Four-Dimensional Estimates of Mutual Information in CoherentOptical Communication Experiments

Tobias A. Eriksson(1), Tobias Fehenberger(2), Norbert Hanik(2), Peter A. Andrekson(1),Magnus Karlsson(1), and Erik Agrell(3)

(1) Dept. of Microtechnology and Nanoscience, (3) Dept. of Signals and Systems, Chalmers Universityof Technology, SE-41296 Gothenburg, Sweden, tobias.eriksson@chalmers.se(2) Institute for Communications Engineering, Technische Universitat Munchen, 80333 Munich, Germany.

Abstract Mutual information is experimentally investigated for long-haul coherent transmission. Re-ceivers that consider memoryless four-dimensional noise distributions can achieve significantly higherrates than receivers assuming two-dimensional symmetric distributions.

IntroductionForward error correcting (FEC) codes are an es-sential part of today’s coherent fiber-optical com-munication systems. Traditionally, hard-decision(HD) codes such as the Reed-Solomon (RS)code have been the dominating codes in long-haul optical communication systems1. Recently,more advanced coding schemes applying soft-decision (SD) FEC such as low-density par-ity check (LDPC) codes or turbo-product codes(TPC) have effectively extended the achievabletransmission reach of long-haul systems, often incombination with higher-order modulation formatssuch as polarization-multiplexed 16-ary quadra-ture amplitude modulation (PM-16QAM).

In coherent fiber-optical experiments, data isoften captured and processed off-line using digi-tal signal processing (DSP) structures that shouldresemble what could be done practically in real-time using application-specific integrated circuits(ASICs) or field-programmable gate arrays (FP-GAs). Due to the extremely high sampling rates itis both expensive and time consuming to developreal-time circuits. This, together with the fact thatprogramming languages such as Matlab or C++offer simple reconfigurability and debugging, hasmade off-line processing the norm in fiber-opticaltransmission research experiments. The one partthat is most often omitted in the DSP is the FEC,because it is simply not feasible to capture andprocess enough data with off-line processing toreach post-FEC BER levels that are considered“error free” (often < 10−15).

The common practice in the community today isto measure the uncoded BER and depending onwhich FEC code that is assumed, set a pre-FECBER target that is considered ”error free”. ForHD codes, there is fixed relationship between pre-FEC BER and the BER after decoding2, whichmakes this practice valid. However, for SD-FEC

codes, there is no such relationship, and in factthe decoders work on soft information. Hence,for experiments assuming an SD-FEC code, themutual information (MI) is a more logical measureand has been shown to be more reliable than us-ing the pre-FEC BER3,4.

In this paper, we experimentally compare fourdifferent estimates of MI assuming either in-dependent, identically distributed (iid) Gaussiannoise distribution in 2D or 4D with the same vari-ance for all constellation points or assuming 2Dor 4D correlated Gaussian distributions with sep-arate covariance matrices for each constellationpoint.

Estimate of Mutual Information withMismatched DecodingThe MI for a memoryless channel fY |X is given

by I(X;Y ) , EX,Y

[log2

fY |X(Y |X)

fY (Y )

], where fY

is the probability density function at the output.However, for the fiber optical channel, fY |X is notknown, which forces us to some assumptions.Following5, it can be shown that using a mis-matched decoder, substituting fY |X with an aux-iliary function qY |X gives a lower bound on MI2.For a system transmitting uniformly distributedsymbols x from a constellation X with m bits persymbol, an achievable rate is computed as

R ≈ m+1

N

N∑n=1

∑x∈X

qX|Y (x|yn) log2 qX|Y (x|yn),

where qX|Y is obtained using Bayes’ rule (see2,eq. (4)) and Y1, . . . , YN are N observed out-put samples2. Although this is a lower bound onthe MI, it can be argued that this is the practicallyachievable rate since the decoder needs to as-sume a channel. In many cases, this is assumedto be a memoryless channel with Gaussian distri-butions of the noise. The better qY |X resemblesthe true channel fY |X , the tighter the bound on

the true (although memoryless) MI becomes.In this work, we compare four different assump-

tions for qY |X . The distribution parameters wereestimated from experimental data as follows:

2D-iidG: 2D iid Gaussian noise in each polar-ization. The output distribution corresponding toeach input point is circularly Gaussian with thesame variance for each constellation point.

2D-CG: 2D correlated Gaussian (CG) noise, in-dependent in each polarization. A 2D covariancematrix is estimated for each constellation point.

4D-iidG: 4D iid Gaussian noise, with the samevariance for all 4D constellation points.

4D-CG: 4D CG noise. A 4D covariance matrix isestimated per 4D constellation point.

For the 2D estimates, the total MI is the sum ofthe MI in the x- and the y-polarization.

Note that in an experiment, the channel is notonly the optical fiber itself, but the whole com-munication system from the transmitted constel-lation to recovered constellation after the DSP.In the transmitter, components such as digital-to-analogue converters, electrical amplifiers and op-tical modulators can all affect the transmitted con-stellation with effects such as quantization noiseand nonlinear transfer functions. In the receiver,effects such as limited resolution in the analogue-to-digital converters, non-orthogonality of the op-tical hybrids, and non ideal algorithms in the DSPwill influence the constellation.

Experimental Setup and ResultsThe recirculating loop that was used in all ex-periments is shown in Fig. 1. The spans con-sisted of 80 km standard single-mode fiber (SMF)amplified by erbium-doped fiber amplifiers (ED-FAs). Inline dispersion compensation could beenabled by the use of dispersion-compensatingmodules (DCMs) based on fiber Bragg-gratingswhich compensated the dispersion in each span.For transmission without dispersion compensa-tion, these modules were bypassed. A polariza-tion scrambler that was synchronized to the loop-roundtrip time was used to avoid any nonrealisticaccumulation of polarization effects, and a thirdEDFA was used to compensate for the loss of thescrambler and the loop-switching components.

The first experiment transmitted 7 channelswith a spacing of 50 GHz and a symbol rateof 28 Gbaud to emulate wavelength-divisionmultiplexed (WDM) transmission of either PM

Lo

op

Sw

itch

EDFA

�2DCM

In

Out

VOA80 km SMF

EDFA

Pol.-

Scrambler

Fig. 1: The recirculating loop used in all experiments. Toenable inline dispersion, compensating modules (DCM)

based on fiber Bragg-gratings were used. These could bebypassed for uncompensated transmission.

quadrature phase shift keying (PM-QPSK) orpolarization-switched QPSK (PS-QPSK). For thisexperiment, no inline dispersion compensationwas used. The DSP is based on the constantmodulus algorithm for equalization and Viterbi-Viterbi phase estimation. The achievable rateas a function of transmission distance for thecenter channel using all four estimates for PM-QPSK and the two 4D estimates for PS-QPSK areshown for different launch powers in Fig. 2. Thereis no distinctive difference between the differentestimates, even at the highest launch power. Thisshows that for a dispersion unmanaged link with28 Gbaud WDM channels, assuming an iid Gaus-sian distribution of the noise with the same vari-ance for all constellation points yields as highrates as with a much more complex 4D receiver.For the optimal launch power, we expect this to betrue for higher-order modulation formats and wenote that this model has been used to estimate MIfor QAM formats up to 256-QAM6. However, forhigh launch powers, this needs verification whenthe modulation order is increased.

In the second experiment, we transmitted a sin-gle channel of 14 Gbaud PM-16QAM, this timeusing the inline DCMs. The DSP is based ondecision-directed least mean square for equaliza-tion and blind phase estimation based on 64 test-phases. The launch power was either 0 dBm or−7 dBm. The achievable rate using the four es-timates is shown in Fig. 3. For −7 dBm, thereis no significant difference in the achievable ratebetween the four methods. At 1920 km trans-mission, the difference between the highest es-timate, 4D-CG, and the lowest, 2D-iidG, is ap-proximately 0.01 bit/4D-symbol. However, whenwe increase the launch power to 0 dBm, a cleardifference can be noticed between the four esti-mates. The 4D-CG estimate achieves the high-est rate at all transmission distances. Assum-ing iid Gaussian noise in 2D with the same vari-ance for all constellation points achieves the low-est rate. This is easily understood by the con-

−0dBm −1dBm −2dBm −3dBm −4dBm −5dBm 4D−iidG 4D−CG 2D−iidG 2D−CG

0 2000 4000 6000 8000 10000 120003.7

3.75

3.8

3.85

3.9

3.95

4

Ach

ieva

ble

Ra

te [

bit/4

D−

sym

bo

l]

Transmission Distance [km]

Center Channel 7×28 Gbaud PM−QPSK

(a) PM−QPSK

2000 4000 6000 8000 10000 12000 14000 160002.75

2.8

2.85

2.9

2.95

3

Ach

ieva

ble

Ra

te [

bit/4

D−

sym

bo

l]

Transmission Distance [km]

Center Channel 7×28 Gbaud PS−QPSK

(b) PS−QPSK

Fig. 2: Achievable rate in dispersion unmanaged WDM transmission using the four different estimates for (a) PM-QPSK and thetwo 4D estimates for (b) PS-QPSK. Note that the lines for the different estimates are overlapping at the same launch-power.

400 600 800 1000 1200 1400 1600 1800 2000 22006.8

7

7.2

7.4

7.6

7.8

8

Transmission Distance [km]

Achie

vable

Rate

[bits/4

D−

sym

bol]

0dBm − 4D−iidG

0dBm − 4D−CG

0dBm − 2x2D−CG

0dBm − 2x2D−iidG

−7dBm − 4D−iidG

−7dBm − 4D−CG

−7dBm − 2x2D−iidG

−7dBm − 2x2D−iidG

Fig. 3: Achievable rate for single-channel 14 GbaudPM-16QAM with inline compensation and −7 dBm launch

power (open markers, dashed lines) and 0 dBm (solidmarkers, solid lines).

stellations plotted in Fig. 4, showing the signal ineach polarization after 1920 km transmission with−7 dBm or 0 dBm launch power. It is commonfor many FEC decoders to assume 2D iid Gaus-sian statistics of the noise. However, our resultshere show that for specific channels such as theone we consider, significant gains can be seenby considering 4D-CG distributions, showing 0.2bit/4D-symbol higher achievable rate at a trans-mission distance of 1600 km.

Surprisingly, the 4D-iidG estimate with thesame covariance for all constellation pointsachieves a higher rate than the sum of the 2D esti-mates with correlated Gaussian distributions withindividual estimates of the covariance for eachconstellation point. This shows that for a highlynonlinear channel, the polarizations are no longerindependent and there is loss in information byseparating them.

ConclusionsWe have experimentally compared four differentestimates of MI with mismatched decoders. Inlong-haul WDM transmission without inline dis-persion compensation, assuming iid Gaussiannoise in 2D gives a good estimate for PM-QPSKand iid Gaussian noise in 4D for PS-QPSK. For a

Fig. 4: PM-16QAM constellations after 1920 km for the inlinecompensated link with −7 or 0 dBm launch-power.

highly nonlinear channel, such as single-channelPM-16QAM in an inline dispersion-compensatedlink with high launch power, there is significantgains in the achievable rate by considering noisedistributions over the full 4D optical field. For thischannel, a practical conclusion is that a receiverthat assumes the noise to be correlated 4D Gaus-sian distributed with individual covariance matri-ces per 4D constellation point achieves a higherrate than other memoryless receivers.Acknowledgements: This work was partiallyfunded by the Swedish Research Council.References[1] ITU, Recommendation G.975: Forward error correction for subma-

rine systems, Oct. 2000.[2] T. Fehenberger et al., “On Achievable Rates for Long-Haul Fiber-

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transmission experiments,” Photon. Technol. Lett., vol. 23, p. 1547(2011).

[4] A. Alvarado et al., “LDPC Codes for Optical Channels: Is the “FECLimit” a Good Predictor of Post-FEC BER?,” OFC’15, p. Th3E.5(2015).

[5] D. Arnold et al., “Simulation-based computation of informationrates for channels with memory,” Trans. Inf. Theory, vol. 52, p. 3498(2006).

[6] D. S. Millar et al., “Transceiver-Limited High Spectral EfficiencyNyquist-WDM Systems,” OFC’15, p. Th2A.13 (2015).