Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver. There are NO constraints on the distribution of the transformation matrix and the field size.
Specifically, the optimality of subspace coding over LOCs is investigated. A lower bound on the maximum achievable rate of subspace coding is obtained and it is shown to be tight for some cases. The maximum achievable rate of constant-dimensional subspace coding is characterized and the loss of rate incurred by using constant-dimensional subspace coding is insignificant.
The maximum achievable rate of channel training is close to the lower bound on the maximum achievable rate of subspace coding. Two coding approaches based on channel training are proposed and their performances are evaluated. Our first approach makes use of rank-metric codes and its optimality depends on the existence of maximum rank distance codes. Our second approach applies linear coding and it can achieve the maximum achievable rate of channel training. Our code designs require only the knowledge of the expectation of the rank of the transformation matrix. The second scheme can also be realized ratelessly without a priori knowledge of the channel statistics.
Shannon's mutual information of a random multiple antenna and multipath channel is studied in the general case where the channel impulse response is an ergodic and stationary process which is assumed to be available at the receiver. From this viewpoint, the channel is represented by an ergodic self-adjoint block-Jacobi operator, which is close in many aspects to a block version of a random Schrödinger operator. The mutual information is then related to the so-called density of states of this operator. In this paper, it is shown that under the weakest assumptions on the channel, the mutual info…
Recent research indicates that packet transmission employing random linear network coding can be regarded as transmitting subspaces over a linear operator channel (LOC). In this paper we propose the framework of linear operator broadcast channels (LOBCs) to model packet broadcasting over LOCs, and we do initial work on the capacity region of constant-dimension multiplicative LOBCs(CMLOBCs), a generalization of broadcast erasure channels. Two fundamental problems regarding CMLOBCs are addressed-finding necessary and sufficient conditions for degradation and deciding whether time sharing suffice…
We consider the discrete, time-varying broadcast channel with memory under the assumption that the channel states belong to a set of finite cardinality. We first define the physically degraded finite-state broadcast channel for which we derive the capacity region. We then define the stochastically degraded finite-state broadcast channel and derive the capacity region for this scenario as well. In both scenarios we consider the non-indecomposable finite-state channel as well as the indecomposable one.
A fundamental problem in any communication system is: given a communication channel between a transmitter and a receiver, how many "independent" signals can be exchanged between them? Arbitrary communication channels that can be described by linear compact channel operators mapping between normed spaces are examined in this paper. The (well-known) notions of degrees of freedom at level $ε$ and essential dimension of such channels are developed in this general setting. We argue that the degrees of freedom at level $ε$ and the essential dimension fundamentally limit the number of independent sig…
A single-letter characterization is provided for the capacity region of finite-state multiple access channels. The channel state is a Markov process, the transmitters have access to delayed state information, and channel state information is available at the receiver. The delays of the channel state information are assumed to be asymmetric at the transmitters. We apply the result to obtain the capacity region for a finite-state Gaussian MAC, and for a finite-state multiple-access fading channel. We derive power control strategies that maximize the capacity region for these channels.
In this paper, we propose capacity-achieving communication schemes for Gaussian finite-state Markov channels (FSMCs) subject to an average channel input power constraint, under the assumption that the transmitters can have access to delayed noiseless output feedback as well as instantaneous or delayed channel state information (CSI). We show that the proposed schemes reveals connections between feedback communication and feedback control.
We derive a single-letter formula for the zero-rate reliability (error exponent) of a finite-state channel whose state variable depends deterministically (and recursively) on past channel inputs, where the code complies with a given channel input constraint. Special attention is then devoted to the important special case of the Gaussian channel with inter-symbol interference (ISI), where more explicit results are obtained.
We study the determination problem of the channel capacity for the discrete memoryless channels in the finite blocklength regime. We derive explicit lower and upper bounds of the capacity. We shall demonstrate that the information spectrum approach is quite useful for investigating this problem.
Channel Estimation for Flexible Intelligent Metasurfaces: From Model-Based Approaches to Neural Operators
Central China Normal University（华中师范大学）；Beijing University of Posts and Telecommunications（北京邮电大学）；China Unicom Group（中国联通集团）；Tsinghua University（清华大学）；Henan Polytechnic University（河南理工大学）；Nanyang Technological University（南洋理工大学）
We derive novel upper and lower finite-length bounds of the error probability in joint source-channel coding when the source obeys an ergodic Markov process and the channel is a Markovian additive channel or a Markovian conditional additive channel. These bounds are tight in the large and moderate deviation regimes.
We consider a finite-state memoryless channel with i.i.d. channel state and the input Markov process supported on a mixing finite-type constraint. We discuss the asymptotic behavior of entropy rate of the output hidden Markov chain and deduce that the mutual information rate of such a channel is concave with respect to the parameters of the input Markov processes at high signal-to-noise ratio. In principle, the concavity result enables good numerical approximation of the maximum mutual information rate and capacity of such a channel.
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.
We consider Additive White Gaussian Noise channels and Discrete Memoryless channels when the transmitter harvests energy from the environment. These can model wireless sensor networks as well as Internet of Things. By providing a unifying framework that works for any energy harvesting channel, we study these channels assuming an infinite energy buffer and provide the corresponding achievability and converse bounds on the channel capacity in the finite blocklength regime. We additionally provide moderate deviation asymptotic bounds as well.
In a multi-user scenario where users belong to different operators, any interference mitigation method needs unavoidably some degree of cooperation among service providers. In this paper we propose a cooperation strategy based on the exchange of mutual interference among operators, rather than of decoded data, to let every operator to recover an augmented degree of diversity either for channel estimation and multi-user detection. In xDSL scenario where multiple operators share the same cable binder the interference-cooperation (IC) approach outperforms data-exchange methods and preserves to ce…
Nanopore sequencing is an emerging DNA sequencing technology that has been proposed for use in DNA storage systems. We propose the noisy nanopore channel model for nanopore sequencing. This model captures duplications, inter-symbol interference, and noisy measurements by concatenating an i.i.d. duplication channel with a finite-state semi-Markov channel. Compared to previous models, this channel models the dominant distortions of the nanopore while remaining tractable. Anticipating future coding schemes, we derive MAP detection algorithms and estimate achievable rates. Given that finite-state …
Inspired by the ideas from the field of stochastic approximation, we propose a randomized algorithm to compute the capacity of a finite-state channel with a Markovian input. When the mutual information rate of the channel is concave with respect to the chosen parameterization, we show that the proposed algorithm will almost surely converge to the capacity of the channel and derive the rate of convergence. We also discuss the convergence behavior of the algorithm without the concavity assumption.
We propose two modified versions of the classical gradient ascent method to compute the capacity of finite-state channels with Markovian inputs. For the case that the channel mutual information is strongly concave in a parameter taking values in a compact convex subset of some Euclidean space, our first algorithm proves to achieve polynomial accuracy in polynomial time and, moreover, for some special families of finite-state channels our algorithm can achieve exponential accuracy in polynomial time under some technical conditions. For the case that the channel mutual information may not be str…
An achievable rate region, based on lattice interference alignment, is derived for a class of time-invariant Gaussian interference channels with more than two users. The result is established via a new coding theorem for the two-user Gaussian multiple-access channel where both users use a single linear code. The class of interference channels treated is such that all interference channel gains are rational. For this class of interference channels, beyond recovering the known results on the degrees of freedom, an explicit rate region is derived for finite signal-to-noise ratios, shedding light …
We consider an additive White Gaussian channel where the transmitter is powered by an energy harvesting source. For such a system, we provide a lower bound on the maximal code book at finite code lengths that improves upon previously known bounds.
Finite blocklength and second-order (dispersion) results are presented for the arbitrarily-varying channel (AVC), a classical model wherein an adversary can transmit arbitrary signals into the channel. A novel finite blocklength achievability bound is presented, roughly analogous to the random coding union bound for non-adversarial channels. This finite blocklength bound, along with a known converse bound, is used to derive bounds on the dispersion of discrete memoryless AVCs without shared randomness, and with cost constraints on the input and the state. These bounds are tight for many channe…
A converse for the Discrete Memoryless Multiple Access Channel is given. The result in [13] is refined, and the third order term is obtained. Moreover, our proof is much simpler than [13]. With little modification, the region can be further improved.
This paper studies the three-user finite-field multi-way relay channel, where the users exchange messages via a relay. The messages are arbitrarily correlated, and the finite-field channel is linear and is subject to additive noise of arbitrary distribution. The problem is to determine the minimum achievable source-channel rate, defined as channel uses per source symbol needed for reliable communication. We combine Slepian-Wolf source coding and functional-decode-forward channel coding to obtain the solution for two classes of source and channel combinations. Furthermore, for correlated source…
This article studies the zero-error feedback capacity of {\em causal} discrete channels with memory. First, by extending the classical zero-error feedback capacity concept, a new notion of {\em uniform zero-error feedback capacity} $ C_{0f} $ for such channels is introduced. Using this notion a tight condition for {bounded} stabilization of unstable {noisy} linear systems via causal channels is obtained, assuming no {channel} state information at either end of the channel.
A single-letter characterization is provided for the capacity region of finite-state multiple-access channels, when the channel state process is an independent and identically distributed sequence, the transmitters have access to partial (quantized) state information, and complete channel state information is available at the receiver. The partial channel state information is assumed to be asymmetric at the encoders. As a main contribution, a tight converse coding theorem is presented. The difficulties associated with the case when the channel state has memory are discussed and connections to …
We study the maximal achievable rate R*(n, ε) for a given block-length n and block error probability εover Rayleigh block-fading channels in the noncoherent setting and in the finite block-length regime. Our results show that for a given block-length and error probability, R*(n, ε) is not monotonic in the channel's coherence time, but there exists a rate maximizing coherence time that optimally trades between diversity and cost of estimating the channel.
This paper investigates the maximal secrecy rate over a wiretap channel subject to reliability and secrecy constraints at a given blocklength. New achievability and converse bounds are derived, which are shown to be tighter than existing bounds. The bounds also lead to the tightest second-order coding rate for discrete memoryless and Gaussian wiretap channels.
The error exponent of Markov channels with feedback is studied in the variable-length block-coding setting. Burnashev's classic result is extended and a single letter characterization for the reliability function of finite-state Markov channels is presented, under the assumption that the channel state is causally observed both at the transmitter and at the receiver side. Tools from stochastic control theory are used in order to treat channels with intersymbol interference. In particular the convex analytical approach to Markov decision processes is adopted to handle problems with stopping time…
We study the performance of nonbinary low-density parity-check (LDPC) codes over finite integer rings over two channels that arise from the Lee metric. The first channel is a discrete memory-less channel (DMC) matched to the Lee metric. The second channel adds to each codeword an error vector of constant Lee weight, where the error vector is picked uniformly at random from the set of vectors of constant Lee weight. It is shown that the marginal conditional distributions of the two channels coincide, in the limit of large block length. Random coding union bounds on the block error probability a…
We consider the time-continuous doubly-dispersive channel with additive Gaussian noise and establish a capacity formula for the case where the channel correlation operator is represented by a symbol which is periodic in time and fulfills some further integrability and smoothness conditions. The key to this result is a new Szegö formula for certain pseudo-differential operators. The formula justifies the water-filling principle along time and frequency in terms of the time--continuous time-varying transfer function (the symbol).
In this work, we obtain lower and upper bounds on the maximal transmission rate at a given codeword length $n$, average probability of error $ε$ and power constraint $\bar{P}$, over a finite valued, block fading additive white Gaussian noise (AWGN) channel with channel state information (CSI) at the transmitter and the receiver. These bounds characterize deviation of the finite blocklength coding rates from the channel capacity which is in turn achieved by the water filling power allocation across time. The bounds obtained also characterize the rate enhancement possible due to the CSI at the t…
We consider the capacity of memoryless finite-state multiple access channel (FS-MAC) with causal asymmetric noisy state information available at both transmitters and complete state information available at the receiver. Single letter inner and outer bounds are provided for the capacity of such channels when the state process is independent and identically distributed. The outer bound is attained by observing that the proposed inner bound is tight for the sum-rate capacity.
This work explores how degrees of freedom (DoF) results from wireless networks can be translated into capacity results for their finite field counterparts that arise in network coding applications. The main insight is that scalar (SISO) finite field channels over $\mathbb{F}_{p^n}$ are analogous to n x n vector (MIMO) channels in the wireless setting, but with an important distinction -- there is additional structure due to finite field arithmetic which enforces commutativity of matrix multiplication and limits the channel diversity to n, making these channels similar to diagonal channels in t…
The finite state Markov channel (FSMC), where the channel transition probability is controlled by a state undergoing a Markov process, is a useful model for the mobile wireless communication channel. In this paper, we investigate the security issue in the mobile wireless communication systems by considering the FSMC with an eavesdropper, which we call the finite state Markov wiretap channel (FSM-WC). We assume that the state is perfectly known by the legitimate receiver and the eavesdropper, and through a noiseless feedback channel, the legitimate receiver sends his received channel output and…
The study investigated the allocation of transmission power and bits for a point-to-point orthogonal frequency-division multiplexing channel assuming perfect channel information at the receiver, but imperfect channel information at the transmitter. Channel information was quantized at the receiver and was sent back to the transmitter via a finite-rate feedback channel. Based on limited feedback from the receiver, the corresponding transmitter adapted the power level and/or modulation across subcarriers. To reduce the amount of feedback, subcarriers were partitioned into different clusters and …
We present nonasymptotic bounds on the maximum coding rate achievable over a Rician block-fading channel for a fixed packet size and a fixed packet error probability. Our bounds, which apply to the scenario where no a priori channel state information is available at the receiver, allow one to quantify the tradeoff between the rate gains resulting from the exploitation of time-frequency diversity and the rate loss resulting from fast channel variations and pilot-symbol overhead.
In previous work, an ordering result was given for the symbolwise probability of error using general Markov channels, under iterative decoding of LDPC codes. In this paper, the ordering result is extended to mutual information, under the assumption of an iid input distribution. For certain channels, in which the capacity-achieving input distribution is iid, this allows ordering of the channels by capacity. The complexity of analyzing general Markov channels is mitigated by this ordering, since it is possible to immediately determine that a wide class of channels, with different numbers of stat…
Motivated by physical-layer network coding, this paper considers communication in multiplicative matrix channels over finite chain rings. Such channels are defined by the law $Y =A X$, where $X$ and $Y$ are the input and output matrices, respectively, and $A$ is called the transfer matrix. It is assumed a coherent scenario in which the instances of the transfer matrix are unknown to the transmitter, but available to the receiver. It is also assumed that $A$ and $X$ are independent. Besides that, no restrictions on the statistics of $A$ are imposed. As contributions, a closed-form expression fo…
This paper elaborates on the achievable schemes of interference alignment in constant channels via finite amplify-and-forward (AF) relays. Consider $K$ sources communicating with $K$ destinations without direct links besides the relay connections. The total number of relays is finite. The objective is to achieve interference alignment for all user pairs to obtain half of their interference-free degrees of freedom. In general, two strategies are employed: coding at the edge and coding in the middle, in which relays show different roles. The contributions are that two fundamental and critical el…
This paper studies the capacities of input-driven finite-state channels, i.e., channels whose current state is a time-invariant deterministic function of the previous state and the current input. We lower bound the capacity of such a channel using a dynamic programming formulation of a bound on the maximum reverse directed information rate. We show that the dynamic programming-based bounds can be simplified by solving the corresponding Bellman equation explicitly. In particular, we provide analytical lower bounds on the capacities of $(d, k)$-runlength-limited input-constrained binary symmetri…
In this note, we show that the operator theoretic concept of Kolmogorov numbers and the number of degrees of freedom at level $ε$ of a communication channel are closely related. Linear communication channels may be modeled using linear compact operators on Banach or Hilbert spaces and the number of degrees of freedom of such channels is defined to be the number of linearly independent signals that may be communicated over this channel, where the channel is restricted by a threshold noise level. Kolmogorov numbers are a particular example of $s$-numbers, which are defined over the class of boun…
As the mobile application landscape expands, wireless networks are tasked with supporting various connection profiles, including real-time communications and delay-sensitive traffic. Among many ensuing engineering challenges is the need to better understand the fundamental limits of forward error correction in non-asymptotic regimes. This article seeks to characterize the performance of block codes over finite-state channels with memory. In particular, classical results from information theory are revisited in the context of channels with rate transitions, and bounds on the probabilities of de…
Though network coding is traditionally performed over finite fields, recent work on nested-lattice-based network coding suggests that, by allowing network coding over certain finite rings, more efficient physical-layer network coding schemes can be constructed. This paper considers the problem of communication over a finite-ring matrix channel $Y = AX + BE$, where $X$ is the channel input, $Y$ is the channel output, $E$ is random error, and $A$ and $B$ are random transfer matrices. Tight capacity results are obtained and simple polynomial-complexity capacity-achieving coding schemes are provid…
The generalized degrees of freedom (GDoF) characterization of the symmetric K-user interference channel is obtained under finite precision channel state information at the transmitters (CSIT). The symmetric setting is where each cross channel is capable of carrying degrees of freedom (DoF) while each direct channel is capable of carrying 1 DoF. Remarkably, under finite precision CSIT the symmetric K-user interference channel loses all the GDoF benefits of interference alignment. The GDoF per user diminish with the number of users everywhere except in the very strong (optimal for every receiver…
We present finite-blocklength achievability bounds for the unsourced A-channel. In this multiple-access channel, users noiselessly transmit codewords picked from a common codebook with entries generated from a $q$-ary alphabet. At each channel use, the receiver observes the set of different transmitted symbols but not their multiplicity. We show that the A-channel finds applications in unsourced random-access (URA) and group testing. Leveraging the insights provided by the finite-blocklength bounds and the connection between URA and non-adaptive group testing through the A-channel, we propose …
In this paper, the Dynamic-Assignment Blahut-Arimoto (DAB) algorithm identifies finite-support probability mass functions (PMFs) with small cardinality that achieve capacity for amplitude-constrained (AC) Additive White Gaussian Noise (AWGN) Channels, or approach capacity to within less than 1% for power-constrained (PC) AWGN Channels. While a continuous Gaussian PDF is well-known to be a theoretical capacity-achieving distribution for the PC-AWGN channel, DAB identifies PMFs with small-cardinality that are, for practical purposes, indistinguishable in performance. We extend the results of Oza…
In this paper, we consider the problem of power allocation in MIMO wiretap channel for secrecy in the presence of multiple eavesdroppers. Perfect knowledge of the destination channel state information (CSI) and only the statistical knowledge of the eavesdroppers CSI are assumed. We first consider the MIMO wiretap channel with Gaussian input. Using Jensen's inequality, we transform the secrecy rate max-min optimization problem to a single maximization problem. We use generalized singular value decomposition and transform the problem to a concave maximization problem which maximizes the sum secr…
The three-user finite field multi-way relay channel with correlated sources is considered. The three users generate possibly correlated messages, and each user is to transmit its message to the two other users reliably in the Shannon sense. As there is no direct link among the users, communication is carried out via a relay, and the link from the users to the relay and those from the relay to the users are finite field adder channels with additive noise of arbitrary distribution. The problem is to determine the set of all possible achievable rates, defined as channel uses per source symbol for…
The capacity region of the Finite-State Multiple Access Channel (FS-MAC) with feedback that may be an arbitrary time-invariant function of the channel output samples is considered. We provided a sequence of inner and outer bounds for this region. These bounds are shown to coincide, and hence yield the capacity region, of FS-MACs where the state process is stationary and ergodic and not affected by the inputs, and for indecomposable FS-MAC when feedback is not allowed. Though the capacity region is `multi-letter' in general, our results yield explicit conclusions when applied to specific scenar…
The exponential server timing channel is known to be the simplest, and in some sense canonical, queuing timing channel. The capacity of this infinite-memory channel is known. Here, we discuss practical finite-length restrictions on the codewords and attempt to understand the maximal rate that can be achieved for a target error probability. By using Markov chain analysis, we prove a lower bound on the maximal channel coding rate achievable at blocklength $n$ and error probability $ε$. The bound is approximated by $C- n^{-1/2} σQ^{-1}(ε)$ where $Q$ denotes the Q-function and $σ^2$ is the asympto…
In this work we study the secrecy capacity of Gaussian multiple-input multiple-output (MIMO) wiretap channels (WTCs) with a finite memory, subject to a per-symbol average power constraint on the MIMO channel input. MIMO channels with finite memory are very common in wireless communications as well as in wireline communications (e.g., in communications over power lines). To derive the secrecy capacity of the Gaussian MIMO WTC with finite memory we first construct an asymptotically-equivalent block-memoryless MIMO WTC, which is then transformed into a set of parallel, independent, memoryless MIM…
Recently, it has been shown that the time-varying multiple-access channel (MAC) with perfect channel state information (CSI) at the receiver and delayed feedback CSI at the transmitters can be modeled as the finite state MAC (FS-MAC) with delayed state feedback, where the time variation of the channel is characterized by the statistics of the underlying state process. To study the fundamental limit of the secure transmission over multi-user wireless communication systems, we re-visit the FS-MAC with delayed state feedback by considering an external eavesdropper, which we call the finite state …
In this paper, a free deterministic equivalent is proposed for the capacity analysis of the multi-input multi-output (MIMO) multiple access channel (MAC) with a more general channel model compared to previous works. Specifically, a MIMO MAC with one base station (BS) equipped with several distributed antenna sets is considered. Each link between a user and a BS antenna set forms a jointly correlated Rician fading channel. The analysis is based on operator-valued free probability theory, which broadens the range of applicability of free probability techniques tremendously. By replacing independ…
The secrecy capacity of a discrete memoryless Gaussian Wire-Tap Channel when the input is from a finite complex constellation is studied. It is shown that the secrecy capacity curves of a finite constellation plotted against the SNR, for a fixed noise variance of the eavesdropper's channel has a global maximum at an internal point. This is in contrast to what is known in the case of Gaussian codebook input where the secrecy capacity curve is a bounded, monotonically increasing function of SNR. Secrecy capacity curves for some well known constellations like BPSK, 4-QAM, 16-QAM and 8-PSK are plo…
In this paper, the linear programming (LP) decoder for binary linear codes, introduced by Feldman, et al. is extended to joint-decoding of binary-input finite-state channels. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the LP joint decoder also has the maximum-likelihood (ML) certificate property and all in…
In this paper, we investigate the design of linear precoders for the multiple-input multiple-output (MIMO) multiple access channel (MAC). We assume that statistical channel state information (CSI) is available at the transmitters and consider the problem under the practical finite alphabet input assumption. First, we derive an asymptotic (in the large system limit) expression for the weighted sum rate (WSR) of the MIMO MAC with finite alphabet inputs and Weichselberger's MIMO channel model. Subsequently, we obtain the optimal structures of the linear precoders of the users maximizing the asymp…
