On Probabilistic Shaping and Learned Decoders with Application to Fiber-Optic Communications
Doctoral thesis, 2020

We live in an ubiquitously connected world whose backbone are optical fibers. Achieving high spectral efficiencies and hence high transmission rates requires a careful combination of forward error correction (FEC) and higher-order modulation. This is referred to as coded modulation. In this thesis, we focus on probabilistic shaping which aims to shape the distribution of the transmitted symbols to the capacity-achieving distribution. For symmetric distributions, probabilistic amplitude shaping (PAS) has been proposed by Böcherer et al.. Certain fiber-optic systems, however, have non-symmetric capacity-achieving distributions and hence, PAS can not be applied.
In the first part of this thesis, we focus on probabilistic shaping for asymmetric distributions. For a nonlinear Fourier transform-based transmission scheme, we introduce probabilistic eigenvalue shaping, where the coded symbols are partially distributed according to the capacity-achieving distribution and partially uniformly distributed. Further, for an intensity modulation with direct-detection (IM/DD) system, we uncover a hidden symmetry of the capacity-achieving distribution. We propose to extend the PAS scheme with a compound construction of a low-density generator matrix code and a low-density parity-check code (LDGM/LDPC) to incorporate this hidden symmetry. For both shaping schemes, we demonstrate significant improvements over the state of the art.
In the second part of this thesis, we address low-complexity, near-maximum-likelihood (ML) decoding of short linear block codes, which play an important role in FEC in fiber-optic communication systems as component codes in staircase codes or generalized LDPC codes. Our work extends neural belief propagation (NBP) introduced by Nachmani et al. where belief propagation decoding is unrolled and weights are placed on the edges. In particular, we consider NBP decoding over an overcomplete parity-check matrix and use the weights of NBP as a measure of the importance of the check nodes (CNs) to decoding. Unimportant CNs are successively pruned. This typically results in a different parity-check matrix in each iteration. We demonstrate that for codes with a dense parity-check matrix such as algebraic codes, our proposed decoder performs close to ML decoding. To further improve the decoding of short LDPC codes, we introduce a two-stage decimation process to NBP decoding. First, we create a list by iterating between a conventional NBP decoder and guessing the least reliable bit. This is followed by iterating between a conventional NBP decoder and learned decimation, where we use a neural network to decide the decimation value for each bit. For short LDPC codes, this results in a significant performance gain.

Error correcting codes

learned decoders

machine learning

iterative decoding

probabilistic shaping

fiber-optic communications


Opponent: Prof. Guido Montorsi, Department of Electronics and Telecommunications, Politecnico di Torino, Italy


Andreas Buchberger

Chalmers, Electrical Engineering, Communication, Antennas and Optical Networks

Probabilistic Eigenvalue Shaping for Nonlinear Fourier Transform Transmission

Journal of Lightwave Technology,; Vol. 36(2018)p. 4799-4807

Journal article

Buchberger, A., Graell i Amat, A., Schmalen, L., Probabilistic Shaping for IM/DD Systems Based on LDGM/LDPC Codes

Pruning and Quantizing Neural Belief Propagation Decoders

IEEE Journal on Selected Areas in Communications,; Vol. 39(2021)p. 1957-1966

Journal article

Learned Decimation for Neural Belief Propagation Decoders

ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings,; Vol. 2021-June(2021)p. 8273-8277

Paper in proceeding

Optical fibers form the backbone of today’s communication networks. Every phone call, text message, email, or website will, at some point, be transmitted through an optical fiber where light carries data from the transmitter to the receiver. Data is usually represented in binary form as a sequence of zeros and ones, both being equally likely (i.e., if we have a long sequence of bits, we know that about half of them are zero and half of them are one). However, it is known that transmitting such a stream of equiprobable bits is not optimal and does not result in the highest possible transmission rate. In the first part of this thesis, we design a system that transforms the sequence of equiprobable bits into a sequence where zeros are more likely than ones. This way, we can increase the transmission rate.
Every communication system is subject to noise caused by different physical phenomena such as random thermal movements of electrons. When transmitting data, it hence may happen that we receive a zero when a one was sent and vice versa. By using error correcting codes, some of these transmission errors can be corrected at the receiver side. In the second part of this thesis, we use machine learning to implement these error correcting codes at the receiver and improve the reliability of the transmission.

Coding for Optical communications In the Nonlinear regime (COIN)

European Commission (EC) (EC/H2020/676448), 2016-03-01 -- 2020-02-28.

Subject Categories


Communication Systems


C3SE (Chalmers Centre for Computational Science and Engineering)



Doktorsavhandlingar vid Chalmers tekniska högskola. Ny serie: 4895




Opponent: Prof. Guido Montorsi, Department of Electronics and Telecommunications, Politecnico di Torino, Italy

More information

Latest update