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Neural networks (NNs) inspired by the forward-backward algorithm (FBA) are used as equalizers for bandlimited channels with a memoryless nonlinearity. The NN-equalizers are combined with successive interference cancellation (SIC) to approach the information rates of joint detection and decoding (JDD) with considerably less complexity than JDD and other existing equalizers. Simulations for short-haul optical fiber links with square-law detection illustrate the gains., Comment: Accepted at IEEE Intern. Symp. on Inf. Theory 2024 in Athens. arXiv admin note: substantial text overlap with arXiv:2401.09217

Reliable communication over bandlimited and non-linear channels usually requires equalization to simplify receiver processing. Equalizers that perform joint detection and decoding (JDD) achieve the highest information rates but are often too complex to implement. To address this challenge, model-based neural network (NN) equalizers that perform successive interference cancellation (SIC) are shown to approach JDD information rates for bandlimited channels with a memoryless nonlinearity and additive white Gaussian noise. The NNs are chosen to have a periodically time-varying and recurrent structure that imitates the forward-backward algorithm (FBA) in every SIC stage. Simulations for short-haul fiber-optic links with square-law detection show that NN-SIC nearly doubles current spectral efficiencies, and bipolar or complex-valued modulations achieve energy gains of up to 3dB compared to state-of-the-art intensity modulation. Moreover, NN-SIC is considerably less complex than equalizers that perform JDD, mismatched FBA processing, and Gibbs sampling., Comment: Submitted to IEEE Trans. Commun. on January 11, 2024; revised June 9, 2024; revised August 9, 2024; accepted August 11, 2024

Neuromorphic computing implementing spiking neural networks (SNN) is a promising technology for reducing the footprint of optical transceivers, as required by the fast-paced growth of data center traffic. In this work, an SNN nonlinear demapper is designed and evaluated on a simulated intensity-modulation direct-detection link with chromatic dispersion. The SNN demapper is implemented in software and on the analog neuromorphic hardware system BrainScaleS-2 (BSS-2). For comparison, linear equalization (LE), Volterra nonlinear equalization (VNLE), and nonlinear demapping by an artificial neural network (ANN) implemented in software are considered. At a pre-forward error correction bit error rate of 2e-3, the software SNN outperforms LE by 1.5 dB, VNLE by 0.3 dB and the ANN by 0.5 dB. The hardware penalty of the SNN on BSS-2 is only 0.2 dB, i.e., also on hardware, the SNN performs better than all software implementations of the reference approaches. Hence, this work demonstrates that SNN demappers implemented on electrical analog hardware can realize powerful and accurate signal processing fulfilling the strict requirements of optical communications., Comment: 9 pages, 5 figures, accepted for publication by the IEEE/Optica Publishing Group Journal of Lightwave Technology

Concatenated forward error correction is studied using an outer KP4 Reed-Solomon code with hard-decision decoding and inner single parity check (SPC) codes with Chase/Wagner soft-decision decoding. Analytical expressions are derived for the end-to-end frame and bit error rates for transmission over additive white Gaussian noise channels with binary phase-shift keying (BPSK) and quaternary amplitude shift keying (4-ASK), as well as with symbol interleavers and quantized channel outputs. The BPSK error rates are compared to those of two other inner codes: a two-dimensional product code with SPC component codes and an extended Hamming code. Simulation results for unit-memory inter-symbol interference channels and 4-ASK are also presented. The results show that the coding schemes achieve similar error rates, but SPC codes have the lowest complexity and permit flexible rate adaptation., Comment: Submitted to IEEE Journal of Lightwave Technology on December 20, 2022; revised March 29, 2023; accepted April 5, 2023

The last decade has witnessed the breakthrough of deep neural networks (DNNs) in many fields. With the increasing depth of DNNs, hundreds of millions of multiply-and-accumulate (MAC) operations need to be executed. To accelerate such operations efficiently, analog in-memory computing platforms based on emerging devices, e.g., resistive RAM (RRAM), have been introduced. These acceleration platforms rely on analog properties of the devices and thus suffer from process variations and noise. Consequently, weights in neural networks configured into these platforms can deviate from the expected values, which may lead to feature errors and a significant degradation of inference accuracy. To address this issue, in this paper, we propose a framework to enhance the robustness of neural networks under variations and noise. First, a modified Lipschitz constant regularization is proposed during neural network training to suppress the amplification of errors propagated through network layers. Afterwards, error compensation is introduced at necessary locations determined by reinforcement learning to rescue the feature maps with remaining errors. Experimental results demonstrate that inference accuracy of neural networks can be recovered from as low as 1.69% under variations and noise back to more than 95% of their original accuracy, while the training and hardware cost are negligible., Comment: Accepted by DATE 2023 (Design, Automation and Test in Europe)

For 200Gb/s net rates, cap probabilistic shaped PAM-8 with different Gaussian orders are experimentally compared against uniform PAM-8. In back-to-back and 5km measurements, cap-shaped 85-GBd PAM-8 with Gaussian order of 5 outperforms 71-GBd uniform PAM-8 by up to 2.90dB and 3.80dB in receiver sensitivity, respectively., Comment: submitted to 2022 European Conference on Optical Communication (ECOC)

An entropy-regularized mean square error (MSE-X) cost function is proposed for nonlinear equalization of short-reach optical channels. For a coherent optical transmission experiment, MSE-X achieves the same bit error rate as the standard MSE cost function and a significantly higher achievable information rate.

A spiking neural network (SNN) non-linear equalizer model is implemented on the mixed-signal neuromorphic hardware system BrainScaleS-2 and evaluated for an IM/DD link. The BER 2e-3 is achieved with a hardware penalty less than 1 dB, outperforming numeric linear equalization.

For 200Gbit/s net rates, uniform PAM-4, 6 and 8 are experimentally compared against probabilistic shaped PAM-8 cap and cup variants. In back-to-back and 20km measurements, cap shaped 80GBd PAM-8 outperforms 72GBd PAM-8 and 83GBd PAM-6 by up to 3.50dB and 0.8dB in receiver sensitivity, respectively, Comment: Originally published in ECOC-2021. We have updated Figure 3. The change also affects the overall outcome. In contrast to the published version, compared to uniform PAM-8 72 GBd, PS-PAM-8 80 GBd performance is updated to 3.50 dB instead of 5.17 dB, while for PAM-6 83 GBd the gain becomes 0.8 dB instead of 2.17 dB. The changes are adapted in all sections except the experimental setup and DSP section

A spiking neural network (SNN) equalizer model suitable for electronic neuromorphic hardware is designed for an IM/DD link. The SNN achieves the same bit-error-rate as an artificial neural network, outperforming linear equalization.

We report an industry leading optical dense wavelength division multiplexing (DWDM) field trial with line rates per channel exceeding 1.66 Tb/s using 130 GBaud dual-polarization probabilistic constellation shaping 256-ary quadrature amplitude modulation (DP-PCS256QAM) in a high capacity data center interconnect (DCI) scenario. This research trial was performed on 96.5 km of field-deployed standard single mode G.652 fiber infrastructure of Deutsche Telekom in Germany employing Erbium-doped fiber amplifier (EDFA)-only amplification. A total of 34 channels were transmitted with 150 GHz spacing for a total fiber capacity of 56.51 Tb/s and a spectral efficiency higher than 11bit/s/Hz. In the single-channel transmission scenario 1.71 Tb/s was achieved over the same link. In addition, we successfully demonstrate record net bitrates of 1.88 Tb/s in back-to-back (B2B) using 130 GBaud DP-PCS400QAM., Comment: This work has been submitted to the IEEE Photonics Technology Letters (PTL) for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

Coded modulation with probabilistic amplitude shaping (PAS) is considered for intensity modulation/direct detection channels with a transmitter peak-power constraint. PAS is used to map bits to a uniform PAM-6 distribution and outperforms PAM-8 for rates up to around 2.3 bits per channel use. PAM-6 with PAS also outperforms a cross-shaped QAM-32 constellation by up to 1 dB and 0.65 dB after bit-metric soft- and hard decoding, respectively. An alternative PAM-6 scheme based on a framed cross-shaped QAM-32 constellation is proposed that shows similar gains., Comment: submitted to 11th International Symposium on Topics in Coding

Linear layered probabilistic shaping (LLPS) is proposed, an architecture for linear codes to efficiently encode to shaped code words. In the previously proposed probabilistic amplitude shaping (PAS) architecture, a distribution matcher (DM) maps information bits to shaped bits, which are then systematically encoded by appending uniformly distributed parity bits. LLPS extends PAS by probabilistic parity shaping (PPS), which uses a syndrome DM to calculate shaped parity bits. LLPS enables the transmission with any desired distribution using linear codes, furthermore, by LLPS, a given linear code with rate $R_\text{fec}$ can be operated at any rate $R\leq R_\text{fec}$ by changing the distribution. LLPS is used with an LDPC code for dirty paper coding against an interfering BPSK signal, improving the energy efficiency by 0.8 dB., Comment: Draft based on talk given at 2019 Oberpfaffenhofen Workshop on High Throughput Coding (OWHTC)

A polar-coded modulation scheme for deep-space optical communication is proposed. The photon counting Poisson channel with pulse position modulation (PPM) is considered. We use the fact that PPM is particularly well suited to be used with multilevel codes to design a polar-coded modulation scheme for the system in consideration. The construction of polar codes for the Poisson channel based on Gaussian approximation is demonstrated to be accurate. The proposed scheme uses a cyclic redundancy check outer code and a successive cancellation decoder with list decoding and it is shown that it outperforms the competing schemes., Comment: 7 pages, 9 figures

A new approach for combining non-binary low-density parity-check (NB-LDPC) codes with higher-order modulation and probabilistic amplitude shaping (PAS) is presented. Instead of symbol-metric decoding (SMD), a bit-metric decoder (BMD) is used so that matching the field order of the non-binary code to the constellation size is not needed, which increases the flexibility of the coding scheme. Information rates, density evolution thresholds and finite-length simulations show that the flexibility comes at no loss of performance if PAS is used., Comment: Accepted for IEEE Communication Letters

An unsupervised learning approach based on expectation maximization is proposed to obtain the parameters of a soft decision forward error correction decoding metric for probabilistic shaping. The algorithm depends only on the channel observations and does not require transmitted data., Comment: Accepted for ECOC 2018 (Rome), Th1H.3

A flexible incremental redundancy hybrid auto- mated repeat request (IR-HARQ) scheme for polar codes is proposed based on dynamically frozen bits and the quasi-uniform puncturing (QUP) algorithm. The length of each transmission is not restricted to a power of two. It is applicable for the binary input additive white Gaussian noise (biAWGN) channel as well as higher-order modulation. Simulation results show that this scheme has similar performance as directly designed polar codes with QUP and outperforms LTE-turbo and 5G-LDPC codes with IR-HARQ., Comment: 6 pages, accepted to 2018 IEEE Wireless Communications and Networking Conference Workshops (WCNCW): Polar Coding for Future Networks: Theory and Practice, presented on 15. April

Parallel, additive white Gaussian noise (AWGN) channels with an average sum power constraint are considered. It is shown how the waterfilling Shannon capacity can be approached by higher order modulation and probabilistic amplitude shaping (PAS). This is achieved by a new distribution matching approach called product distribution matching (PDM). The asymptotic performance of PDM is analyzed by achievable rates. A heuristic for optimizing the input distribution is proposed, which enables signaling at a target spectral efficiency with a fixed-rate forward error correction (FEC) code, while the optimal power allocation is ensured by mercury-waterfilling and a simple bit-loading strategy. Finite blocklength simulation results with 5G low-density parity-check codes show power savings of around 1 dB compared to a conventional scheme with uniform input distributions., Comment: Invited paper at CISS 2018, special session "Optimal Communications with Discrete Input Signals"

The rate flexibility and probabilistic shaping gain of $4$-dimensional signaling is experimentally tested for short-reach, unrepeated transmission. A rate granularity of 0.5 bits/QAM symbol is achieved with a distribution matcher based on a simple look-up table., Comment: Presented at OFC'18, San Diego, CA, USA

This work shows how non-binary low-density parity-check codes over GF($2^p$) can be combined with probabilistic amplitude shaping (PAS) (B\"ocherer, et al., 2015), which combines forward-error correction with non-uniform signaling for power-efficient communication. Ultra-sparse low-density parity-check codes over GF(64) and GF(256) gain 0.6 dB in power efficiency over state-of-the-art binary LDPC codes at a spectral efficiency of 1.5 bits per channel use and a blocklength of 576 bits. The simulation results are compared to finite length coding bounds and complemented by density evolution analysis., Comment: Accepted for Globecom 2017

A heuristic construction of polar codes for successive cancellation list (SCL) decoding with a given list size is proposed to balance the trade-off between performance measured in frame error rate (FER) and decoding complexity. Furthermore, a construction based on dynamically frozen bits with constraints among the "low weight bits" (LWB) is presented. Simulation results show that the LWB-polar codes outperform the CRC-polar codes and the eBCH-polar codes under SCL decoding., Comment: 6 pages

For a layered probabilistic shaping (PS) scheme with a general decoding metric, an achievable rate is derived using Gallager's error exponent approach and the concept of achievable code rates is introduced. Several instances for specific decoding metrics are discussed, including bit-metric decoding, interleaved coded modulation, and hard-decision decoding. It is shown that important previously known achievable rates can also be achieved by layered PS. A practical instance of layered PS is the recently proposed probabilistic amplitude shaping (PAS)., Comment: Compared to v4, a simpler proof of Proposition 1 (achievable encoding rate) is given and an interpretation of the uncertainty as a cross-entropy is added

This paper studies the impact of probabilistic shaping on effective signal-to-noise ratios (SNRs) and achievable information rates (AIRs) in a back-to-back configuration and in unrepeated nonlinear fiber transmissions. For back-to-back, various shaped quadrature amplitude modulation (QAM) distributions are found to have the same implementation penalty as uniform input. By demonstrating in transmission experiments that shaped QAM input leads to lower effective SNR than uniform input at a fixed average launch power, we experimentally confirm that shaping enhances the fiber nonlinearities. However, shaping is ultimately found to increase the AIR, which is the most relevant figure of merit as it is directly related to spectral efficiency. In a detailed study of these shaping gains for the nonlinear fiber channel, four strategies for optimizing QAM input distributions are evaluated and experimentally compared in wavelength division multiplexing (WDM) systems. The first shaping scheme generates a Maxwell-Boltzmann (MB) distribution based on a linear additive white Gaussian noise channel. The second strategy uses the Blahut-Arimoto algorithm to optimize an unconstrained QAM distribution for a split-step Fourier method based channel model. In the third and fourth approach, MB-shaped QAM and unconstrained QAM are optimized via the enhanced Gaussian noise (EGN) model. Although the absolute shaping gains are found to be relatively small, the relative improvements by EGN-optimized unconstrained distributions over linear AWGN optimized MB distributions are up to 59%. This general behavior is observed in 9-channel and fully loaded WDM experiments., Comment: 9 pages, 7 figures, 7 tables

Product distribution matching (PDM) is proposed to generate target distributions over large alphabets by combining the output of several parallel distribution matchers (DMs) with smaller output alphabets. The parallel architecture of PDM enables low-complexity and high-throughput implementation. PDM is used as a shaping device for probabilistic amplitude shaping (PAS). For 64-ASK and a spectral efficiency of 4.5 bits per channel use (bpcu), PDM is as power efficient as a single full-fledged DM. It is shown how PDM enables PAS for parallel channels present in multi-carrier systems like digital subscriber line (DSL) and orthogonal frequency-division multiplexing (OFDM). The key feature is that PDM shares the DMs for lower bit-levels among different sub-carriers, which improves the power efficiency significantly. A representative parallel channel example shows that PAS with PDM is 0.93 dB more power efficient than conventional uniform signaling and PDM is 0.35 dB more power efficient than individual per channel DMs., Comment: Including 7 diagrams and 5 figures with simulation results

An efficient algorithm for the construction of polar codes for higher-order modulation is presented based on information-theoretic principles. The bit reliabilities after successive demapping are estimated using the LM-rate, an achievable rate for mismatched decoding. The successive demapper bit channels are then replaced by binary input Additive White Gaussian Noise (biAWGN) surrogate channels and polar codes are constructed using the Gaussian approximation (GA). This LM-rate Demapper GA (LM-DGA) construction is used to construct polar codes for several demapping strategies proposed in literature. For all considered demappers, the LM-DGA constructed polar codes have the same performance as polar codes constructed by Monte Carlo (MC) simulation. The proposed LM-DGA construction is much faster than the MC construction. For 64-QAM, spectral efficiency 3 bits/s/Hz, and block length 1536 bits, simulation results show that LM-DGA constructed polar codes with cyclic redundancy check and successive cancellation list decoding are 1 dB more power efficient than state-of-the-art AR4JA low-density parity-check codes., Comment: 11 figures

In this work, geometric shaping (GS) and probabilistic shaping (PS) for the AWGN channel is reviewed. Both approaches are investigated in terms of symbol-metric decoding (SMD) and bit-metric decoding (BMD). For GS, an optimization algorithm based on differential evolution is formulated. Achievable rate analysis reveals that GS suffers from a 0.4 dB performance degradation compared to PS when BMD is used. Forward-error correction simulations of the ATSC 3.0 modulation and coding formats (modcods) confirm the theoretical findings. In particular, PS enables seamless rate adaptation with one single modcod and it outperforms ATSC 3.0 GS modcods by more than 0.5 dB for spectral efficiencies larger than 3.2 bits per channel use., Comment: Submitted to SCC 2017, minor wording and spelling updates compared to v1

These lecture notes on channel coding were developed for a one-semester course for graduate students of electrical engineering. Chapter 1 reviews the basic problem of channel coding. Chapters 2-5 are on linear block codes, cyclic codes, Reed-Solomon codes, and BCH codes, respectively. The notes are self-contained and were written with the intent to derive the presented results with mathematical rigor. The notes contain in total 68 homework problems, of which 20% require computer programming., Comment: 5 chapters, 68 homework problems

Different aspects of probabilistic shaping for a multi-span optical communication system are studied. First, a numerical analysis of the additive white Gaussian noise (AWGN) channel investigates the effect of using a small number of input probability mass functions (PMFs) for a range of signal-to-noise ratios (SNRs), instead of optimizing the constellation shaping for each SNR. It is shown that if a small penalty of at most 0.1 dB SNR to the full shaping gain is acceptable, just two shaped PMFs are required per quadrature amplitude modulation (QAM) over a large SNR range. For a multi-span wavelength division multiplexing (WDM) optical fiber system with 64QAM input, it is shown that just one PMF is required to achieve large gains over uniform input for distances from 1,400 km to 3,000 km. Using recently developed theoretical models that extend the Gaussian noise (GN) model and full-field split-step simulations, we illustrate the ramifications of probabilistic shaping on the effective SNR after fiber propagation. Our results show that, for a fixed average optical launch power, a shaping gain is obtained for the noise contributions from fiber amplifiers and modulation-independent nonlinear interference (NLI), whereas shaping simultaneously causes a penalty as it leads to an increased NLI. However, this nonlinear shaping loss is found to have a relatively minor impact, and optimizing the shaped PMF with a modulation-dependent GN model confirms that the PMF found for AWGN is also a good choice for a multi-span fiber system., Comment: 11 pages, 9 figures, 5 tables. Accepted for publication in IEEE/OSA Journal of Lightwave Technology

The approximation of a discrete probability distribution $\mathbf{t}$ by an $M$-type distribution $\mathbf{p}$ is considered. The approximation error is measured by the informational divergence $\mathbb{D}(\mathbf{t}\Vert\mathbf{p})$, which is an appropriate measure, e.g., in the context of data compression. Properties of the optimal approximation are derived and bounds on the approximation error are presented, which are asymptotically tight. It is shown that $M$-type approximations that minimize either $\mathbb{D}(\mathbf{t}\Vert\mathbf{p})$, or $\mathbb{D}(\mathbf{p}\Vert\mathbf{t})$, or the variational distance $\Vert\mathbf{p}-\mathbf{t}\Vert_1$ can all be found by using specific instances of the same general greedy algorithm., Comment: 5 pages

We implemented a flexible transmission system operating at adjustable data rate and fixed bandwidth, baudrate, constellation and overhead using probabilistic shaping. We demonstrated in a transmission experiment up to 15% capacity and 43% reach increase versus 200 Gbit/s 16-QAM., Comment: Accepted for presentation as postdeadline paper at ECOC 2015

A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform input and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels for the code design are used. The constructed codes for uniform inputs perform as good as the multi-edge type codes of Zhang and Kschischang (2013). For 8-ASK and 64-ASK with probabilistic shaping, codes of rates 2/3 and 5/6 with blocklength 64800 are designed, which operate within 0.63dB and 0.69dB of continuous AWGN capacity for a target frame error rate of 1e-3 at spectral efficiencies of 1.38 and 4.25 bits/channel use, respectively., Comment: 9 pages, 10 figures. arXiv admin note: substantial text overlap with arXiv:1501.05595

Distribution matching transforms independent and Bernoulli(1/2) distributed input bits into a sequence of output symbols with a desired distribution. Fixed-to-fixed length, invertible, and low complexity encoders and decoders based on constant composition and arithmetic coding are presented. Asymptotically in the blocklength, the encoder achieves the maximum rate, namely the entropy of the desired distribution. Furthermore, the normalized divergence of the encoder output and the desired distribution goes to zero in the blocklength., Comment: 5 pages, 5 figures

A new coded modulation scheme is proposed. At the transmitter, the concatenation of a distribution matcher and a systematic binary encoder performs probabilistic signal shaping and channel coding. At the receiver, the output of a bitwise demapper is fed to a binary decoder. No iterative demapping is performed. Rate adaption is achieved by adjusting the input distribution and the transmission power. The scheme is applied to bipolar amplitude shift keying (ASK) constellations with equidistant signal points and it is directly applicable to two-dimensional quadrature amplitude modulation (QAM). The scheme is implemented by using the DVB-S2 low-density parity-check (LDPC) codes. At a frame error rate of 1e-3, the new scheme operates within less than 1 dB of the AWGN capacity 0.5log2(1+SNR) at any spectral efficiency between 1 and 5 bits/s/Hz by using only 5 modes, i.e., 4-ASK with code rate 2/3, 8-ASK with 3/4, 16-ASK and 32-ASK with 5/6 and 64-ASK with 9/10., Comment: 13 pages, 11 figures, 10 tables

A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform input and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels are constructed and used for code design. The constructed codes perform as good as multi-edge type codes of Zhang and Kschischang (2013). For 64-ASK with probabilistic shaping, a blocklength 64800 code is constructed that operates within 0.69 dB of 0.5log(1+SNR) at a spectral efficiency of 4.2 bits/channel use and a frame error rate of 1e-3., Comment: 5 pages, 8 figures

An LDPC coded modulation scheme with probabilistic shaping, optimized interleavers and noniterative demapping is proposed. Full-field simulations show an increase in transmission distance by 8% compared to uniformly distributed input., Comment: 3 pages, 3 figures. Paper is accepted for presentation at OFC 2015. Following the submission to OFC, the blue box in Fig. 1 has been corrected and reference [3] has been updated

A new achievable rate for bit-metric decoding (BMD) is derived using random coding arguments. The rate expression can be evaluated for any input distribution, and in particular the bit-levels of binary input labels can be stochastically dependent. Probabilistic shaping with dependent bit-levels (shaped BMD), shaping of independent bit-levels (bit-shaped BMD) and uniformly distributed independent bit-levels (uniform BMD) are evaluated on the additive white Gaussian noise (AWGN) channel with Gray labeled bipolar amplitude shift keying (ASK). For 32-ASK at a rate of 3.8 bits/channel use, the gap to 32-ASK capacity is 0.008 dB for shaped BMD, 0.46 dB for bit-shaped BMD, and 1.42 dB for uniform BMD. These numerical results illustrate that dependence between the bit-levels is beneficial on the AWGN channel. The relation to the LM rate and the generalized mutual information (GMI) is discussed., Comment: Revised version. Compared to version v5, the LM rate is now discussed

In this work, arithmetic distribution matching (ADM) is presented. ADM invertibly transforms a discrete memoryless source (DMS) into a target DMS. ADM can be used for probabilistic shaping and for rate adaption. Opposed to existing algorithms for distribution matching, ADM works online and can transform arbitrarily long input sequences. It is shown analytically that as the input length tends to infinity, the ADM output perfectly emulates the target DMS with respect to the normalized informational divergence and the entropy rate. Numerical results are presented that confirm the analytical bounds., Comment: 6 pages

A scheme is proposed that combines probabilistic signal shaping with bit-metric decoding. The transmitter generates symbols according to a distribution on the channel input alphabet. The symbols are labeled by bit strings. At the receiver, the channel output is decoded with respect to a bit-metric. An achievable rate is derived using random coding arguments. For the 8-ASK AWGN channel, numerical results show that at a spectral efficiency of 2 bits/s/Hz, the new scheme outperforms bit-interleaved coded modulation (BICM) without shaping and BICM with bit shaping (i Fabregas and Martinez, 2010) by 0.87 dB and 0.15 dB, respectively, and is within 0.0094 dB of the coded modulation capacity. The new scheme is implemented by combining a distribution matcher with a systematic binary low-density parity-check code. The measured finite-length gains are very close to the gains predicted by the asymptotic theory., Comment: 7 pages; compared to v2, the title has slightly changed. More references to related work have been added. A shortened version will appear in the proceedings of the ISIT 14

Rooted trees with probabilities are used to analyze properties of a variable length code. A bound is derived on the difference between the entropy rates of the code and a memoryless source. The bound is in terms of normalized informational divergence. The bound is used to derive converses for exact random number generation, resolution coding, and distribution matching., Comment: 5 pages. With proofs and illustrating example

Finite precision approximations of discrete probability distributions are considered, applicable for distribution synthesis, e.g., probabilistic shaping. Two algorithms are presented that find the optimal $M$-type approximation $Q$ of a distribution $P$ in terms of the variational distance $| Q-P|_1$ and the informational divergence $\mathbb{D}(Q| P)$. Bounds on the approximation errors are derived and shown to be asymptotically tight. Several examples illustrate that the variational distance optimal approximation can be quite different from the informational divergence optimal approximation., Comment: Submitted to the IEEE Transactions on Information Theory

The number of random bits required to approximate a target distribution in terms of un-normalized informational divergence is considered. It is shown that for a variable-to-variable length encoder, this number is lower bounded by the entropy of the target distribution. A fixed-to-variable length encoder is constructed using M-type quantization and Tunstall coding. It is shown that the encoder achieves in the limit an un-normalized informational divergence of zero with the number of random bits per generated symbol equal to the entropy of the target distribution. Numerical results show that the proposed encoder significantly outperforms the optimal block-to-block encoder in the finite length regime., Comment: Essentially the ITW 2013 final version. Compared to v1, minor typos were corrected and Fig. 1 with an example variable length encoder was added

In this work, binary block-to-block distribution matching is considered. m independent and uniformly distributed bits are mapped to n output bits resembling a target product distribution. A rate R is called achieved by a sequence of encoder-decoder pairs, if for m,n to infinity, (1) m/n approaches R, (2) the informational divergence per bit of the output distribution and the target distribution goes to zero, and (3) the probability of erroneous decoding goes to zero. It is shown that the maximum achievable rate is equal to the entropy of the target distribution. A practical encoder-decoder pair is constructed that provably achieves the maximum rate in the limit. Numerical results illustrate that the suggested system operates close to the limits with reasonable complexity. The key idea is to internally use a fixed-to-variable length matcher and to compensate underflow by random mapping and to cast an error when overflow occurs., Comment: 5 pages

Rooted trees with probabilities are convenient to represent a class of random processes with memory. They allow to describe and analyze variable length codes for data compression and distribution matching. In this work, the Leaf-Average Node-Sum Interchange Theorem (LANSIT) and the well-known applications to path length and leaf entropy are re-stated. The LANSIT is then applied to informational divergence. Next, the differential LANSIT is derived, which allows to write normalized functionals of leaf distributions as an average of functionals of branching distributions. Joint distributions of random variables and the corresponding conditional distributions are special cases of leaf distributions and branching distributions. Using the differential LANSIT, Pinsker's inequality is formulated for rooted trees with probabilities, with an application to the approximation of product distributions. In particular, it is shown that if the normalized informational divergence of a distribution and a product distribution approaches zero, then the entropy rate approaches the entropy rate of the product distribution., Comment: This an evolving draft currently in the form of slides

Fixed-to-variable length (f2v) matchers are used to reversibly transform an input sequence of independent and uniformly distributed bits into an output sequence of bits that are (approximately) independent and distributed according to a target distribution. The degree of approximation is measured by the informational divergence between the output distribution and the target distribution. An algorithm is developed that efficiently finds optimal f2v codes. It is shown that by encoding the input bits blockwise, the informational divergence per bit approaches zero as the block length approaches infinity. A relation to data compression by Tunstall coding is established., Comment: 5 pages, essentially the ISIT 2013 version

Consider the estimation of an unknown parameter vector in a linear measurement model. Centralized sensor selection consists in selecting a set of k_s sensor measurements, from a total number of m potential measurements. The performance of the corresponding selection is measured by the volume of an estimation error covariance matrix. In this work, we consider the problem of selecting these sensors in a distributed or decentralized fashion. In particular, we study the case of two leader nodes that perform naive decentralized selections. We demonstrate that this can degrade the performance severely. Therefore, two heuristics based on convex optimization methods are introduced, where we first allow one leader to make a selection, and then to share a modest amount of information about his selection with the remaining node. We will show that both heuristics clearly outperform the naive decentralized selection, and achieve a performance close to the centralized selection., Comment: 6 pages, to be presented at GLOBECOM 2012

The construction of optimal non-uniform mappings for discrete input memoryless channels (DIMCs) is investigated. An efficient algorithm to find optimal mappings is proposed and the rate by which a target distribution is approached is investigated. The results are applied to non-uniform mappings for additive white Gaussian noise (AWGN) channels with finite signal constellations. The mappings found by the proposed methods outperform those obtained via a central limit theorem approach as suggested in the literature., Comment: 6 pages. Compared to v2, the former Section IV.B is removed because the statement was wrong. A formatting problem is resolved

Bit-interleaved coded modulation (BICM) is a practical approach for reliable communication over the AWGN channel in the bandwidth limited regime. For a signal point constellation with 2^m points, BICM labels the signal points with bit strings of length m and then treats these m bits separately both at the transmitter and the receiver. BICM capacity is defined as the maximum of a certain achievable rate. Maximization has to be done over the probability mass functions (pmf) of the bits. This is a non-convex optimization problem. So far, the optimal bit pmfs were determined via exhaustive search, which is of exponential complexity in m. In this work, an algorithm called bit-alternating convex concave method (Bacm) is developed. This algorithm calculates BICM capacity with a complexity that scales approximately as m^3. The algorithm iteratively applies convex optimization techniques. Bacm is used to calculate BICM capacity of 4,8,16,32, and 64-PAM in AWGN. For PAM constellations with more than 8 points, the presented values are the first results known in the literature., Comment: 5 pages, to be presented at ISIT 2012. Compared to version v1, several parts were clarified

The design of the channel part of a digital communication system (e.g., error correction, modulation) is heavily based on the assumption that the data to be transmitted forms a fair bit stream. However, simple source encoders such as short Huffman codes generate bit streams that poorly match this assumption. As a result, the channel input distribution does not match the original design criteria. In this work, a simple method called half Huffman coding (halfHc) is developed. halfHc transforms a Huffman code into a source code whose output is more similar to a fair bit stream. This is achieved by permuting the codewords such that the frequency of 1s at the output is close to 0.5. The permutations are such that the optimality in terms of achieved compression ratio is preserved. halfHc is applied in a practical example, and the resulting overall system performs better than when conventional Huffman coding is used., Comment: submitted to ICSPCS 2011, Honolulu

In this work, a coding technique called cost constrained Geometric Huffman coding (ccGhc) is developed. ccGhc minimizes the Kullback-Leibler distance between a dyadic probability mass function (pmf) and a target pmf subject to an affine inequality constraint. An analytical proof is given that when ccGhc is applied to blocks of symbols, the optimum is asymptotically achieved when the blocklength goes to infinity. The derivation of ccGhc is motivated by the problem of encoding a text to a sequence of slats subject to architectural design criteria. For the considered architectural problem, for a blocklength of 3, the codes found by ccGhc match the design criteria. For communications channels with average cost constraints, ccGhc can be used to efficiently find prefix-free modulation codes that are provably capacity achieving., Comment: 5 pages, submitted to ISWCS 2011