Staircase Codes. Error-correction for High-Speed Fiber-Optic Channels

Size: px

Start display at page:

Download "Staircase Codes. Error-correction for High-Speed Fiber-Optic Channels"

Laurence Bradford
6 years ago
Views:

1 Staircase Codes Error-correction for High-Speed Fiber-Optic Channels Frank R. Kschischang Dept. of Electrical & Computer Engineering University of Toronto Talk at Delft University of Technology, Delft, The Netherlands April 23rd,

2 Acknowledgements Joint work with: Benjamin P. Smith University of Toronto Andrew Hunt and John Lodge Communications Research Centre, Ottawa Arash Farhood Cortina Systems Inc., Sunnyvale/Ottawa Thank you to Drs. Nader Alagha and Fernando Kuipers for the invitation! 2

3 Fiber-Optic Communication Systems Physics: Enabling Technologies 1 Low-loss optical fiber ( 54 THz bandwidth) 2 Optical amplifiers 3 Laser transmitters and Mach-Zehnder modulators 3

4 Fiber-Optic Communication Systems: Challenges Reliability P e <

5 Fiber-Optic Communication Systems: Challenges Reliability P e < Speed 100 Gb/s per-channel data rates 4

6 Fiber-Optic Communication Systems: Challenges Reliability P e < Speed 100 Gb/s per-channel data rates Non-Linearity The fiber-optic channel is non-linear in the input power 4

7 Outline of Talk Part I Binary Error-Correcting Codes for High-Speed Communications Syndrome-based interative decoding Part II Performance optimization for product-like codes Staircase codes FPGA-based simulation results Analytical error-floor predictions Spectrally-Efficient Fiber-Optic Communications Memoryless capacity estimates, with and without digital backpropagation Pragmatic coded modulation via staircase codes 5

8 Part I Binary Error-Correcting Codes for High-Speed Communications most reliable least reliable Π Π Π 6

9 Outline This part is about... FEC for the binary symmetric channel (BSC) After optical (and/or) electrical compensation, suitable as forward-error-correction (FEC) for 100Gb/s PD-QPSK systems without soft information 7

10 Outline This part is about... Outline FEC for the binary symmetric channel (BSC) After optical (and/or) electrical compensation, suitable as forward-error-correction (FEC) for 100Gb/s PD-QPSK systems without soft information 1 Existing solutions (G.975, G.975.1) 2 Implementation considerations 3 Staircase codes 4 Generalizations and conclusions 7

11 Existing Solutions 8

12 Performance Measure Net Coding Gain Given a particular BER, we can obtain the corresponding Q via Q = 2erfc 1 (2 BER). The coding gain (CG) of an error-correcting code is defined as CG (in db) = 20 log 10 (Q out /Q in ), and the Net Coding Gain (NCG) is NCG (in db) = CG + 10 log 10 (R), where R is the rate of the code. 9

13 ITU-T G.975 (1996) Reed-Solomon (255,239) Code Coding Symbols are bytes Depth-16 interleaving corrects (some) bursts up to 1024 bits Coding Gain = 6.1 db Net Coding Gain = 5.8 db P[e] Shannon Limit (C=239/255) 2-PSK Limit (C=239/255) BSC Limit (C=239/255) RS(255,239) 6.1 db uncoded 2-PSK log 10 (Q) (db) Constraint: Rate and framing structure fixed for future generations 10

14 ITU-T G (2004) Concatenated Codes Product-like codes with algebraic component codes k a n a C C k b Information Symbols Row Parity Π n b Column Parity Parity on Parity 11

15 ITU-T G (2004) Code NCG Notes I Outer RS, Inner CSOC I Outer BCH (t = 3), Inner BCH (t = 10) I Outer RS, Inner BCH (t = 8) I Outer RS, Inner Product (t = 1) I LDPC I Outer BCH (t = 4), Inner BCH (t = 11) I RS(2720,2550) I Product BCH (t = 3), Erasure Dec? 12

16 Objectives Increased Net Coding Gain NCG Shannon Limit of BSC (9.97 db at ) Error Floor Error floor Lower error floor Insurance in presence of correlated errors Block Length n or less Low Implementation Complexity Dataflow considerations at 100Gb/s 13

17 Implementation Considerations 14

18 Hardware Considerations: Product vs. LDPC Product Code C Π C Algebraic component codes Syndrome-based decoding LDPC Code Π SPC component codes Belief propagation decoding 15

19 Syndromes The codewords of a linear (n, k) block code C are the solutions of a homogeneous system of simultaneous linear equations, i.e., v C vh = 0. Let v C be sent and let r be received. Define the error pattern e as e = r v so that r = v + e. Then the syndrome corresponding to r is s = rh = (v + e)h = }{{} vh +eh = eh. 0 An error-correcting decoder must infer the most likely error pattern from the syndrome. 16

20 Syndrome-based Decoding of Product Codes Decoding an (n, k) component codeword n received symbols n k symbol syndrome R = 239/255, n 1000, n k 32 For high-rate codes, syndromes provide compression 3 decodings/component 96 bits/decoding components/symbol bits/symbol Syndrome Algebraic Decoder Error Locators Update Syndromes Total Dataflow At 100Gb/s, 76.8 Gb/s internal dataflow 17

21 LDPC Belief Propagation Decoding 15 iterations 2 messages/iteration edge 5 bits/message 3 edges/symbol 450 bits/symbol Π Total Dataflow At 100Gb/s, 45Tb/s internal dataflow! 18

22 Dataflow Comparison 76.8 Gb/s 45 Tb/s 2 3 orders of magnitude (huge implementation challenge for soft message-passing LDPC decoders). 19

23 Our Solution 20

24 Coding with Algebraic Component Codes C 1 C 2 C 3 C l Π Graph Optimization Degrees of Freedom Mixture of component codes (e.g., Hamming, BCH) Multi-edge-type structures 21

25 Staircase Codes: Construction Consider a sequence of m-by-m matrices B i m B 1 B 0 B 1 B 2 m and a linear, systematic, (n = 2m, k = 2m r) component code C 2m r Information r Parity Encoding Rule i 0, all rows of [ B T i 1 B i] are codewords in C 22

26 Staircase Codes: Construction B 1 B T 0 B 1 B T 2 B 3 23

27 Staircase Codes: Construction B 1 B T 0 B 1 B T 2 B 3 23

28 Staircase Codes: Construction B 1 B T 0 B 1 B T 2 B 3 23

29 Staircase Codes: Construction B 1 B T 0 B 1 B T 2 B 3 23

30 Staircase Codes: Construction B 1 B T 0 B 1 B T 2 B 3 23

31 Staircase Codes: Construction B 1 B T 0 B 1 B T 2 B 3 23

32 Staircase Codes: Construction B 1 B T 0 B 1 B T 2 B 3 23

33 Staircase Codes: Properties Hybridization of recursive convolution coding and block coding Recurrent Codes of Wyner-Ash (1963) Info i B i 1 Algebraic Encoder Delay B i Rate : R = 1 r/m Variable-latency (sliding-window) decoder B i 1 B i B i+1 B i+2 B i+3 B i+4 24

34 Staircase Codes: Properties B T i Bi+1 B T i+2 Bi+3 The multiplicity of (minimal) stalls of size (t + 1) (t + 1) is K = ( ) m t + 1 t+1 j=1 ( )( ) m m, j t + 1 j and the corresponding contribution to the error floor, for transmission over a binary symmetric channel with crossover probability p, is BER floor = K (t + 1)2 m 2 p (t+1)2. 25

35 General Stall Patterns B T i B i+1 B T i+2 B i+3 (5,5)-stall K L Contribution Contribution of (K, L)-stalls, p =

36 Multi-Edge-Type Representation most reliable least reliable ΠB ΠB ΠB m m B i 1 B i B i+1 B i+2 B i+3 C C Π 27

37 Movie Time B i 1 B i B i+1 B i+2 B i+3 B i+4 28

38 Braided Block Codes: Construction I P P Felström, Truhachev, Lentmaier, Zigangirov, Braided Block Codes, IEEE Trans. on Info. Theory,

39 Braided Block Codes: Construction I P P I P P Felström, Truhachev, Lentmaier, Zigangirov, Braided Block Codes, IEEE Trans. on Info. Theory,

40 Braided Block Codes: Construction I P P I P P I P P Felström, Truhachev, Lentmaier, Zigangirov, Braided Block Codes, IEEE Trans. on Info. Theory,

41 Braided Block Codes: Performance Our experiments show: But: For high-rate (239/255), braided block codes yield approximately the same performance as the corresponding product code Multi-edge gain diminished, since fewer bits (previous parity only) serve as clean Spatially-coupled generalized LDPC ensembles with iterative hard-decision decoding can approach capacity at high rates: Y.-Y. Jian, H. Pfister, K. Narayanan, Approaching Capacity at High Rates with Iterative Hard-decision Decoding, Proc. IEEE Int. Symp. Inform. Theory, July,

42 FPGA-based Simulation Results Code Parameters m = 510, r = 32, triple-error-correcting BCH component code 20 log 10 (Q) (db) BER out BSC Limit (C=239/255) Staircase I.9 G codes I.3 I.4 I.5 RS(255,239) BER in

43 Part II Spectrally-Efficient Fiber-Optic Communications most reliable least reliable Π Π Π 32

$The Kerr Effect Kerr electro-optic effect (DC Kerr effect) An effect discovered by John Kerr in 1875 It produces a change of refractive index in the$

44 The Kerr Effect Kerr electro-optic effect (DC Kerr effect) An effect discovered by John Kerr in 1875 It produces a change of refractive index in the direction parallel to the externally applied electric field The change of index is proportional to the square of the magnitude of the external field 33

45 The Kerr Effect in Optical Fibers Optical Kerr Effect (or AC Kerr effect) No externally applied electric field is necessary The signal light itself produces the electric field that changes the index of refraction of the material (fused silica) The change in index in turn changes the signal field The change in index of refraction is proportional to the square of the field magnitude P Changes index of refraction WDM channel Optical Fiber Channel of interest Generates distortion f A signal in a certain frequency band can distort the signal in a different frequency band without spectral overlap 34

46 Nonlinear Effects in Fibers intra-channel inter-channel signal-noise signal-signal signal-noise signal-signal NL phase noise parametric amplification SPM NL phase noise WDM nonlinearities SPM-induced NL phase noise MI isolated pulse SPM IXPM IFWM XPM-induced NL phase noise XPM FWM NL=nonlinear; SPM=self-phase modulation; MI=modulation instability; XPM=cross-phase modulation; IXPM=intrachannel XPM FWM=four-wave mixing; IFWM=intrachannel FWM; WDM=wavelength division multiplexing 35

47 System Model coherent fiber-optic communication system standard-single-mode fiber ideal distributed Raman amplification But, could also consider systems with inline dispersion-compensating fiber lumped amplification N spans TX A(0, t) A(L, t) AMP SSMF RX 36

48 Generalized Nonlinear Schrödinger Equation A(z, t) is the complex baseband representation of the signal (the full field, representing co-propagating DWDM signals Transmitter sends A(0, t) Receiver gets A(L, t), where L is the total system length Evolution of A(z, t) The generalized non-linear Schrödinger (GNLS) equation expresses the evolution of A(z, t): A z + jβ 2 2 A 2 t 2 jγ A 2 A = n(z, t). No loss term (since ideal distributed Raman amplification is assumed). n(z, t) is a circularly symmetric complex Gaussian noise process with autocorrelation E [ n(z, t)n (z, t ) ] = αhν s K T δ(z z, t t ), 37

49 System Parameters Second-order dispersion β ps 2 /km Loss α m 1 Nonlinear coefficient γ 1.27 W 1 km 1 Center carrier frequency ν s THz Phonon occupancy factor K T

50 Solving the GNLS Equation Throughout propagation over an optical fiber, stochastic effects (noise), linear effects (dispersion) and nonlinear effects (Kerr nonlinearity) interact. Noise Nonlinearity Dispersion Even in the absence of noise, solving the GNLS equation requires numerical techniques. 39

51 Split-Step Fourier Method divide fiber length into short segments consider each segment as the concatenation of (separable) nonlinear and linear transforms for distributed amplification, an additive noise is added after the linear step. A(z 0, t) A(z 0 + h, t) step size h 40

52 Nonlinear Step In the absence of linear effects, the GNLS equation has the form with solution A z = jγ A 2 A, A(z 0 + h, t) = A(z 0, t) exp(jγ A(z 0, t) 2 h). 41

53 Linear Step We now use the previous solution as the input to the the linear step, i.e., let Â(z 0, t) = A(z 0, t) exp(jγ A(z 0, t) 2 h) be the input to the linear step. The linear form of the GNLS equation is A z = α 2 A jβ 2 2 A 2 t 2, which can be efficiently solved in the frequency domain. Defining it can be shown that A(z, t) = 1 2π Ã(z 0 + h, ω) = Ã(z 0, ω) exp Ã(z, ω) exp(jωt)dω, (( j β 2 2 ω2 α ) ) h. 2 42

54 Split-Step Propagator Putting this together, we have } (( A(z 0 + h, t) = F {F 1 {Â(z0, t) exp j β 2 2 ω2 α ) )} h, 2 where F is the Fourier transform operator, and where Â(z 0, t) = A(z 0, t) exp(jγ A(z 0, t) 2 h) 43

55 Digital Backpropagation Digital backpropagation = split-step Fourier method, using a negative step-size h, performed at the receiver Full compensation (involving multiple WDM channels) generally impossible, even in absence of noise (due to wavelength routing) Noise is neglected (cf. zero-forcing equalizer) Single-channel backpropagation typically performed, after extraction of desired channel using a filter 44

56 Capacity Estimation See: R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, B. Goebel, Capacity limits of optical fiber networks, J. Lightw. Technol., vol. 28, pp , Sept./Oct System Model TX A(0, t) Channel A(L, t) 1 2Ts LPF 1 2Ts t = kt s DSP ˆφk,0 45

57 Transmitted Signal Channel l signal: where sinc(θ) = X l (t) = sin πθ πθ. k= φ k,l Ts sinc ( t kts T s ), φ k,l are elements of a discrete-amplitude continuous-phase input constellation M, i.e, for N rings, θ [0, 2π), and r 0, M = {m r exp (jθ) m {1, 2,..., N}}. Each ring is assumed equiprobable, and for a given ring, the phase distribution is uniform. 46

58 Multi-channel systems In the general case of a multi-channel system having 2B + 1 channels with a channel spacing 1/T s Hz, the input to the fiber has the form A(z = 0, t) = B k= l= B φ k,l Ts sinc ( t kts T s ) e j2πlt/ts. 47

59 Towards a Probability Model Back-rotation: φ k,l = ˆφ k,l exp ( j(φ XPM + φ k,l )), where Φ XPM is a constant (input-independent) phase rotation contributed by cross-phase modulation (XPM). 8 x x Imag (mw 1/2 ) Imag (mw 1/2 ) Real (mw 1/2 ) x Real (mw 1/2 ) x

60 Gaussian fitting For each i and a fixed l (the channel of interest), we calculate the mean µ i and covariance matrix Ω i (of the real and imaginary components) of those φ k,l corresponding to the i-th ring, and model the distribution of those φ k,l by N (µ i, Ω i ). From the rotational invariance of the channel, the channel is modeled as f (y x = r i exp (jφ)) N (µ i exp (jφ), Ω i ), where the (constant) phase rotation due to Φ XPM is ignored, since it can be canceled in the receiver. 49

61 Capacity Estimation The mutual information of the (assumed) memoryless channel is f (y x) I (X ; Y ) = f (x, y) log 2 dx dy, f (y) where f (x) represents the input distribution on M with equiprobable rings and a uniform phase distribution, which provides an estimate of the capacity of an optically-routed fiber-optic communication system. 50

62 Signaling Parameters Baud rate 1/T s 100 GHz Channel bandwidth W 101 GHz Number of rings N 64 Number of channels 2B + 1 = 5 51

63 (BP adds 0.55 to 0.75 bits/s/hz relative to EQ) 52 Achievable Rates from Memoryless Capacity Estimate 10 9 Spectral Efficiency (bits/s/hz) L=2000 km, Eq. only L=2000 km, BP L=1000 km, Eq. only L=1000 km, BP L=500 km, Eq. only L=500 km, BP Shannon Limit (AWGN) SNR (db)

64 Pragmatic Coded Modulation via Staircase Codes Approach: BICM + shaping, modulation 2 K+2 -QAM, hard-decisions K R bits/sym Binary FEC Encoder K bits/sym Binary FEC Decoder K R bits/sym 1 bit/sym [HU 1 ] T Viterbi Decoder 2 bits/sym Modulator X Channel Y Demodulator H T U 1 bit/sym Syndrome-former matrix H T U = [1 + D + D2, 1 + D 2 ] T. 53

65 Achievable Rates P in Fiber System K p avg (dbm) (bits/s/hz) L = 500 km, EQ L = 500 km, BP L = 1000 km, EQ L = 1000 km, BP L = 2000 km, EQ L = 2000 km, BP (These achieve within 0.4 to 0.6 bits/s/hz of estimated channel capacity.) I P 54

66 Staircase code design First, design a collection of staircase codes of various (appropriate) rates: R=77/95 R=239/255 R=3/4 R=11/15 R=146/ BER out BER in 55

67 Coded Modulation Performance Then, simulate their performance on the actual channel: 10 9 Spectral Efficiency (bits/s/hz) L=2000 km, Eq. only 3 L=2000 km, BP L=1000 km, Eq. only 2 L=1000 km, BP L=500 km, Eq. only 1 L=500 km, BP Shannon Limit (AWGN) SNR (db)

68 Code Parameters Spec. Eff. Fiber System m t R (bits/s/hz) L = 500 km, EQ / L = 500 km, BP / L = 1000 km, EQ / L = 1000 km, BP / L = 2000 km, EQ / L = 2000 km, BP /

69 Conclusions At rate 239/255, staircase codes provide best-in-class 9.41 db NCG For high-spectral efficiency modulation, a pragmatic coding approach using staircase codes yields performance with 0.6 bits/s/hz (at L = 2000 km) with practical decoding complexity 58

70 Future Directions Can these capacity estimates be improved? Can soft-decisions be incorporated while retaining low complexity? Are there (well-principled) alternatives to digital backpropagation? 59

71 Future Directions Can these capacity estimates be improved? Can soft-decisions be incorporated while retaining low complexity? Are there (well-principled) alternatives to digital backpropagation? Thank you for your attention! 59

Staircase Codes. for High-Speed Optical Communications

Staircase Codes. for High-Speed Optical Communications Staircase Codes for High-Speed Optical Communications Frank R. Kschischang Dept. of Electrical & Computer Engineering University of Toronto (Joint work with Lei Zhang, Benjamin Smith, Arash Farhood, Andrew