Information Theory for Dispersion-Free Fiber Channels with Distributed Amplification

Transcription

1 Information Theory for Dispersion-Free Fiber Channels with Distributed Amplification Gerhard Kramer Department of Electrical and Computer Engineering Technical University of Munich CLEO-PR, OECC & PGC Singapore August 3, 2017

2 1) Fiber Models 2) Per-Sample Model 3) Filter-and-Sample Model 4) Capacity Bounds 5) Conclusions Outline Detailed derivations: arxiv: , May

3 Generalized nonlinear Schrödinger equation (NLSE)* with distributed and band-limited optical amplifier (OA) noise (see next slide): Dispersion-free model: 1) Fiber Models u z = α 2 u i 2 β 2 2 u T 2 + iγ u 2 u + n u z = α 2 u + iγ u 2 u + n Advantages: closed-form solution for the input-output relations; get insight into non-linearities that seem difficult to obtain otherwise *Ch. 2 in G.P. Agrawal, Nonlinear Fiber Optics, 3rd ed., 2001

4 Details on OA Noise Model Consider Raman amplification with power spectral-distance density (PSDD) of N A [W/Hz/m] and bandwidth B [Hz] Noise power distance density (PDD) K=N A B [W/m] is proportional to B The spatiotemporal autocorrelation function: B E N z,t ( ) N ( z ʹ, tʹ) * ( ( )) = K δ ( z zʹ )sinc B t tʹ f Accumulated noise at time t is K ½ times a spatial Wiener process W l ( z,t) = lim l z 1 l i=1 l N i ( t) where the N i (.), i=1,2,..., are i.i.d., complex, circularly-symmetric, bandlimited Gaussian noise processes with mean 0 and autocorrelation function ρ(t-t ) = E[N i (t)n i (t ) * ] = sinc(b(t-t )) 4

5 Ideal Distributed Amplification Suppose distributed optical amplification (OA) eliminates loss: Solution (that is statistically-equivalent* for one fixed time t): u( z,t) = u(0,t) + K w( z,t) exp jγ z u(0,t) + K w( z ʹ,t) 2 d z ʹ Consider the quadratic term in the exponential that has three terms: - a self-phase modulation (SPM) term: u(0,t) 2 - a noise term: K w(z,t) 2 u z = iγ u 2 u + n - a signal-noise mixed term: 2 K ½ Re{u 0 (t) w(z,t) * } ( ) 0 Large u 0 (t) thus causes large and random phase variations, which causes uncontrolled spectral broadening * This is subtle; for more detail, see Mecozzi,

6 2) Per-Sample Model n Previous work has focused on a per-sample model*: the receiver is assumed to be able to sample the field at any time t: ( ) 0 u( z,t) = u(0,t) + K w( z,t) exp jγ z u(0,t) + K w( z ʹ,t) 2 d z ʹ n Advantage: immediate capacity lower bound of C 1 2 log( SNR) 1 log(2) bits/symbol 2 by using amplitude modulation and instantaneous direct detection n Criticisms: n n n Spectral broadening: C is too optimistic Receiver has bandwidth: C is too optimistic Receiver ignores noise correlations: C is pessimistic! * Mecozzi, 1994; Turitsyn et al., 2001; Yousefi-Kschischang,

7 A Modeling Problem: Large Capacity Consider a rectangular pulse g(t) of length T s and a small time-offset T. Transmit the information symbol x via u(0,t) = x g( t T 2) 2x g( t 3T 2) 0 x T 2T -x T s +T/2 t Notes: - Launch energy T s x 2 - Bandwidth ~ 1/T s (and not 1/T) For small T, we have W(z,0) W(z,T) W(z,2T) and can compute x as precisely as desired using y k = u(z,kt) 2 for k=0,1,2 7

8 The samples are (with as much precision as required with small T) y 0 = K ( w 2 2 R + w ) I y 1 = ( x + Kw ) 2 2 R + Kw I y 2 = ( x + Kw ) 2 2 R + Kw I We can form: x 2 = y + y y 0 The capacity is thus unbounded for any launch power. Explanation: the noise is bandlimited.* And we will have the same issue with dispersion. Models thus need, e.g., thermal noise at the receiver or in the channel; they also need projection receivers, e.g., bandlimited receivers * A. D. Wyner, BSTJ, Mar

9 3) Filter-and-Sample Model n n n Starting point: the receiver filters u(z,t) and adds receiver noise modeled as filtered white noise Study (1) band-limited and (2) time-resolution limited receivers Analytics seem difficult. Approach: - Compute autocorrelation function for jointly Wiener noise statistics (main analytic result; but noise statistics need more research); - Use this function to bound the power spectral density (PSD) - and the power inside a fixed frequency band or time interval; - use these power bounds to lower bound spectral broadening - and the capacity and spectral efficiency. Warning: upper bounds are loose, but valid for any launch signal Further insights: (1) model is unrealistic* beyond some launch power; (2) making model realistic leads to a nonlinear Shannon limit due to signal-noise mixing, as predicted in Essiambre et al. (2010) * NLSE with dispersion also has this problem 9

10 Conditional Autocorrelation Function Let A(t,t )=E[U(z,t)U(z,t) * U 0 (.)=u 0 (.)] where u 0 (t)=u(0,t) is the launch signal Result* assuming jointly Wiener noise statistics; u 0 = u 0 (t), u 0 =u 0 (t ) A(t,t') = T R Kρ + S R u 0 + j where S R =Re(S), S I =Im(S), T R =Re(T), T I =Im(T) and S I 1 ρ 2 S 2 exp jγt R u 2 0 u 0ʹ 2 exp γ ( u 0 ρu 0ʹ ) S u R 0ʹ + j S I 1 ρ 2 u 0ʹ ρu 0 ( ) T I u 2 1 ρ u 0ʹ 2 2ρR u 0 u 0ʹ ( ) * S = sech( 2cz), T = tanh( 2cz), c = jγ(k / 2) 1 ρ 2 Note: 1 st exp. describes SPM; 2 nd exp. describes signal-noise mixing; large u 0 (.) makes 2 nd term small, i.e., there is spectral broadening * * Generalizes the approach and results of Mecozzi, JLT

11 Example 1: Isolated Rectangular Pulse Consider an isolated rectangular pulse of height P ½ and width T s =10 ps Plots: 10 log 10 A(t,0) and 10 log 10 A(t, 5.1 ps)... note log 10 scaling 10 log 10 A(t,t') P=10 mw, t'=0 P=10 mw, t'=5.1 ps P=100 mw, t'=0 P=100 mw, t'=5.1 ps P=200 mw, t'=0 P=200 mw, t'=5.1 ps P=400 mw, t'=0 P=400 mw, t'=5.1 ps t [ps] Fiber Parameters - Kerr coefficient: 1.27 (W-km) -1 - Fiber length: 2000 km - Temperature: 300 K - PSDD N A : x W/Hz - Bandwidth B: 500 GHz Notes - At low power A(t,0) is the pulse and A(t,5.1 ps) is Kρz - For P = 10 mw (or 10 dbm), there is already a bulge at t=0: spectral broadening starts - Large power: A(t,0) develops a sinc pulse shape in log scale: strong spectral broadening 11

12 Example 2: PAM with Ring Modulation Consider PAM with rectangular pulses (T s =10 ps) and ring modulation Plots: PSDs for large launch power; observe 100 GHz rect. PSD for ɣ= P=10 mw, =0 P=10 mw P=50 mw P=100 mw P=500 mw P=1 W f [GHz] 12

13 Power Bound Using A(t,t ), and several bounding steps, the average received power of a bandlimited receiver (bandwidth W) satisfies P r (W ) c 1 + 8W B f ( P + P o) where c 1 and P o are constants and P is the average launch power. For constants δ and κ, the function f(.) is non-decreasing and concave for P P o : f P ( ) = Kz + P +δ Note: erf(y) 1 for large y ( ) 2 For large P, the avg. receive power thus scales at most as P ½ ; the power loss is due to spectral broadening π 2 erf κp 8 ( ) κp 8 13

14 A Modeling Problem: Spectral Broadening Using the above power bound: the (99% in-band power) propagating signal bandwidth W scales at least as P ½ But the model is accurate only if B W Conclusion: model loses practical relevance beyond some P Same conclusion will be true for NLSE Nevertheless: we study capacity bounds for large P by scaling B as P ½ 14

15 4) Capacity Bounds 15

16 Capacity Bounds with Fixed B Same fiber parameters as above, Tx & Rx bandwidth is W=B=500 GHz Rate [bits/s/hz] Upper bound Upper bound 2 Lower bound bound P [dbm] Notes - Threshold for B W is 20 Watts... loose! - Upper bound normal. by W and is loose at small P due to band-limited noise and lack of tools - Upper bound 2 assumes simulation accurate inside band W - Spectral efficiency η normal. by propagating signal bandwidth and is small beyond threshold: nonlinear Shannon limit - Lower bound from Essiambre et al.,

17 Capacity Bounds with B ~ P ½ Same fiber parameters as above, Tx & Rx bandwidth is W=500 GHz Rate [bits/s/hz] Upper bound Upper bound 2 Lower bound bound Notes - Capacity maximum at a finite P: a non-linear Shannon limit for both capacity and spectral efficiency - Capacity scaling is at best C P -¼ for large P: Explanation: noise has power proportional to B - Spectral efficiency scaling is also worse than before P [dbm] 17

18 5) Conclusions 1) Per-sample model is unrealistic at any launch power 2) Autocorrelation function derived for jointly Wiener noise statistics 3) Propagating signal bandwidth grows at least* as P ½ with launch power P; filter-and-sample models are thus unrealistic* at large P 4) Non-linear Shannon limit established* for both capacity and spectral efficiency of models where B W is required 5) We still need: better understanding of noise statistics due to amplification better capacity bounds and understanding at high SNR analysis for frequency-dependent noise, loss, dispersion, and non-linearity * This insight should carry over to fiber models with dispersion 18

19 Extra Slides 19

20 Discussion Definition 1: A channel* is said to have a non-linear Shannon limit if its capacity (or spectral efficiency) is bounded as a function of P Criticism: definition is interesting for abstract models only, since every practical channel* has a non-linear Shannon limit. Definition 2: maximal capacity (or spectral efficiency) is at finite P (for abstract models) and below a maximum P (for practical models) The latter type of limit (Definition 2) was established if: (1) Tx & Rx bandwidths are fixed, and (2) B W is required Note: one can redefine W to permit flash power-waste signaling, which can flatten capacity and spectral efficiency curves Criticisms: this may require transmitter bandwidth; returns us to Definition 1; a peak power constraint returns us to Definition 2 * What is a channel? J.L. Kelly: the part of a communication system one is unable or unwilling to change 20