A Study of Capacity and Spectral Efficiency of Fiber Channels

Transcription

1 A Study of Capacity ad Spectral Efficiecy of Fiber Chaels Gerhard Kramer (TUM) based o joit work with Masoor Yousefi (TUM) ad Frak Kschischag (Uiv. Toroto) MIO Workshop Muich, Germay December 8, 2015 Istitute for Commuicatios Egieerig

2 1) Mai Message A upper boud o the spectral efficiecy of a stadard optical fiber model η log( 1+ SNR) [bits/sec/hz] this is the first upper boud o a full model the boud is tight at low SNR; the boud may be extremely loose at high SNR; but it s better tha othig Istitute for Commuicatios Egieerig 2

3 2) Iformatio Theory Basics Capacity C of a chael P Y X (.) is the maximum I(X;Y) uder costraits put o X Example: real-alphabet additive white Gaussia oise (AWGN) chael Y = X + Z with Var[Z]=N ad a iput power costrait E[X 2 ] P has ( ) C = 1 2 log $ 1+ P # N I X;Y " % ' & Complex alphabet AWGN chaels: C = log(1+p/n) N is usually take as N 0 W where N 0 is the (oe-sided) oise PSD ad W is the badwidth Spectral efficiecy is η=c if oe uses sic-pulses of badwidth W

4 Maximum Etropy Maximum Etropy: cosider R X =E[X X ] where X has legth L. The h X " # ( ) log ( π e) L det R X $ % with equality if ad oly if X is Gaussia ad circularly symmetric For a complex square matrix M we have h( M X) = h( X) + 2log det( M) I particular, if M is uitary the h( M X ) = h( X )

5 Etropy Power Iequality Etropy Power: V ( X) = e h( X ) L ( π e) Etropy Power Iequality: for idepedet X ad Y we have V ( X + Y ) V ( X) + V ( Y ) Coditioal versio: for coditioally idepedet X ad Y we have ( ) = e h ( X U ) L ( π e) ( ) V ( X U) + V ( Y U) V X U V X + Y U

6 3) Fiber Chael(s) To simulate, split the fiber legth z* ito K small steps (Δz) ad the time T ito L small steps (Δt) time sigal: Split-step Fourier method at distace z k, k=0,1,...,k vector of legth L E(z k ) F D L F -1 E N (z k+1 ) D N E(z k+1 ) Fiber Loss/Gai Liear Noliear Noise Ideal Rama amplificatio: removes the loss but adds oise F = Fourier trasform D L = diagoal matrix with fixed etries of uit amplitude (all-pass filter) D N = diagoal matrix with uit amplitude etries; the (l,l)-etry phase shift is proportioal to the magitude-squared of the l th etry of E N (z k+1 )

7 E(z k ) 4) A Upper Boud E N (z k+1 ) E(z F D L F -1 k+1 ) D N Liear Noliear Noise Mai Observatios The liear step coserves eergy ad etropy The o-liear step also coserves eergy ad etropy! j arg(a) + jf( a ) $ h# a e & = h a,arg(a) + f( a ) " % ( ) + h arg(a) + f( a ) a ( ) + E log a = h a ( )!##### "##### $ + E '( log a ) * = h(a) h a,arg(a) ( ) '( ) *

8 Eergy Recursio E(z k ) E N (z k+1 ) E(z F D L F -1 k+1 ) D N Liear Noliear Noise Eergy after K steps: Eergy Lauch + KN. We thus have: ( ) log" π e h E ( z K ) L log" # π e R i,i E z K i=1 # ( ( )) ( ) L det R E ( z K ) ( ( )) $ % ( ) L $ % maximum etropy Hadamard's iequality L log"# π e Eergy Lauch + KN $ % Jese's iequality

9 Etropy Recursio E(z k ) E N (z k+1 ) E(z F D L F -1 k+1 ) D N Etropy recursio: Liear Noliear Noise We thus have: ( ) E ( z 0 ) ( ) V ( E z ) k + N L V E z k+1 V E z K ( ) E ( z 0 ) ( ) KN L or h E z K ( ) E ( z 0 ) ( ) E ( z 0 ) ( ) ( ) Llog π e KN L

10 E(z k ) E N (z k+1 ) E(z F D L F -1 k+1 ) D N So for every step we have: Sigal eergy grows by the oise variace: ca upper boud h( E(z K ) ) Etropy power grows by at least the oise variace: ca lower boud h( E(z K ) E(z 0 ) ) Result*: Liear I E( z 0 );E( z K ) Noliear Noise ( ) = h( E( z K )) h E( z K ) E( z 0 ) ( ) L log 1+ SNR ( ) *SNR = receiver sigal-to-oise ratio

11 1 L I ( E( z );E( z ) 0 K ) log( 1+ SNR) Let B = 1/Δt be the badwidth of the simulatio So L = T/Δt = TB is the time-badwidth product The spectral efficiecy is thus bouded by η log( 1+ SNR) [bits/sec/hz]

12 Discussio η log( 1+ SNR) [bits/sec/hz] Q1: Why ormalize by the simulatio badwidth B? The real badwidth W ca be smaller. A1: B ca be chose (this is eve desirable) as the smallest badwidth for which simulatios give accurate results Q2: What about capacity? A2: Ay real fiber has a maximal badwidth B max. A capacity upper boud follows by multiplyig B max by log(1+snr)

13 Discussio η log( 1+ SNR) [bits/sec/hz] Q3: What about MIMO fiber? A3: If eergy is preserved by the liear ad o-liear steps, ad the oise is AWGN the the above boud remais valid per mode Q4: What about frequecy-depedet (or mode-depedet) loss? A4: Ope research! Q5: What about lower bouds? A5: Apply etropy recursio to V( E(z k ) ) ad eergy recursio to h( E(z k ) E(z 0 ) ). Issues (looks solvable): badwidth expasio bouds

14 Coclusios 1) Noliear cascade models are fu to study... may other applicatios 2) Spectral efficiecy of SMF with liear polarizatio is log(1+snr) 3) May extesios are possible: lumped amplificatio, 3 rd -order dispersio, delayed Kerr effect uiform loss, liear filters (for capacity results) MIMO fiber (MMF or MCF) if the liear ad o-liear steps coserve eergy ad etropy, ad the oise is Gaussia ad white 4) More difficult: better bouds ad uderstadig at high SNR frequecy-depedet loss, dispersio, o-liearity 5) Network iformatio theory for fiber should be developed Istitute for Commuicatios Egieerig 14