Unitary Matrices in Fiber Optical Communications: Applications

Size: px

Start display at page:

Download "Unitary Matrices in Fiber Optical Communications: Applications"

Gilbert Spencer
5 years ago
Views:

1 Uniay Maices in Fibe Opical Communicaions: Applicaions Ais Mousaas A. Kaadimiais Ahens P. Vivo KCL R. Couille Pais-Cenal L. Sanguinei Pisa A. Mulle Huawei H. Hafemann Huawei Ais Mousaas, Univesiy of Ahens

Inoducion eed fo high speed infasucue Bandwidh BW hungy echnologies emeging ~2 db incease pe yea -Wied: IPTV, elepesence, online gaming, live seaming ec.

2 Inoducion eed fo high speed infasucue Bandwidh BW hungy echnologies emeging ~2 db incease pe yea -Wied: IPTV, elepesence, online gaming, live seaming ec. - Capaciy- cunch eminen Soluions so fa Opical ewos WDM-DWDM have exploied many degees of feedom BW, available powe, polaizaion divesiy excep one: spaial. Soon he online devices will exceed he numbe of global populaion! Use of opical Space Division Muliplexing SDM is compelling. Ideally: wihou changing he aleady infasucue Ais Mousaas, Univesiy of Ahens 2

Inoducion Wha is Opical MIMO? Jus lie in wieless domain, in opical we can use paallel ansmission pahs o gealy enhance he capaciy of he sysem. Fis pape on Opical MIMO: H. R.

3 Inoducion Wha is Opical MIMO? Jus lie in wieless domain, in opical we can use paallel ansmission pahs o gealy enhance he capaciy of he sysem. Fis pape on Opical MIMO: H. R. Sua, Dispesive muliplexing in mulimode opical fibe, Science , Muli-Mode Fibes MMF: Use muliple modes o cay infomaion Muli-Coe Fibes MCF: Uilize diffeen opical pahs of diffeen coes in he same fibe wihin he same cladding Ais Mousaas, Univesiy of Ahens 3

4 Inoducion Poblem: Cossal phenomenon aises Cossaling beween adjacen coes in MCF Ligh beam scaeing esuling in cossaling in MMF Due o : Exensive fibe lengh Bending of fibe Limied aea wih muliple powe disibuions Ligh beam scaeing on-lineaiies Ais Mousaas, Univesiy of Ahens 4

5 Inoducion Poblem: Cossal phenomenon Two Appoaches: Figh i Secion of 7-coe opical fibe Ais Mousaas, Univesiy of Ahens 5

6 Inoducion Poblem: Cossal phenomenon Two Appoaches: Tae advanage of i Classic MIMO echniques equied bu wih some wiss: Low powe consains o avoid non-linea behavio. Opical channel maix jus a subse of a uniay maix. Ais Mousaas, Univesiy of Ahens 6

7 Sysem Model Conside a single-segmen -channel lossless opical fibe sysem: ansmiing channels excied, eceiving channels coheenly. The 2 2 scaeing maix is S T SS T Only Haa-disibued I sub-maix is of inees is ~. Geneally, < : Ohe channels may be used fom diffeen, paallel ansceives Modelling of loss: addiional enegy los duing popagaion Ais Mousaas, Univesiy of Ahens 7

8 Sysem Model Only Haa-disibued I sub-maix is of inees is ~ Define x maix U as T U P SP whee P pojecion opeao: P I - Ais Mousaas, Univesiy of Ahens 8

9 Sysem Model Channel Equaion: Assume no diffeenial delays beween channels fequency fla fadinghe muual infomaion is I y Ux I U U log U log de λ z Gaussian noise z Receive nows he channel pilo Tansmie does no now he channel w.l.o.g. Ais Mousaas, Univesiy of Ahens 9

10 Ais Mousaas, Univesiy of Ahens Infomaion Meics Ouage Pobabiliy: Opimal: Assumes infinie codewods Wha is he pice of finie codelenghs? Gallage eo bound fo M-lengh code: [ ] Pob U U ou I E I P Θ < < U U I U log de max exp M E P e

11 Coulomb Gas Analogy Join pobabiliy disibuion of eigenvalues of UU U λ λ,..., λ exp log λ log λ 2 log λ P, 2 Ν λm 2 Exponen is enegy of poin chages epelling logaihmically in he pesence of exenal field 2 S [ p] P e Lage : chages coalesce o densiy Dyson Ben Aous/Maida S dx p x Veff x dxdy p y p xlog x y Veff x n log x β log x whee n β Minimizing convex S w.. px gives he Maceno-Pasu disibuion p MP x x a b x 2πx x a, b n ± β n β n β 2 > m Ais Mousaas, Univesiy of Ahens

12 Ouage Pobabiliy We need ails of disibuion: Opical Comms opeae a P ou I < E [ Θ I ] Pob U U p ou 8 Fouie ansfom o use lage deviaions agumens S S V eff x V eff dx p x x log log x x plays ole of sengh of logaihmic aacion/epulsion a x > shifs chage densiy o lage values R>Reg, < o smalle ones R<Reg Minimizing S w.. px gives he genealized Maceno Pasu disibuion Equivalen o balancing foces on he chage locaed a x: p x' n β 2P dx x x' x x Ais Mousaas, Univesiy of Ahens 2

13 Genealized MP equaion Use Ticomi heoem o calculae px Obain closed fom exp. fo enegy S[p] E.g. β>; n> p x b x x a 2πx x n x a b β x ab a, b, calculaed fom p b p a b x dxp xlog a b dxp x a Ais Mousaas, Univesiy of Ahens 3

14 Genealized MP equaion In geneal S b S ab S S a a b< a> b< a b a> b n; β < c -- c < < c2 > c2 n>; β < c3 > c n; β> -- < c4 -- > c4 n>; β> -- all Phase ansiions a o a> ec ae hid ode disconinuous S' '' Relaion o Tacy-Widom? c Ais Mousaas, Univesiy of Ahens 4

15 Ais Mousaas, Univesiy of Ahens 5 Disibuion Densiy of Finally whee is he vaiance a he pea of he disibuion. [ ] eg S S v e f eg 2π log '' b a a b S v eg eg

16 umeical Simulaions β > and n > The LD appoach demonsaes bee behavio, following Mone Calo. Ais Mousaas, Univesiy of Ahens 6

17 umeical Simulaions β > and n > The LD appoach demonsaes bee behavio, following Mone Calo. Fo small values of, and, he discepancy is minimal Ais Mousaas, Univesiy of Ahens 7

18 Ais Mousaas, Univesiy of Ahens 8 Finie Bloc-Lengh Eo Pobabiliy Hee we need whee ow is bounded in [,] Also when < ohewise no consain x x V x V x x p dx S S eff eff log log α α < j j e M E P λ log max exp U x x x x dxp b a log Ν Μ α

19 umeical Simulaions β > and n Thee ae wo ypes of phase ansiion poins hee: c : A poin disconinuous S' ' c2 : Disconinuous S' '' c2 c Eo Pobabiliy vs Rae; P; β3; n α2 α α5 op 4 2 -logpe/ Ais Mousaas, Univesiy of Ahens 9

20 Moe Realisic Channel Chaoic caviy picue: S I 2πiW π H i WW W Assuming vey low bacscaeing a edges we obain U P SP cp H G iγ P Deeminisic mode enegy Random complex Gaussian Diagonal loss maix Γ H G Ais Mousaas, Univesiy of Ahens 2

21 Ais Mousaas, Univesiy of Ahens 2 Moe Realisic Channel Muual Infomaion meic: I Disibuion mean-vaiance can be obained using eplica heoy I 2 same wih also expessions fo vaiance [ ] i i I P Γ G H P P Γ G H P I G log de [ ] [ ] log de p p I I I I E δ H G p T p p T p p T δ δ δ δ H H H H

22 umeical Simulaions Ais Mousaas, Univesiy of Ahens 22

23 Conclusions MIMO: pomising idea in Opical Communicaions In his wo: Simple model fo Opical MIMO channel Lage Deviaion Appoach povides ails fo MIMO muual infomaion Mehod povides meic fo ouage houghpu and finie bloclengh eo Many issues sill open: Channel modeling sill a is infancy Tansmie/Receive Achiecues Muliple fibe segmens onlineaiies Ais Mousaas, Univesiy of Ahens 23

24 Inoducion Moivaion: Cap cunch, degees of feedom Channel Model: Uniay, andom, lossyefeence fo exa channel model on-lineaiies, MDL Meics: Capaciy Equaion, dof explanaion Eo exponen: Poof by Gallaghe Eigenvalue picue: Mapping o effecive enegy minimize o maximize pobabiliy: MP esul fo mos pobable Coulomb appoach: effecive densiy Inegal equaion and soluion Examples of mos pobable densiies compaison beween opimal and nonopimal ouage Ais Mousaas, Univesiy of Ahens 24

25 Inoducion Phase ansiions: Second ode and hid ode ansiions in he Gallaghe exponens Genealizaion o ohe appoaches Deeminisic plus andom channel Lossy channel Amplify and fowad channel Ais Mousaas, Univesiy of Ahens 25

Low-complexity Algorithms for MIMO Multiplexing Systems

Low-complexity Algorithms for MIMO Multiplexing Systems Low-complexiy Algoihms fo MIMO Muliplexing Sysems Ouline Inoducion QRD-M M algoihm Algoihm I: : o educe he numbe of suviving pahs. Algoihm II: : o educe he numbe of candidaes fo each ansmied signal. :