A correlated phase-and-additive-noise (CPAN) mismatched model is developed for wavelength division multiplexing over optical fiber channels governed by the nonlinear Schrödinger equation. Both the phase and additive noise processes of the CPAN model are Gauss-Markov whereas previous work uses Wiener phase noise and white additive noise. Second order statistics are derived and lower bounds on the capacity are computed by simulations. The CPAN model characterizes nonlinearities better than existing models in the sense that it achieves better information rates. For example, the model gains 0.35 dB in power at the peak data rate when using a single carrier per wavelength. For multiple carriers per wavelength, the model combined with frequency-dependent power allocation gains 0.14 bits/s/Hz in rate and 0.8 dB in power at the peak data rate.
We show that the mismatched capacity of binary discrete memoryless channels can be improved by channel combining and splitting via Arıkan's polar transformations. We also show that the improvement is possible even if the transformed channels are decoded with a mismatched polar decoder.
Arıkan's polar coding, is by now a well studied technique that allows achieving the symmetric capacity of binary input memoryless channels with low complexity encoding and decoding, provided that the polar decoding architecture is used and the decoding metric is matched to the true channel. In this paper, we analyze communication rates that are achievable when the polar coding/decoding architecture is used with the decoder using an incorrect model of the channel. We define the `polar mismatched capacity' as an analogue of the classical mismatched capacity, give an expression for it, and derive…
Regular perturbation is applied to the Manakov equation and motivates a generalized correlated phase-and-additive noise model for wavelength-division multiplexing over dual-polarization optical fiber channels. The model includes three hidden Gauss-Markov processes: phase noise, polarization rotation, and additive noise. Particle filtering is used to compute lower bounds on the capacity of multi-carrier communication with frequency-dependent powers and delays. A gain of 0.17 bits/s/Hz/pol in spectral efficiency or 0.8 dB in power efficiency is achieved with respect to existing models at their p…
We develop a novel framework for proving converse theorems for channel coding, which is based on the analysis technique of multicast transmission with an additional auxiliary receiver, which serves as a genie to the original receiver. The genie provides the original receiver a certain narrowed list of codewords to choose from that includes the transmitted one. This technique is used to derive upper bounds on the mismatch capacity of discrete memoryless channels as well as the reliability function with a mismatched decoding metric. Unlike previous works, our bounding technique exploits also the…
The problem of mismatched decoding for discrete memoryless channels is addressed. A mismatched cognitive multiple-access channel is introduced, and an inner bound on its capacity region is derived using two alternative encoding methods: superposition coding and random binning. The inner bounds are derived by analyzing the average error probability of the code ensemble for both methods and by a tight characterization of the resulting error exponents. Random coding converse theorems are also derived. A comparison of the achievable regions shows that in the matched case, random binning performs a…
We derive a sphere-packing error exponent for coded transmission over discrete memoryless channels with a fixed decoding metric. By studying the error probability of the code over an auxiliary channel, we find a lower bound to the probability of error of mismatched decoding. The bound is shown to decay exponentially for coding rates smaller than a new upper bound to the mismatch capacity which is established in this paper. For rates higher than the new upper bound, the error probability is shown to be bounded away from zero. The new upper bound is shown to improve over previous upper bounds to…
We extend the notion of list decoding to {\em ratio list decoding} which involves a list decoder whose list size is specified as a function of the number of messages $M_n$ and the block length $n$. We present necessary and sufficient conditions on $M_n$ for the existence of code sequences which enable reliable list decoding with respect to the desired list size $L(M_n,n)$. It is shown that the ratio-capacity, defined as the supremum of achievable normalized logarithms of the ratio $r(M_n,n)=M_n/L(M_n,n)$ is equal to the Shannon channel capacity $C$, for both stochastic and deterministic encodi…
The common approach of designing a communication device is to maximize a well-defined objective function, e.g., the channel capacity and the cut-off rate. We propose easy-to-implement solutions for Gaussian channels that approximate the optimal results for these maximization problems. Three topics are addressed. First, we consider the case where the channel output is quantized, and we find the quantization thresholds that maximize the mutual information. The approximation derived from the asymptotic solution has a negligible loss on the entire range of SNR when 2-PAM modulation is used, and it…
Shaping gain is attained in schemes where a shaped subcode is chosen from a larger codebook by a codeword selection process. This includes the popular method of Trellis Shaping (TS), originally proposed by Forney for average power reduction. The decoding process of such schemes is mismatched, since it is aware of only the large codebook. This study models such schemes by a random code construction and derives achievable bounds on the transmission rate under matched and mismatched decoding. For matched decoding the bound is obtained using a modified asymptotic equipartition property (AEP) theor…
We derive a single-letter upper bound to the mismatched-decoding capacity for discrete memoryless channels. The bound is expressed as the mutual information of a transformation of the channel, such that a maximum-likelihood decoding error on the translated channel implies a mismatched-decoding error in the original channel. In particular, a strong converse is shown to hold for this upper-bound: if the rate exceeds the upper-bound, the probability of error tends to 1 exponentially when the block-length tends to infinity. We also show that the underlying optimization problem is a convex-concave …
We study a nonlinear fiber optical channel impaired by cross-phase modulation and dispersion from the viewpoint of an interference channel. We characterize an outer bound on the capacity region of simultaneously achievable rate pairs, assuming a two-user perturbative channel model.
The problem of mismatched decoding with an additive metric $q$ for a discrete memoryless channel $W$ is addressed. The "product-space" improvement of the random coding lower bound on the mismatch capacity, $C_q^{(\infty)}(W)$, was introduced by Csiszár and Narayan. We study two kinds of decoders. The {\it $δ$-margin mismatched decoder} outputs a message whose metric with the channel output exceeds that of all the other codewords by at least $δ$. The {\it $τ$-threshold decoder} outputs a single message whose metric with the channel output exceeds a threshold $τ$. Both decoders declare an error …
We consider the optical fiber channel modelled by the nonlinear Shrödinger equation with zero dispersion and additive Gaussian noise. Using Feynman path-integral approach for the model we find corrections to conditional probability density function, output signal distribution, conditional and output signal entropies, and the channel capacity at large signal-to-noise ratio. We demonstrate that the correction to the channel capacity is positive for large signal power. Therefore, this correction increases the earlier calculated capacity for a nondispersive nonlinear optical fiber channel in the i…
The fundamental limits of channels with mismatched decoding are addressed. A general formula is established for the mismatch capacity of a general channel, defined as a sequence of conditional distributions with a general decoding metrics sequence. We deduce an identity between the Verdú-Han general channel capacity formula, and the mismatch capacity formula applied to Maximum Likelihood decoding metric. Further, several upper bounds on the capacity are provided, and a simpler expression for a lower bound is derived for the case of a non-negative decoding metric. The general formula is special…
This paper considers the problem of channel coding over Gaussian intersymbol interference (ISI) channels with a given metric decoding rule. Specifically, it is assumed that the mismatched decoder has an incorrect assumption on the impulse response function. The mismatch capacity is the highest achievable rate for a given decoding rule. Existing lower bounds to the mismatch capacity for channels and decoding metrics with memory (as in our model) are presented only in the form of multi-letter expressions that have not been calculated in practice. Consequently, they provide little insight on the …
The per-sample zero-dispersion channel model of the optical fiber is considered. It is shown that capacity is uniquely achieved by an input probability distribution that has continuous uniform phase and discrete amplitude that takes on finitely many values. This result holds when the channel is subject to general input cost constraints, that include a peak amplitude constraint and a joint average and peak amplitude constraint.
We consider multi-input multi-output (MIMO) communications over multi-mode fibers (MMFs). Current MMF standards, such as OM3 and OM4, use fibers with core radii of 50 μm, allowing hundreds of modes to propagate. Unfortunately, due to physical and computational complexity limitations, we cannot couple and detect hundreds of data streams into and out of the fiber. In order to circumvent this issue, we present input-output coupling schemes that allow the user to couple and extract a reasonable number of signals from a fiber with many modes. This approach is particularly attractive as it is scalab…
An upper bound to the information capacity of a wavelength-division multi- plexed optical fiber communication system is derived in a model incorporating the nonlinear propagation effects of cross-phase modulation (XPM). This work is based on the paper by Mitra et al., finding lower bounds to the channel capacity, in which physical models for propagation are used to calculate statistical properties of the conditional probability distribution relating input and output in a single WDM channel. In this paper we present a tractable channel model incorporating the effects of cross phase modulation. …
By taking advantage of the temporal correlations of the nonlinear phase noise in WDM systems we show that the capacity of a nonlinear fiber link is notably higher than what is currently assumed. This advantage is translated into the doubling of the link distance for a fixed transmission rate.
The concept of a fiber aided wireless network architecture (FAWNA) is introduced in [Ray et al., Allerton Conference 2005], which allows high-speed mobile connectivity by leveraging the speed of optical networks. In this paper, we consider a single-input, multiple-output (SIMO) FAWNA, which consists of a SIMO wireless channel and an optical fiber channel, connected through wireless-optical interfaces. We propose a scheme where the received wireless signal at each interface is quantized and sent over the fiber. Though our architecture is similar to that of the classical CEO problem, our problem…
A novel technique to optimize the input distribution and compute a lower bound for the capacity of the nonlinear optical fiber channel is proposed. The technique improves previous bounds obtained with the additive white Gaussian noise decoding metric.
A number of simplified models, based on perturbation theory, have been proposed for the fiber-optical channel and have been extensively used in the literature. Although these models are mainly developed for the low-power regime, they are used at moderate or high powers as well. It remains unclear to what extent the capacity of these models is affected by the simplifying assumptions under which they are derived. In this paper, we consider single channel data transmission based on three continuous-time optical models i) a regular perturbative channel, ii) a logarithmic perturbative channel, and …
Information-Theoretic Foundations of Mismatched Decoding
National University of Singapore（新加坡国立大学）；Institució Catalana de Recerca i Estudis Avançats (ICREA)（加泰罗尼亚高级研究学院）；Universitat Pompeu Fabra（庞培法布拉大学）；University of Cambridge（剑桥大学）；Bar-Ilan University（巴伊兰大学）
The emerging cell-free massive multiple-input multiple-output (CF-mMIMO) is a promising scheme to tackle the capacity crunch in wireless networks. Designing the optimal fronthaul network in the CF-mMIMIO is of utmost importance to deploy a cost- and energy-efficient network. In this paper, we present a framework to optimally design the fronthaul network of CF-mMIMO utilizing optical fiber and free space optical (FSO) technologies. We study an uplink data transmission of the CF-mMIMO network wherein each of the distributed access points (APs) is connected to a central processing unit (CPU) thro…
The capacity of the channel defined by the stochastic nonlinear Schrödinger equation, which includes the effects of the Kerr nonlinearity and amplified spontaneous emission noise, is considered in the case of zero dispersion. In the absence of dispersion, this channel behaves as a collection of parallel per-sample channels. The conditional probability density function of the nonlinear per-sample channels is derived using both a sum-product and a Fokker-Planck differential equation approach. It is shown that, for a fixed noise power, the per-sample capacity grows unboundedly with input signal. …
In our previous paper [Phys. Rev. E 95, 062122 (2017)] we considered the optical channel modelled by the nonlinear Schrödinger equation with zero dispersion and additive Gaussian noise. We found per-sample channel capacity rof this model. In the present paper we extend per-sample model by introducing the initial signal dependence on time and the output signal detection procedure. The proposed model is a closer approximation of the realistic communication link than the per-sample model where there is no dependence of the initial signal on time. For the proposed model we found the correlators of…
Optical fiber signals with high power exhibit spectral broadening that seems to limit capacity. To study spectral broadening, the autocorrelation function of the output signal given the input signal is derived for a simplified fiber model that has zero dispersion, distributed optical amplification (OA), and idealized spatial noise processes. The autocorrelation function is used to upper bound the output power of bandlimited or time-resolution limited receivers, and thereby to bound spectral broadening and the capacity of receivers with thermal noise. The output power scales at most as the squa…
Technology is moving towards space division multiplexing in optical fiber to keep up the trend in rate increase over time and to avoid an imminent capacity crunch. Thus, it is of paramount interest to estimate the potential gains of this approach. As more spatial channels are being packed into a single fiber, the increased crosstalk necessitates the use of MIMO to guarantee reliable operation. In this paper, we exploit the analogy between an optical fiber and a model from mesoscopic physics - a chaotic cavity - to obtain a novel channel model for the optical fiber. The model captures both rand…
The optical fiber multiple-input multiple-output (MIMO) channel with intensity modulation and direct detection (IM/DD) per spatial path is treated. The spatial dimensions represent the multiple modes employed for transmission and the cross-talk between them originates in the multiplexers and demultiplexers, which are polarization dependent and thus timevarying. The upper bounds from free-space IM/DD MIMO channels are adapted to the fiber case, and the constellation constrained capacity is constructively estimated using the Blahut-Arimoto algorithm. An autoencoder is then proposed to optimize a…
It is shown that signal energy is the only available degree-of-freedom (DOF) for fiber-optic transmission as the input power tends to infinity. With $n$ signal DOFs at the input, $n-1$ DOFs are asymptotically lost to signal-noise interactions. The main observation is that, nonlinearity introduces a multiplicative noise in the channel, similar to fading in wireless channels. The channel is viewed in the spherical coordinate system, where signal vector $\underline{X}\in\mathbb{C}^n$ is represented in terms of its norm $|\underline{X}|$ and direction $\underline{\hat{X}}$. The multiplicative nois…
The capacity of a discrete-time model of optical fiber described by the split-step Fourier method (SSFM) as a function of the signal-to-noise ratio $\text{SNR}$ and the number of segments in distance $K$ is considered. It is shown that if $K\geq \text{SNR}^{2/3}$ and $\text{SNR} \rightarrow \infty$, the capacity of the resulting continuous-space lossless model is lower bounded by $\frac{1}{2}\log_2(1+\text{SNR}) - \frac{1}{2}+ o(1)$, where $o(1)$ tends to zero with $\text{SNR}$. As $K\rightarrow \infty$, the inter-symbol interference (ISI) averages out to zero due to the law of large numbers a…
This paper focuses on delivering quantified results for the evaluation of the aggregated impact of stochastic antenna misalignment, multipath fading and hardware imperfections on the terahertz (THz) wireless fiber extenders. In this line, we present the appropriate signal model that accommodates the different technical and environmental parameters. In particular, it takes into consideration the antenna gains, the central frequency, the transmission range, the environmental conditions, i.e. temperature, humidity and pressure, the spatial jitter between the transmitter and receiver antennas, whi…
Constellation shaping is a practical and effective technique to improve the performance and the rate adaptivity of optical communication systems. In principle, it could also be used to mitigate the impact of nonlinear effects, possibly increasing the information rate beyond the current limit dictated by fiber nonlinearity. However, this appealing idea is frustrated by the difficulty of designing an effective shaping strategy that takes into account the nonlinearity and long memory of the fiber channel, as well as the possible interplay with other nonlinearity mitigation strategies. As a result…
The optimal decoder achieving the outage capacity under imperfect channel estimation is investigated. First, by searching into the family of nearest neighbor decoders, which can be easily implemented on most practical coded modulation systems, we derive a decoding metric that minimizes the average of the transmission error probability over all channel estimation errors. This metric, for arbitrary memoryless channels, achieves the capacity of a composite (more noisy) channel. Next, according to the notion of estimation-induced outage capacity (EIO capacity) introduced in our previous work, we c…
We study the capacity regions of two-user interference channels where transmitters base their transmission schemes on local views of the channel state. Under the local view model, each transmitter knows only a subset of the four channel gains, which may be mismatched from the other transmitter. We consider a set of seven local views, and find that for five out of the seven local views, TDM is sufficient to achieve the qualified notion of capacity region for the linear deterministic interference channel which approximates the Gaussian interference channel. For these five local views, the qualif…
The optimal decoder achieving the outage capacity under imperfect channel estimation is investigated. First, by searching into the family of nearest neighbor decoders, which can be easily implemented on most practical coded modulation systems, we derive a decoding metric that minimizes the average of the transmission error probability over all channel estimation errors. Next, we specialize our general expression to obtain the corresponding decoding metric for fading MIMO channels. According to the notion of estimation-induced outage (EIO) capacity introduced in our previous work and assuming n…
MIMO processing techniques in fiber optical communications have been proposed as a promising approach to meet increasing demand for information throughput. In this context, the multiple channels correspond to the multiple modes and/or multiple cores in the fiber. In this paper we characterize the distribution of the mutual information with Gaussian input in a simple channel model for this system. Assuming significant cross talk between cores, negligible backscattering and near-lossless propagation in the fiber, we model the transmission channel as a random complex unitary matrix. The loss in t…
We study the capacity region of two-user erasure interference channels with local delayed channel state information at the transmitters. In our model, transmitters have local mismatched outdated knowledge of the channel gains. We propose a transmission strategy that only relies on the delayed knowledge of the {\it outgoing} links at each transmitter and achieves the outer-bound for the scenario in which transmitters learn the entire channel state with delay. Our result reveals the subset of the channel state information that affects the capacity region the most. We also identify cases in which…
This paper compares and evaluates a set of non-parametric mutual information estimators with the goal of providing a novel toolset to progress in the analysis of the capacity of the nonlinear optical channel, which is currently an open problem. In the first part of the paper, the methods of the study are presented. The second part details their application to several optically-related channels to highlight their features.
The achievable information rates of optical communication networks have been widely increased over the past four decades with the introduction and development of optical amplifiers, coherent detection, advanced modulation formats, and digital signal processing techniques. These developments promoted the revolution of optical communication systems and the growth of Internet, towards the direction of high-capacity and long-distance transmissions. The performance of long-haul high-capacity optical fiber communication systems is significantly degraded by transmission impairments, such as chromatic…
A fiber optic channel is modeled in a variety of ways; from the simple additive white complex Gaussian noise model, to models that incorporate memory in the channel. Because of Kerr nonlinearity, a simple model is not a good approximation to an optical fiber. Hence we study a fiber optic channel with finite memory and provide an achievable bound on channel capacity that improves upon a previously known bound.
A memoryless interference network model is introduced that is based on non-linear phenomena observed when transmitting information over optical fiber using wavelength division multiplexing. The main characteristic of the model is that amplitude variations on one carrier wave are converted to phase variations on another carrier wave, i.e., the carriers interfere with each other through amplitude-to-phase conversion. For the case of two carriers, a new technique called interference focusing is proposed where each carrier achieves the capacity pre-log 1, thereby doubling the pre-log of 1/2 achiev…
Capacity is the eternal pursuit for communication systems due to the overwhelming demand of bandwidth hungry applications. As the backbone infrastructure of modern communication networks, the optical fiber transmission system undergoes a significant capacity growth over decades by exploiting available physical dimensions (time, frequency, quadrature, polarization and space) of the optical carrier for multiplexing. For each dimension, stringent orthogonality must be guaranteed for perfect separation of independent multiplexed signals. To catch up with the ever-increasing capacity requirement, i…
A new variant of bit interleaved coded modulation (BICM) is proposed. In the new scheme, called Parallel BICM, L identical binary codes are used in parallel using a mapper, a newly proposed finite-length interleaver and a binary dither signal. As opposed to previous approaches, the scheme does not rely on any assumptions of an ideal, infinite-length interleaver. Over a memoryless channel, the new scheme is proven to be equivalent to a binary memoryless channel. Therefore the scheme enables one to easily design coded modulation schemes using a simple binary code that was designed for that binar…
In multi-(core/mode) optical fiber communication, the transmission channel can be modeled as a complex sub-matrix of the Haar-distributed unitary matrix (complex Jacobi unitary ensemble). In this letter, we present new analytical expressions of the upper and lower bounds for the ergodic capacity of multiple-input multiple-output Jacobi-fading channels. Recent results on the determinant of the Jacobi unitary ensemble are employed to derive a tight lower bound on the ergodic capacity. We use Jensen's inequality to provide an analytical closed-form upper bound to the ergodic capacity at any signa…
An upper bound on the capacity of a cascade of nonlinear and noisy channels is presented. The cascade mimics the split-step Fourier method for computing waveform propagation governed by the stochastic generalized nonlinear Schroedinger equation. It is shown that the spectral efficiency of the cascade is at most log(1+SNR), where SNR is the receiver signal-to-noise ratio. The results may be applied to optical fiber channels. However, the definition of bandwidth is subtle and leaves open interpretations of the bound. Some of these interpretations are discussed.
We consider a model nondispersive nonlinear optical fiber channel with additive white Gaussian noise at large $\mathrm{SNR}$ (signal-to-noise ratio) in the intermediate power region. Using Feynman path-integral technique we for the first time find the optimal input signal distribution maximizing the channel's per-sample mutual information. The finding of the optimal input signal distribution allows us to improve previously known estimates for the channel capacity. The output signal entropy, conditional entropy, and per-sample mutual information are calculated for Gaussian, half-Gaussian and mo…
The mutual information between a complex-valued channel input and its complex-valued output is decomposed into four parts based on polar coordinates: an amplitude term, a phase term, and two mixed terms. Numerical results for the additive white Gaussian noise (AWGN) channel with various inputs show that, at high signal-to-noise ratio (SNR), the amplitude and phase terms dominate the mixed terms. For the AWGN channel with a Gaussian input, analytical expressions are derived for high SNR. The decomposition method is applied to partially coherent channels and a property of such channels called "s…
