a FM O-D yem. To characerize he performance of he O-D, we follow cloely he analyi of [6] on Mach Zehnder opical dicriminaion and exend hee reul o derive ome ueful formula for he SNR of FMCD wih an O-D. Uing he maximum achievable SNR - a he poin of he hrehold ha we udied - for convenional FM when given a cerain CNR, we hen proceed o employ FMCD in a remoe anenna cheme. Remoe anenna or phaed array anenna uing opical fiber have ued moly IMDD or no modulaion a all in lieraure [4], [4]. The novel par of our approach i ha we employ FMCD operaing near he hrehold, hu providing he maximum SNR ha FM can achieve. We fir conider a wo-anenna and wo-uer remoe anenna eup (a hown in Fig. 4-) and find he neceary condiion uch ha an inerferer can be nulled. We exend hi obervaion o a more generalized cheme involving N anenna and N uer (Figure 4-9). 7

18 A we increae he number of uer and anenna, we oberve he imilariie our reearch bear wih ha of adapive anenna yem [8]-[0]. Thee yem have demonraed everal advanage, among which i increaed eniiviy of he received ignal o inerfering ource. The adapaion proce in uch yem i baed on he minimizaion of he mean-quare error by he lea-mean-quare (LMS) algorihm [0]. The difference in our yem i ha we have frequency modulaed ignal. The weigh required by he LMS algorihm, which ypically are applied eiher direcly o he ignal or o he ampliude modulaed ignal, canno be applied in FM in he ame manner. A differen approach ha o be followed. We how hi o be he change of he FM modulaion index, which i baically he dynamic FM acion of changing he ignal ampliude in AM.. Thei Ouline In hi hei, we udy he ue of FMCD in a remoe anenna cheme combined wih opical/elecrical proceing. The hei i organized a follow: In Chaper we dicu he difference beween IMDD and CD Syem. In addiion, we derive he SNR of IMDD. Chaper i devoed enirely o FMCD, aring by iniially deriving he CNR of FMCD, aking ino accoun opical noie ource. Thi i followed by he weak noie analyi of convenional FM yem. The very well known formula for FM yem i re-derived uing he previouly calculaed CNR. Thi formula i valid only when here i no anomaly in he phae of he ignal, which i rue when he CNR i high. In order o find he SNR for he whole range of CNR value, we follow he Rician analyi; and by conidering he probabiliy of anomaly [9], we derive a complee formula. The nex par of hi chaper i our analyi on FMCD wih a Mach Zehnder inerferomeer performing a an opical dicriminaor. An analyical derivaion of he correponding SNR i provided and he cae of a inuoidal ignal i inveigaed. The chaper coninue wih a novel comparion of all of he modulaion cheme ha were dicued o far and end wih a dicuion ecion. Chaper 4 inroduce a remoe anenna cheme employing FMCD wih wo uer and wo anenna. We derive he neceary condiion for nulling one of he wo uer and reric our analyi in hi 8

19 hei o nulling. A hi poin we alo preen a heerodyne eparae deecion verion of he above cheme which ha beer performance. Nex, we proceed wih he inveigaion of nulling and we how ha he neceary condiion for nulling applie alo for IMDD for which we derive he reuling SNR. The SNR of FMCD O-D in he remoe anenna cheme i alo derived in hi ecion. The chaper coninue wih a comparion among he modulaion cheme - analogou o ha of chaper - for he cae of he remoe anenna eup. We hen generalize he wo anenna cheme for N anenna and N uer. We propoe configuraion ha can ue adapive beamforming for remoe anenna eup ha operae under FMCD. Finally, chaper 5 i devoed o a complee examinaion of all he reul in all of he preceding chaper, and offer direcion for fuure reearch. 9

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21 Chaper Modulaion of Microwave Signal ono Opical Carrier. Inroducion Thi chaper inroduce he differen kind of analog modulaion, namely IMDD and CD. In he re of he hei, we are inereed only in CD, and pecifically in conjuncion wih FM. The curren chaper begin wih an inroducion o he wo modulaion followed by a udy of he IMDD performance. The performance of CD will be dicued in greaer deail ubequenly in he hei. Finally, wih he help of he exiing lieraure, we dicu he advanage and diadvanage of coheren and direc deecion yem.. IMDD and CD RF opical ranmiion i defined a he echnique by which an RF ignal i impoed on an opical carrier, wih opical fiber being he ranmiion medium. The original RF ignal i recovered a he receiver by performing deecion on he opical ignal. Two baic approache o opical-ignal modulaion and recovery are poible. The fir and impler alernaive i ineniy-modulaion direc-deecion (IMDD) a hown in Figure -. In hi cae, he opical-ource ineniy i eiher direcly modulaed by he inpu microwave ignal, or pae hrough an exernal ineniy modulaor. The reuling ineniymodulaed ignal pae hrough he opical fiber o he phoodiode, where he modulaing ignal i reurned in he elecrical domain.

22 Figure -: IMDD ranmiion yem, [7]. In a coheren yem Figure -, he opical ource can be modulaed in ineniy, frequency, or phae by he inpu microwave ignal, eiher direcly or by paage hrough an exernal modulaor. The modulaing ignal pae hrough he opical fiber o he receiver, where i i combined wih he inpu from a local-ocillaor (LO) laer. The combined ignal illuminae he phoodiode o produce an elecrical ignal cenered on an inermediae frequency (IF) beween he unmodulaed opical ource and he LO laer. Thi ignal i furher proceed o recover he analog inpu ignal. Figure -: Opical Coheren Syem, [7].. Noie Mechanim in Opical Link Sho noie and hermal noie are he wo fundamenal noie mechanim reponible for curren flucuaion in all-opical receiver even when he inpu opical power i conan. Of coure, addiional noie i generaed if he inpu power ielf i flucuaing becaue of ineniy noie aociaed wih he ranmier.

23 The main caegorie of noie [5] are: Thermal noie: A a finie emperaure, elecron move randomly in any conducor. Random hermal moion of elecron in a reior i manifeed a a flucuaing curren even in he abence of an applied volage. The load reior in he fron end of an opical receiver add uch flucuaion o he curren generaed by he phoodiode. Thi addiional componen i referred o a hermal noie. Mahemaically, hermal noie i modeled a a aionary Gauian proce wih a pecral deniy, I h, ha i frequency-independen up o THz (nearly whie noie) and i given by 4kT I h B R (.) L where k i Bolzmann conan, T i he abolue emperaure, ignal bandwidh. R L i he load reior, and B i he Sho noie: Thi kind of noie i a manifeaion of he fac ha he elecric curren coni of a ream of elecron ha are generaed a random ime. Mahemaically, i can be modeled a a aionary random proce wih Poion aiic, which in pracice can be approximaed by Gauian aiic I q( I d I )B (.) h + dk where I d i he mean opically generaed curren and I dk i he phoodiode dark curren. Opical ource relaive ineniy noie: Thi i he exce ineniy noie due o noncoheren behavior of he ranmier, and i given by I I RIN B (.) nrin d where RIN i he ource relaive ineniy noie value.

24 .4 Ineniy Modulaion Direc Deecion (IMDD) In hi ecion, we focu on he mo popular analog modulaion cheme ineniy modulaion []-[6] and we re-derive he SNR of hi cheme []. In an IMDD yem, a depiced in Figure -, he analog ignal o be ranmied can be repreened by m(). The opical power a he deecor i proporional o he power of ampliude of he ignal properly weighed by he modulaion index γ P o [ + m( ) ] P γ (.4) u where P u i he mean received opical power and γ i he modulaion index ( m( ) > ) γ. The mean quare ignal curren a he deecor oupu i he ignal par of he mean received power (.). Tha i, I ( RP ) m ( ) γ (.5) u where R ηq i he reponiiviy of he p-i-n phoodiode, hv η: quanum efficiency of he phoodiode q: elecrical charge h: Planck conan v: he frequency of ligh Noie arie from he ranmier and he phoodeecor. Fundamenally he noie i limied by he quanum naure of phoon-deecion. We repreen he one-ided cumulaive pecral deniy of noie a N 0, coniing of he um of he variance of he noie erm ha were menioned in he previou ecion - ho, hermal, relaive ineniy noie - when no amplifier are preen. Taking ino accoun ha he ignal power i given by (.) and ha he noie power i N 0 B, we have he following expreion for SNR: 4

25 SNR IMDD ( RP γ ) u N 0 m B ( ) (.6) where 4kT N e 0 RL ( I I ) B he meage m() bandwidh, dk I dk he diode dark curren, uually negligible., The able on he following page preen he ho noie limi, in which cae he ho noie i aumed o dominae and we derive he correponding SNR accordingly. 5

26 For opical power below he RIN limi, ho-noie limied recepion can be achieved if he hermal conribuion i mall, giving SNR DD RP u γ m ( ) qb γ m ( ) ηpu γ m ( ) CNR hvb (.7) when N 0 q(rp u ) Table -: IMDD-SNR Limi of pracical inere 6

27 .5 Comparion beween Coheren and Direc-Deecion Dyem Coheren-ranmiion yem offer hree main advanage over yem uing direc deecion []:. Sho-noie limied recepion can be achieved, even a low received-ignal power, imply by increaing he LO power.. Ineniy, frequency, or phae modulaion mode can be ued, wherea DD yem are limied o ineniy modulaion.. The excellen frequency eleciviy ha can be achieved uing elecrical po-phoo-deecor filer i ranlaed ino he opical domain by he coheren-deecion echnique. Thi enable he realizaion of dene wavelengh-diviion-muliplexing cheme for muli-channel ranmiion, or channel-elecion cheme. We now review hee advanage in urn. The fir i of reduced imporance for yem operaing a a wavelengh of 550 nm, now ha effecive opical amplifier are available. However, if we conider paive opical nework, hi remain a valuable characeriic of coheren yem. I i alo of imporance o yem operaing a he 00 nm wavelengh, in order o ake advanage of he ilica-fiber diperion minimum, and he low noie and high oupu power of emiconducorlaer-pumped Nd-YAG laer. Thi i alo of inere in yem operaing a a 850 nm wavelengh, for compaibiliy wih GaA microwave monolihic-inegraed-circui (MMIC) echnology. Effecive opical-fiber amplifier are no available for eiher of hee wavelengh []. The alernaive raegy for ho-noie-limied IMDD yem of increaing he ource power i limied by he one of imulaed-brillouin caering (SBS) and oher nonlinear effec in opical fiber. Thu, coheren ranmiion remain of inere for long-diance yem, where high SNR i required. The econd advanage enable SBS o be reduced by broadening he opical-ignal bandwidh beyond he SBS linewidh (~0 MHz a he 550 nm wavelengh), uing frequency or phae modulaion. Ue of frequency or phae modulaion alo enable a radeoff o be made beween opical-ignal bandwidh and received SNR; hi i he opic of Chaper and coniue he moivaion for finding he FM hrehold. 7

28 The imporance of he hird advanage depend upon wheher he abiliy o wich among many ource carried on he ame fiber i required. There are hree main diadvanage of coheren-ranmiion yem, relaive o hoe uing direc deecion:. The frequencie of he LO laer and ignal mu be conrolled o differ by he required IF, wherea in he DD yem, i i only neceary ha he ource laer frequency be uiable for he phoodiode ued.. The linewidh of ource and LO laer mu be uiable for he modulaion mode ued, wherea in DD yem, he required ource linewidh i mainly deermined by he opical fiber diperion penaly.. The polarizaion ae of he LO and ignal mu be mached a he phoodiode. However, none of he above hree iue are fundamenal and can be circumvened wih oday echnology. Polarizaion maching can be achieved by acive-polarizaion conrol of he LO ignal for maximum deeced-ignal oupu [7], or uing polarizaion-diveriy recepion [5]. While he diadvanage of coheren-ranmiion yem can all be overcome, he penaly i an increae in yem complexiy relaive o DD yem. Wheher he coheren-yem approach hould be ued in a paricular applicaion herefore depend upon wheher he performance advanage are ufficien o juify he increae in complexiy. 8

29 Chaper Frequency Modulaion Coheren Deecion (FMCD). Inroducion A crucial par of hi hei i o inveigae furher he FM ype of CD. In he foregoing chaper we preened he advanage of CD and we concluded ha he increaed complexiy of CD wih repec o DD yem ha o be juified by uperior performance. In hi par of he hei, we will inveigae he performance of FM. We udy boh he convenional form, which ue an elecrical dicriminaor, and he all-opical form, which employ an opical dicriminaor (O-D). The chaper ar in Secion. by inroducing he reader o he fundamenal characeriic of FMCD, deriving he known formula for he FM ignal field, he reuling phoocurren, and he CNR of FMCD. Afer hi ecion, he chaper i divided ino wo par. Secion. focue on he convenional FM dicriminaor, which ha been udied o ome exen in he lieraure [7] and Secion.4 focue on FMCD wih O-D in which a Mach Zehnder inerferomeer i ued, which i a ignifican exenion of he iniial approach of [6]. Wihin Secion., we follow cloely he previouly udied FMCD analyi in [7], re-deriving he implified formula for he SNR of FM auming ha he CNR i high enough for ome approximaion o be made (weak noie aumpion). Thi formula i acually he ame for radio frequency FM receiver [9]. Nex, by conidering he Rician approach ha i menioned in [9] and combining i wih he implified formula of [7] we are able o produce a full formula for he SNR of FM and deermine he FM hrehold. Thi i a ignifican reul (in Secion..5) and provide he maximum SNR for a given opical bandwidh, or inverely, he maximum SNR for fixed CNR. In Secion..6 we commen on he reul obained for a ignal of pecific econd order characeriic, and how ha he hrehold doe no depend rongly on he econd order 9

30 characeriic of he ignal. Then, in Secion..7, we dicu he lieraure [9], [], [] on FM feedback yem and highligh ha he hrehold exenion phenomenon, which i common pracice in FM receiver, can alo be ued for FMCD yielding a lower hrehold and herefore allowing u o ue more bandwidh expanion under he ame CNR. We end he convenional FM ecion in..8 by calculaing how much accuracy we acrifice in he re of he hei when we ue he known impliic SNR formula of FM, inead of he full one ha we derived, o decribe he operaion of FM near i peak performance. In he econd half of hi chaper, Secion.4, we anwer he queion of where he elecricalopical inerface in FMCD hould be. By aemping o ue an O-D inead of an elecrical dicriminaor, we exend he analyi of [6] on O-D o derive ha he performance of he FMCD O-D and how ha i i acually he ame a IMDD and much wore han ha of convenional FMCD. Thi i expeced ince we loe all he benefi of weak noie uppreion becaue we demodulae our ignal before deecing and hu he noie i no uppreed a in he convenional cae. A comparion among he differen modulaion cheme i offered in Secion.5 where convenional FMCD i compared wih IMDD and FMCD O-D and a brief numerical example i given. The chaper conclude wih Secion.6 on a dicuion of he reul. 0

31 . Fundamenal of FMCD An opical FMCD yem i a direc emulaion of RF FM broadca and communicaion yem in he opical domain and ue he ame baic concep. In an opical link, he laer oupu provide a carrier of very high frequency ( THz). A wih RF carrier, here are wo independen characeriic - ampliude and phae or frequency - which can be modulaed by he ignal o be ranmied. In opical FM (OFM), he carrier frequency deviae from i cenral frequency proporionally o he ampliude of he ignal o be ranmied. Boh random and impule noie can be reduced by uing a wide paband receiver conaining an ampliude limier, which coniue a convenional receiver which we will inveigae horoughly in Secion.. Therefore, regarding convenional FM and received CNR, much greaer han uniy, FM yem can provide improved SNR relaive o ampliude modulaed yem [], [7], [0], [], [4]. The OFM yem ha hree main feaure which diinguih i from i RF equivalen. Fir, he opical carrier frequency i very high relaive o he bandwidh of he ignal o be ranmied. A a reul, a large frequency deviaion relaive o he ignal bandwidh can eaily be achieved wih good lineariy. On he oher hand, a mall relaive carrier frequency drif will be ignifican compared wih he ignal bandwidh. Second, becaue no phoodeecor ha a bandwidh maching he opical carrier frequency, eiher demodulaion in he opical domain or down-converion of he received ignal from he opical frequency o a much lower inermediae frequency i needed. Third, due o he high opical carrier frequency, he laer generae far more phae/frequency noie han an RF ocillaor and we conider ha hi phae noie equal o noie ince i can be cancelled a repored in []. In an FMCD link, a hown in Fig. -, he ignal elecric field a he phoodiode i E E exp j ω + π f m( τ ) dτ + φ ( ) 0 (.) where f i he maximum frequency deviaion, and in our cae, correpond o he opical modulaion bandwidh; φ () i a random iniial phae and m() i he normalized ignal varying from - o.

32 The local ocillaor elecric field i E LO { j[ ω + φ ( ) ]} E exp (.) LO LO LO wih ω LO being he local ocillaor frequency, and φ ( ) LO he local ocillaor phae. Defining he inermediae frequency (IF), ω IF ω ω, he ignal inciden on he phoodiode i LO V in E exp j π f m( τ ) dτ φ ( ) ELO exp{ j( ω IF φlo () )} exp{ jω} (.) For ω << ω o ha he receiver can handle he inciden opical bandwidh, he oupu curren IF from he phoodiode i proporional o E LO P LO, he curren i given by * V inv in, o ha, aking ino accoun ha E Pu and I ( ) I 0 + I co ω IF + π f m LO ) 0 ( τ ) dτ + φ ( ) ϕ ( (.4) where I 0 R( P LO + Pu ), I R PLO Pu, and he reponiiviy of he diode R ηq. hv The above equaion can be rewrien in erm of ignal and LO power a: ( P + P ) + R P P co + π f m( τ ) dτ + φ ( ) ϕ ( I( ) R LO u LO u ω IF LO ). 0 (.5) The fir erm repreen direc deecion of he ignal and LO, repecively. The econd erm i of more inere for wo reaon. I ampliude i proporional o he quare roo of he localocillaor power. Thu, he deeced ignal can be made larger imply by increaing he local

33 ocillaor power. Secondly, becaue he deeced ignal i proporional o he quare roo of he ource-oupu power, i i obviou ha AM and PM can alo be deployed, bu hi i beyond he cope of hi chaper. We now define a quaniy which will be widely ued in he re of he hei: he FM modulaion index f β B where f i he opical bandwidh, and B i he ignal bandwidh (i.e. he bandwidh of m()). In Fig. -, we preen he bandwidh expanion ha occur in he FM cae. In hi figure, we oberve ha FM ha he abiliy of being able o expand he ignal bandwidh, and, a we will ee in he following ubecion, he performance gain by doing o can be ignifican. We alo noe ha he bandwidh can converely be uppreed and herefore i i poible o have β<. The modulaion index canno, on he oher hand, increae infiniely. Thi la poin i relaed o he FM hrehold phenomenon, which we will dicu in grea deail in laer ecion. Figure -: FM Bandwidh Expanion We proceed now o define he econd mo imporan quaniy relaed o FM, which i he carriero-noie raio (CNR). The CNR i defined a he power of he carrier divided by he power of he

34 noie induced in he bandwidh of he ignal [], [9]. The ource of noie in a coheren yem are imilar o hoe in a direc-deecion yem, which are menioned in Secion. Recalling (.4), he CNR afer phoo-deecion i CNR I N / 0 B (.6) where I i he ampliude of he modulaed carrier in he IF, N 0 i he one ided noie pecral deniy, and B i he ignal m() bandwidh, a hown in Fig. -. The noie pecral deniy N 0 i he um of he hermal, ho, and ineniy noie componen, a dicued in Secion. kt N q( I 0 + I dk ) (.7) R L Where I R( + ) P LO P u 0, dk power, ho noie limied recepion i obained, giving I i he negligible dark curren. A a reul, by increaing he LO CNR I / R P lim N B PLO 0 qr LO LO P u RP RPu ηq Pu hv ( P + P ) B PLO q( P + P ) B qb qb u lim LO LO P u u ηpu CNR hvb η E hvb (.8) 4

35 The CNR derived in (.8) how ha he LO power cancel ou for high LO power and he reuling CNR depend only on he iniial ignal power P u and he induced ho noie. Thi derived CNR will be widely ued in he re of hi hei. Some ypical CNR value are preened in Table -. 5

36 CNR (db) P u (dbw) Table -: Sho noie limied CNR for η0.9, Β50MHz according o (.8) 6

37 . Convenional FM Receiver.. Seup The convenional FM receiver ha been exenively udied for radio frequencie [8]. A good reference for convenional FM receiver can be found in [9]. In hi ecion, we ue he rucure of he convenional FM receiver udied in [9] adaped o he opical domain. The following analyi follow cloely ha of [9] and verifie he reul of [7]. Mix down from ω c o ω IF Bandpa filer cenred a inermediae frequency, ω IF Limier Dicriminaor r d () Lowpa filer Figure -: Convenional FM receiver A convenional FM dicriminaor i hown in Fig. -. We fir mix he received ignal down o inermediae frequency ω IF. A bandpa filer wih a bandwidh large enough o pa he modulaed ignal nearly undiored i he nex componen. A limier remove any ampliude variaion. The dicriminaor ha an oupu proporional o he difference beween he inananeou frequency and he inermediae frequency. Finally, a low-pa filer remove a much of he remaining noie a poible... Weak Noie Analyi For he convenional FM receiver decribed above, we now re-derive he claical FM weak noie analyi [7], [9]. A he FM dicriminaor oupu ha i, afer dicriminaing he argumen of (.4) - he mean quared ignal curren auming no deecor gain i: i 4π f m ( ). (.9) 7

38 The noie proce a he inpu o he dicriminaor i bandlimied becaue of he rericed bandwidh ha he dicriminaor. Thu, we can decompoe he noie ino wo low-pa procee ( n ( ), n ( ) ) c muliplying quadraure carrier and uing he repreenaion I I I ( ) I S co[ ω IF + x() ] + n() () [ I + n ( ) ]co[ ω + x() ] + n ( ) in[ ω + x() S () ( I + n ( ) ) + n ( ) co ω + x() S c c IF IF IF + an ] I / n ( ) + n c ( ) (.0) where x() f m( τ ) π dτ and having aume ha φ ( ) 0, ϕ ( ) 0 due o he phae cancellaion repored in []. 0 The one ided pecral deniy of he quadraure componen n ( ), ( ) LO n c i S ( ω) S ( ω) n c n N 0 0, ω < B, ω > B. (.) Nex, he limier remove he envelope variaion. The dicriminaor oupu i he inananeou frequency deviaion of he inuoid fromω IF. Differeniaing he argumen and dropping he ω IF erm, we obain he argumen of (.0) differeniaed and under he weak noie aumpion (.) d () x() r n + I / (). (.) 8

39 9 The weak noie aumpion, which wa menioned and ued in (.), implie ha / 0 I B N <, and coni of he implificaion below. Thi implificaion ha been ued in (.) o implify he invere angen erm in (.0) / ) ( ) ( / ) ( ) ( / ) ( an c c I n n I n n I n + +. (.) The above approximaion i valid excep for cerain improbable ime inerval during which ) ( n and/or ) ( n c aume value much larger han uual. Excluding uch inerval from conideraion, he noie pecral power deniy, afer paing he dicriminaor, can be calculaed a ( ) ( ) ( ) ( ) ( ) ( ) CNR B I B N B B N I B df f j N I df f S n B B n new / / / ) ( π π π π (.4) where () ( ) f S n c i he pecral deniy of he dicriminaed quadraure in-phae componen of he noie, and f j f H π ) ( he dicriminaor ranfer funcion. The SNR of an opical FMCD link uing (.9) and (.4) can now be expreed a ( ) CNR B m f n i SNR new FMCD ) ( 4 π π

40 f m ( ) CNR B SNR FMCD β m ( ) CNR (.5) where CNR i defined a in he ho noie limied recepion of (.8), and expanion a defined before. f β i he bandwidh B Thi i a ypical SNR formula for FM, a derived in [7], [9], and how ha he SNR increae when eiher he CNR increae or he opical bandwidh i expanded. Alo, he dependence of he SNR on he bandwidh expanion i of econd order. Therefore, expanding he opical bandwidh i a waned way of increaing he SNR of FM. However, hi expreion become invalid when he weak noie aumpion fail o hold and hen he approximaion of (.) canno be made. In hi cae, he phae noie canno be calculaed a in he above derivaion and (.5) fail o hold. Thi coni wha i called he even of anomaly and i will be he opic of he following ecion. We alo accoun for he poibiliy of anomaly in ubequen ecion. 40

41 .. Probabiliy of Cycle Skipping for Convenional FM The analyi ha we carried ou above i only valid under he weak noie aumpion; ha i, when N B << I /. Thi aumpion evenually break down a f increae, which can be underood 0 by realizing ha by increaing f, he FM yem inroduce more noie ino he pecrum o ha a degradaion in SNR performance occur, and a hrehold phenomenon i conequenly oberved, which i direcly relaed o he probabiliy of cycle kipping ha i analyzed in hi ecion. Thi wa oberved in he early day of FM. Rice developed a ueful model for he dicriminaor near i hrehold by deriving he probabiliy of anomaly [9]. Hi reul on hrehold behaviour compare favourably wih experimenal reul []. The mechanim leading o anomalie wih convenional FM receiver i inheren in he behaviour of he ignal phae a he limier-dicriminaor inpu. In he abence of modulaion, we have ued he approximaion an I / n ( ) + n c ( ) I / n ( ) + n c ( ) I n ( ) / (.6) Bu hi approximaion i valid only during inerval over which boh ( ) n and n c () are mall in relaion o I, and even wih weak noie, hee condiion will occaionally be violaed. The unmodulaed carrier can be repreened a a roaing phaor, a hown in Fig. -a. The frequency i ju he rae of roaion, a hown in Fig. -b. The noie vecor add ono he ignal vecor, a hown in Fig. -c. Since he ignal vecor i roaing a a conan rae, we can imply plo he relaive roaion of he oal vecor wih repec o he ignal vecor. Thi i hown in Fig. 4d. When he noie vecor i mall (i.e., he weak noie aumpion i invoked), a indicaed in Fig. -e, i caue a mall perurbaion in he inananeou frequency, a hown in Fig. -f. If he noie vecor i large, he received vecor circle he origin, a hown in Fig. -g. Thi caue a π phae error (i.e., a cycle kip). If he encirclemen i rapid, we can rea i a an approximae ep in phae which lead o a narrow pule in inananeou frequency, a hown in Fig. -h. The waveform in Fig -h i he oupu of he low-pa filer. 4

42 Following Rice argumen, we can eimae he probabiliy of cycle kipping (or anomaly), denoed by P [ A] ; ha i, ha a lea one encirclemen occur during he inerval [0, /B]. Thi analyi i exenively carried in [9, p ]. The reuling probabiliy of anomaly i: P[ A] ( β + ) e CNR /( β + ) 4π CNR /( β + ). (.7) Commening on (.7), we oberve ha by increaing β he probabiliy of anomaly become greaer due o he exponenial influence mainly, and herefore increaing he opical bandwidh induce more cycle kip. Thi i omehing expeced ince more noie accumulae inide he opical bandwidh. On he oher hand, by decreaing he CNR we again ener he cycle kipping phae becaue of he low CNR which render he weak noie aumpion menioned in (.) invalid. Thee wo remark repreen wo differen view of he ame hrehold ha lead o he cycle kipping condiion. 4

43 Figure -: Phaor diagram, [] (a) Signal phaor. (b) Inananeou frequency. (c) Signal and noie phaor. (d) Roaing coordinae yem (carrier eliminaed). (e) Small noie (no origin encirclemen). (f) Inananeou frequency. (g) Origin encirclemen, large noie. (h) Inananeou frequency 4

44 44..4 Complee SNR of FMCD A one-parameer characerizaion of he performance of FM may be obained by combining he mean quare error in he abence of anomalou error wih he minimum mean quare error (MMSE) conribued by he anomalie hemelve. When an anomaly occur, ^ m, he eimaion of our ignal m, i equally likely o be anywhere in he inerval [-, ], regardle of he value of m. Furhermore, he anomaly even i independen of m, and hu ^ m i independen of m. [ ] ^ ^ + + m A m E A m E Anomaly m m E (.8) By dividing he mean quare ignal power ( ) m by SNR FMCD (.5) in he abence of anomalie, we obain he mean quare noie power for ignal m(). When dividing he mean quare noie power by he ignal bandwidh B we obain in (.9) an expreion for he minimum mean quare error ( ) B CNR B SNR m anomaly no noie weak m m E FMCD _, ^ β (.9) We now obain he following expreion for he oal mean quare error baed on (.8) and (.9) ( ) ] [ ] [ ] [ ] [ ]) [ ( ] ' [ ^ ^ A P m A P B CNR A P A m m E A P A m m E T β ε (.0) where ' A i he complemen of he even of anomaly. The full formula for he SNR i now found o be: ( ) B A P m A P CNR m B m SNR T + + ] [ ] [ β ε. (.)

45 By ubiuing in ome boundary value for he probabiliy of anomaly, we have: For [ A] 0 P, (.) implie SNR β m CNR ame a in (.5). For P [ A], (.) implie m SNR m +, which i independen of he CNR becaue we have cycle kipping and we are gueing our ignal. The weak noie aumpion i no longer valid and he CNR canno yield any change o he performance inide he phae of cycle kipping. Even for low CNR and more inene cycle kipping, we can only ill gue he ignal and hu even he lowe CNR canno make performance wore. 45

46 ..5 FM Threhold Phenomenon In order o be able o ge ome numerical reul and provide a graphical repreenaion of he FM hrehold, we have o aume pecific econd order characeriic for he ignal m(). Subequenly, in Secion.4.6, we how ha he hrehold acually ha a weak dependence on hee characeriic. For m(), which i normalized in [-,], we aume ha i i alo uniformly diribued in hi range, which implie he econd order characeriic m (). The reuling mean quare error of (.8) i ^ E m m. Then (.) become: Anomaly m SNR ε T β CNR ( P[ A] ) + P[ A] B (.) where, for convenional FM receiver, he probabiliy of error i defined in (.7), bu for compleene we preen i below a well P[ A] ( β + ) CNR /( β + ) e. 4π CNR /( β + ) The SNR in (.) i a funcion only of wo parameer, namely he CNR and he modulaion index β. 46

47 In Figure -4, he SNR i ploed a a funcion of hee wo parameer and we oberve he FM hrehold phenomenon. The hree-dimenional graph allow u o ee ha for higher CNR we can achieve higher SNR by increaing he FM modulaion index and hu furher expanding he opical bandwidh. In Figure -5, we oberve he FM hrehold, which i defined a he value of CNR above which he probabiliy of anomaly i pracically 0. Below he FM hrehold poin, he noie ignal may inananeouly have ampliude greaer han ha of he waned ignal and herefore applicaion may wih o operae wihin a margin above he FM hrehold poin (a few db). For a cerain value of he CNR, he SNR increae a he bandwidh expanion increae up o a maximum value (Figure -6), afer which he noie accumulaed in he expanded bandwidh decreae he SNR, a dicued. Figure -5 and -6 acually repreen cro-ecion of Figure -4. Combining boh of he conrain menioned in he preceding paragraph - a required CNR above he hrehold a preened in Figure -5 and a bandwidh expanion correponding o he peak of Figure -6 - we conclude ha we wih o operae on he righ ide lope of he hree dimenional curve of Figure - 4 and cloe o he peak SNR. For a given CNR, if we wih o maximize he SNR performance we are required o chooe he maximum bandwidh expanion, a can be een in Figure -6. Thi β max can be more formally defined a: β max (, β, m () ) SNR CNR β : 0 β (.) I i very difficul o ge an analyical expreion on β max mainly becaue of he exponenial naure of he probabiliy of anomaly which affec he exponenial dependence of β max on CNR, a can be een in Figure -7. Linear approximaion wihin mall region of he CNR are poible bu no a complee expreion. 47

48 In Figure -7, he maximum value of he bandwidh expanion β i ploed a a funcion of he CNR and he correponding calculaed value are preened in Table -. The β max preened in hi graph ake value cloe o uniy for low CNR, and a he CNR increae we are able o uilize a higher β max. Thu, we can expand he opical bandwidh more wih a higher CNR. Thi ha acually been he moivaion behind FM: expanding β max a much a poible o beer uilize he available enormou bandwidh capaciy of opical fiber, and conequenly achieving higher SNR. Moreover, we can oberve he exponenial naure by which β max increae a a funcion of he CNR. Finally, Table -, - provide numerical value of β max for a range of CNR and fixed ignal bandwidh. Table - correpond o a ignal of Hz and Table - o a ignal of 50MHz. I i clear ha he value of β max for he 50MHz ignal are lower and hi i reaonable becaue a wider bandwidh mean lower CNR according o (.8). Thu, he lower he CNR, he le we can pread our ignal a we have noiced in he previou graph. I i no peculiar ha he value of β are le han in Table - ince we can ee in lieraure ha indexe below are realizable [p. 649, 9]. 48

49 75 SNR HdB L β 0 CNR Figure -4: SNR of FM a a funcion of he CNR and he modulaion index β for a convenional FM receiver (m()uniformly diribued in [-,] and ignal of Hz bandwidh) 49

50 SNR HdB L β β5 β0 β CNR HdB L Figure -5: SNR a a funcion of CNR for differen value of he modulaion index β (m() uniformly diribued in [-,] and ignal of Hz bandwidh) SNR HdB L CNR 0dB CNR 5dB CNR 0dB CNR 5dB β -0 Figure -6: SNR a a funcion of β for differen CNR value (m() uniformly diribued in [-,] and ignal of Hz bandwidh) 50

51 β max Deail CNR HdB L Figure -7: Maximum value of modulaion index β max a a funcion of CNR for convenional FM receiver wih m()uniformly diribued in [-,] (m() uniformly diribued in [-,] and ignal of Hz bandwidh) 5

52 CNR (db) β max Table -: β max for differen CNR value for a uniformly diribued ignal in [-, ] of Hz bandwidh CNR (db) β max Table -: β max for differen CNR value for a uniformly diribued ignal of 50 MHz bandwidh 5

53 ..6 Dependence of Rician Approach on <m ()> In Secion..5, we inveigaed he cae in which m() i uniformly diribued. The econd order aiic of m() eem o play a role in he Rician approach, epecially in he value of β max, judging from (.). The SNR ha been ploed in Figure -8 for differen value of he econd order characeriic of m(). The concluion of hi ecion, judging alo from he figure ha preen he FM hrehold (Figure -9, -0), i ha he hrehold doe no depend rongly on he econd order characeriic. Thu, he whole analyi ha we have carried o far for uniformly diribued ignal m() remain valid for differen diribuion regardle of he econd order characeriic. Regardle of he econd order characeriic, our ignal alway ha o be normalized o [-,] ince he noie i alo aken a normalized. A can be een in Fig. -8, he FM hrehold remain he ame for variaion of m () from 0 o 4. Thu, even if we have lighly differen econd order characeriic, he β max which we will calculae will be cloe o he real one, and very cloe o he peak performance of he FM. Thi weak dependence can inuiively be underood from (.), where he SNR depend linearly on m ( ), wherea i depend exponenially on he CNR and β hrough he probabiliy of anomaly erm. 54 β max <m HL> Figure -8: β max for differen econd order characeriic of he unmodulaed ignal (Hz bandwidh, CNR0 db) 5

54 SNR HdB L <m HL> ê4 <m HL> ê <m HL> ê <m HL> <m HL> <m HL> CNR HdB L Figure -9: SNR for differen econd order characeriic of he unmodulaed ignal (Hz bandwidh, β8 maximum value for 0dB CNR according o Table -) SNR HdB L <m HL> ê4 <m HL> ê <m HL> ê <m HL> <m HL> <m HL> β Figure -0: SNR for differen econd order characeriic of he unmodulaed ignal (Hz bandwidh, CNR0 db) 54

55 ..7 Opimum Angle Modulaion Syem The convenional FM receiver i no an opimum angle modulaion yem. An opimum modulaion yem deign can be approached by opimum linear filering heory, in conjuncion wih eiher Wiener echnique or Kalman-Bucy echnique which ake ino accoun he ype of power pecral deniy of he ignal [], []. However, above he hrehold, he convenional dicriminaor wih an opimum po-dicriminaor filer perform exacly like he opimum anglemodulaion yem. The only difference beween he wo yem i he locaion of he hrehold []. In hi hei, we followed he claic Rician approach wih an analyic expreion for he dicovery of he hrehold by uing he probabiliy of anomaly for FM ha Rice found [9]. There i ubanial work in he bibliography which exend Rice approach conidering he meage a a random proce. For example, Chang udied he hrehold for a Gauian meage wih one-pole pecrum, and Rachel udied i for a econd-order Buerworh pecrum []. The concluion of he analyi [], [] i ha he opimum yem improve he hrehold performance by db for he fir-order Buerworh pecrum and by 6dB for he econd-order Buerworh pecrum. Thee reul have been verified by boh heoreical analyi and imulaion reul []. In pracice, in order o achieve opimaliy, engineer have ried o devie echnique o delay he one of he FM hrehold effec. Thee device are generally known a FM hrehold exenion demodulaor. Technique uch a FM feedback, phae locked loop and frequency locked loop have been ued o achieve hi effec. A an example, hough no a rigorou one becaue of he ue of a lower bound, we demonrae hi effec in Figure - for feedback receiver veru he convenional FM receiver ha we have udied. The claic Rician approach wih an analyical expreion for he probabiliy of anomaly ha alo been deployed for hi plo of he FM feedback receiver (FMFB), by uing he noion ha a lower bound for he probabiliy of cycle kipping for feedback receiver would be he probabiliy of anomaly for Frequency Poiion Modulaion [9, pp.666]. In [9] i i no menioned how igh hi bound migh be. Keeping hi under conideraion, i eem from Figure - poible o achieve beer SNR under he ame combinaion CNR when uing a feedback receiver inead of a convenional one. We alo oberve from Figure - ha above he hrehold he convenional FM 55