Channel Matching: An Adaptive Technique to Increase the Accuracy of Soft Decisions
CChannel Matching: An Adaptive Technique toIncrease the Accuracy of Soft Decisions
Reza Rafie Borujeny and Frank R. Kschischang
Department of Electrical and Computer EngineeringUniversity of Toronto, Toronto, Ontario, M5S 3G4, Canada.rrafi[email protected], [email protected]
Abstract:
Nonlinear interference is modeled by a time-varying conditionally Gaussianchannel. It is shown that approximating this channel with a time-invariant channel imposesconsiderable loss in the performance of channel decoding. An adaptive method to maintaindecoding performance is described. © 2021 The Author(s)
1. Introduction
To meet the stringent reliability requirements of optical networks, some form of forward-error-correction (FEC)is usually employed. One important class of FEC schemes uses an inner low-density parity-check (LDPC) codedmodulation scheme—which may suffer from an error floor—concatenated with an algebraic outer code. Twopopular approaches used to achieve the required bit error rate (BER) seen by the outer code are bit-interleavedcoded modulation (BICM) [1] and multi-level coding (MLC) [2].It is well-known that the fiber nonlinearity causes inter-channel interactions that, eventually, cap the spectralefficiency of a wavelength-division multiplexed (WDM) system [3]. Each WDM channel leaks energy into neigh-boring channels which is impossible to perfectly mitigate due to the absence of the neighboring WDM channels ata given receiver. The resulting inter-channel interference is usually considered as a nuisance and is referred to asnonlinear interference noise (NLIN) [4]. Usually, NLIN is assumed to be additive and is modeled by a Gaussianrandom vector whose covariance is a function of the average power of the neighboring channels, the instantaneouspower of the channel of interest, the modulation and demodulation format and the fiber parameters. Amongstother factors, the average power of the neighboring channels has the most significant contribution in the NLINcovariance [5]. At the same time, it is known that the average power of WDM channels can fluctuate [6, 7] which,in turn, causes fluctuations in the covariance of NLIN. If the fluctuations of the noise parameters are not properlytaken into account, the soft information fed into the FEC decoder may be inaccurate. In this work, we study theimportance of having accurate soft information on the performance of the decoder.We take the inner MLC-based coded modulation scheme of [8] for our baseline system, although similar resultsare expected with BICM. We assume a time-varying conditionally Gaussian channel in which, conditional on theinput, the noise covariance is a function of the average power of the WDM channels. We model the fluctuationsof the average power through the second order statistics of the noise and consider the performance of the innerMLC scheme with two main strategies: with a fixed estimate of the NLIN power and with an adaptive estimate ofthe NLIN power. Our main finding is that if one does not adaptively match the soft information to the conditionsof the channel, considerable degradations in decoding performance are caused. These results are also comparedagainst a hypothetical genie-aided decoder having access to perfect channel state information.It is shown that the simple strategy of re-estimating the noise covariance based on the currently decoded code-word can significantly improve the performance of the inner code. This simple idea is the main ingredient ofturbo equalization [9] and has been previously used to compensate for channels with inter-symbol interference.This includes the compensation of intra-channel effects in optical fiber (see [10] and references therein) as wellas inter-channel nonlinear equalization [11]. The application of turbo equalization for channel estimation has alsobeen considered for some linear channels [12].
2. Simulation Setup and Channel Model
We simulate a single-polarization WDM system with five channels over a fiber of length 4500 km with the adaptivesplit-step Fourier method of [13]. In each run, the split-step model with periodic boundary conditions is solvedfor trains of 3600 symbols. It is assumed that each span of length L A =
50 km is followed by an erbium-dopedfiber amplifier (EDFA). The fiber loss is set to α = . · km − . The nonlinearity coefficient is set to γ = .
27 W − · km − and the chromatic dispersion coefficient is set to β = − . × − s · km − . The amplifiedspontaneous emission (ASE) noise is assumed to be a circularly symmetric white Gaussian process with powerspectral density N EDFAASE = (cid:0) e α L A − (cid:1) h ν n sp where h = . × − J · s is Planck’s constant, ν = .
41 THzis the center frequency, and n sp = a r X i v : . [ c s . I T ] J a n nput IQ Pulse Shaper tp ( t ) WDM f | X ( f ) | Low-Pass FilterBack-propagation f | X ( f ) | Matched Fitler tp ( t ) Output IQ (a) The channel model − − − − − − − − . . . . . . P in (dBm) M u t u a li n f o r m a t i o n ( b i t / s y m b o l ) (b) Mutual information Fig. 1: The blocks in the considered channel model are shown in (a). The mutual information achieved withuniform input distribution for the channel of interest in the simulated WDM system is illustrated as a function ofthe average input power (b).Root-raised-cosine pulses with 6.67% excess bandwidth are used for pulse shaping. As a result, the symbol rate perWDM channel is 43 .
95 Gsymbol · s − . At the receiver, the channel of interest—the center channel—is filtered anddigitally back-propagated, and the result is passed through a matched filter. The obtained points are commonlyback rotated to undo the effect of cross-phase modulation. All WDM channels are assumed to have the sameaverage input power P in . The achieved mutual information under uniform input distribution is shown in Fig. 1b.With a maximum at − . P in must be set to the optimal value of − . X and output random variable Y . The input alphabet X is the 16-QAM constellation in use,while the output alphabet Y is the whole complex plane (see Fig. 1a).Ideally, one wishes to operate on the optimal input power, or very close to it, so that the data rate can bemaximized with aid of an appropriate choice of FEC. Because of the dynamics of the network, some powerfluctuations may happen which in turn result in reductions in the achievable information rates. Our goal here is tocapture such fluctuations and make sure that the FEC block is aware of them. This is achieved by re-estimatingthe noise parameters and making sure that the soft information used by the FEC decoder is accurate.
3. Multi-level Coding
We consider a similar MLC transmission scheme as in [8] with a 16-QAM constellation and an inner LDPCcode of rate 0 .
63 which gives an overall inner coded modulation information rate of 3.63 bits per symbol. Westudy the performance of the inner coded modulation scheme under three scenarios, namely, obtaining noisecovariance based on the optimal input power, a genie-aided decoder and an iterative channel estimation. We definethe survivability of the inner coded modulation as the range of power fluctuations that it can tolerate while stillmaintaining the BER below a prescribed target BER. In our running example, the target BER is set to 10 − . Wecompare these three decoding methods in terms of their survivability. Fixed Noise Covariance:
In this scenario, the inner decoder calculates the soft information based on the optimalinput power, i.e., the noise parameters are those induced by the optimal input power. This means that the noisecovariance is assumed to be fixed, despite the fact that the actual average power experiences fluctuations. Theresults are shown in Fig. 2b. When the receiver uses a fixed set of noise parameters based on the optimal powerwith a target BER of 10 − , the MLC is successful as long as the average input power (in dBm) lies in [ − . , − . ] .That is, the survivability of this method is 3 dB. Perfect Channel State Information:
In this scenario, it is assumed that a genie provides the inner decoder withthe true noise statistics. This corresponds to the availability of perfect channel state information at the receiver.The corresponding power interval that the MLC performs below the target BER of 10 − is [ − . , − . ] dBmwith a survivability of 6.3 dB, i.e., about 3.3 dB more survivability in comparison with having fixed estimates forthe noise statistics. The results are shown in Fig. 2b. Channel Matching:
In this scenario, the inner code initially calculates soft information based on the optimalpower. The decoded codeword is then remapped to the corresponding constellation points similarly to the way thatthe transmitter performs the mapping of the codeword to the constellation points. By considering the differencebetween these symbols and the actually received symbols, maximum likelihood estimates of the noise statisticsare obtained. The updated noise statistics are then used to update the soft information fed to the decoder in a emapper LDPC DecoderLDPC EncoderChannelEstimatorMapper
Channel Output r r (a) The decoder − − − − − − − − − P in (dBm) B E R Genie aided r = 4 ,r = 0 r = 3 ,r = 3 (b) BER vs. average input power Fig. 2: Structure of the adaptive decoder of this work is shown in (a). Performance of the inner coded modulationscheme, in terms of BER, is shown in (b).turbo-equalizer fashion. Fig. 2a shows the structure of the turbo LDPC decoder used in this work. There are twoparameters, namely r and r , that determine the number of iterations performed: r counts the number of iterationsperformed by the LDPC decoder before re-estimating the noise parameters while r counts the number of “turbo”iterations performed to re-estimate the noise parameters. The performance of this method is illustrated in Fig. 2b.Depending on the choice of r and r , different survivabilities can be obtained. This provides an interesting trade-off between decoding complexity and reliability. One can see that with 3 turbo iterations and 3 LDPC decodingiterations, the survivability is virtually identical to that of the genie-aided decoder. References
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