Rate-limited Links Research Articles

Negligible Cooperation: Contrasting the Maximal- and Average-Error Cases

In communication networks, cooperative strategies are coding schemes where network nodes work together to improve performance metrics such as the total rate delivered across the network. This work studies encoder cooperation in the setting of a discrete multiple access channel (MAC) with two encoders and a single decoder. A network node, here called the cooperation facilitator (CF), that is connected to both encoders via rate-limited links, enables the cooperation strategy. Previous work by the authors presents two classes of MACs: (i) one class where the average-error sum-capacity has an infinite derivative in the limit where CF output link capacities approach zero, and (ii) a second class of MACs where the maximal-error sum-capacity is not continuous at the point where the output link capacities of the CF equal zero. This work contrasts the power of the CF in the maximal- and average-error cases, showing that a constant number of bits communicated over the CF output link can yield a positive gain in the maximal-error sum-capacity, while a far greater number of bits, even a number that grows sublinearly in the blocklength, can never yield a non-negligible gain in the average-error sum-capacity.

Distributed Hypothesis Testing: Cooperation and Concurrent Detection

A single-sensor two-detectors system is considered where the sensor communicates with both detectors and Detector 1 communicates with Detector 2, all over noise-free rate-limited links. The sensor and both detectors observe discrete memoryless source sequences whose joint probability mass function depends on a binary hypothesis. The goal at each detector is to guess the binary hypothesis in a way that, for increasing observation lengths, the probability of error under one of the hypotheses decays to zero with largest possible exponential decay, whereas the probability of error under the other hypothesis can decay to zero or to a small positive number arbitrarily slow. For the setting with positive communication rates from the sensor to the detectors and when both detectors are interested in maximizing the error exponent under the same hypothesis, we characterize the set of all possible exponents in a special case of testing against independence. In this case the cooperation link allows Detector 2 to increase its Type-II error exponent by an amount that is equal to the exponent attained at Detector 1. We also provide a general inner bound on the set of achievable error exponents that shows a tradeoff between the exponents at the two detectors in most cases. When the two detectors aim at maximizing the error exponent under different hypotheses and the distribution at the Sensor is different under the two hypotheses, then we show that such a tradeoff does not exist. We propose a general scheme that allows each detector to attain the same exponent as if it was the only detector in the system. For the setting with zero-rate communication on both links, we exactly characterize the set of possible exponents and the gain brought up by cooperation, in function of the number of bits that are sent over the two links. Notice that, for this setting, tradeoffs between the exponents achieved at the two detectors arise only in few particular cases. In all other cases, each detector achieves the same performance as if it were the only detector in the system.

Multiplexing Gains under Mixed-Delay Constraints on Wyner's Soft-Handoff Model.

This paper analyzes the multiplexing gains (MG) achievable over Wyner’s soft-handoff model under mixed-delay constraints, that is, when delay-sensitive and delay-tolerant data are simultaneously transmitted over the network. In the considered model, delay-sensitive data cannot participate or profit in any ways from transmitter or receiver cooperation, but delay-tolerant data can. Cooperation for delay-tolerant data takes place over rate-limited links and is limited to a fixed number of cooperation rounds. For the described setup, inner and outer bounds are derived on the set of MG pairs that are simultaneously achievable for delay-sensitive and delay-tolerant data. The bounds are tight in special cases and allow us to obtain the following conclusions. For large cooperation rates, and when both transmitters and receivers can cooperate, it is possible to simultaneously attain maximum MG for delay-sensitive messages and maximum sum MG for all messages. For comparison, in scheduling schemes (also called time-sharing schemes), the largest achievable sum MG decreases linearly with the MG of delay-sensitive messages. A similar linear decrease is proved for any coding scheme, not only for scheduling schemes, if only transmitters or only receivers can cooperate (but not both) and if delay-sensitive messages have moderate MG. In contrast, if the MG of delay-sensitive messages is small, the maximum sum MG can be achieved even with only transmitter or only receiver cooperation. To summarise, when cooperation rates are high and both transmitters and receivers can cooperate or when delay-sensitive messages have small MG, then transmitting delay-sensitive messages causes no penalty on the sum-MG. In other regimes, this penalty increases proportionally to the delay-tolerant MG in the sense that increasing the delay-sensitive MG by penalises the largest achievable delay-tolerant MG by and thus the sum MG by .

Integer-Forcing Architectures for Uplink Cloud Radio Access Networks

Consider an uplink cloud radio access network where users are observed simultaneously by several base stations, each with a rate-limited link to a central processor, which wishes to decode all transmitted messages. Recent efforts have demonstrated the advantages of compression-based strategies that send quantized channel observations to the central processor, rather than attempt local decoding. We study the setting where channel state information is not available at the transmitters, but known fully or partially at the base stations. We propose an end-to-end integer-forcing framework for compression-based uplink cloud radio access, and show that it operates within a constant gap from the optimal outage probability if channel state information is fully available at the base stations. We demonstrate via simulations that our framework is competitive with state-of-the-art Wyner-Ziv-based strategies.

Coded Caching for Heterogeneous Systems: An Optimization Perspective

In cache-aided networks, the server populates the cache memories at the users during low-traffic periods in order to reduce the delivery load during peak-traffic hours. In turn, there exists a fundamental tradeoff between the delivery load on the server and the cache sizes at the users. In this paper, we study this tradeoff in a multicast network, where the server is connected to users with unequal cache sizes and the number of users is less than or equal to the number of library files. We propose centralized uncoded placement and linear delivery schemes which are optimized by solving a linear program. Additionally, we derive a lower bound on the delivery memory tradeoff with uncoded placement that accounts for the heterogeneity in cache sizes. We explicitly characterize this tradeoff for the case of three end-users, as well as an arbitrary number of end-users when the total memory size at the users is small, and when it is large. Next, we consider a system where the server is connected to the users via rate-limited links of different capacities and the server assigns the users’ cache sizes subject to a total cache budget. We characterize the optimal cache sizes that minimize the delivery completion time with uncoded placement and linear delivery. In particular, the optimal memory allocation balances between assigning larger cache sizes to users with low capacity links and uniform memory allocation.

When to preempt? Age of information minimization under link capacity constraint

In this paper, we consider a scenario where a source continuously monitors an object and sends time-stamped status updates to a destination through a rate-limited link. We assume updates arrive randomly at the source according to a Bernoulli process. Due to the link capacity constraint, it takes multiple time slots for the source to complete the transmission of an update. Therefore, when a new update arrives at the source during the transmission of another update, the source needs to decide whether to skip the new arrival or to switch to it, in order to minimize the expected average age of information (AoI) at the destination. We start with the setting where all updates are of the same size, and prove that within a broadly defined class of online policies, the optimal policy should be a renewal policy, and has a sequential switching property. We then show that the optimal decision of the source in any time slot has threshold structures, and only depends on the age of the update being transmitted and the AoI at the destination. We then consider the setting where updates are of different sizes, and show that the optimal Markovian policy also has a multiple-threshold structure. For each of the settings, we explicitly identify the thresholds by formulating the problem as a Markov Decision Process (MDP), and solve it through value iteration. Special structural properties of the corresponding optimal policy are utilized to reduce the computational complexity of the value iteration algorithm.

On Achievability for Downlink Cloud Radio Access Networks With Base Station Cooperation

This paper investigates the downlink of a cloud radio access network (C-RAN) in which a central processor communicates with two mobile users through two base stations (BSs). The BSs act as relay nodes and cooperate with each other through error-free rate-limited links. We develop and analyze two coding schemes for this scenario. The first coding scheme modifies the Liu–Kang scheme (to make it amenable to a rigorous analysis) and extends it to introduce common codewords and to apply for downlink C-RAN with BS-to-BS cooperation. This first coding scheme enables arbitrary correlation among the auxiliary codewords that are recovered by the BSs. We show that this scheme improves over previous schemes for various instances of Gaussian C-RAN channels. In particular, in many scenarios, the scheme can better exploit the possibility of BS-to-BS cooperation than other schemes. The second coding scheme extends the distributed decode and forward (DDF) scheme by means of Gray–Wyner compression and by exploiting the cooperation links between BSs. In addition and as a separate extension, we provide an improved capacity approximation for the DDF strategy for the capacity of a general $N$ -BS $L$ -user C-RAN model in the memoryless Gaussian case.

The Unbounded Benefit of Encoder Cooperation for the k-User MAC

Cooperation strategies allow communication devices to work together to improve network capacity. Consider a network consisting of $k$ encoders, a multiple access channel (MAC), a decoder, and a node, referred to as a “cooperation facilitator” (CF), that is connected to each encoder via a pair of rate-limited links, with one link going from the encoder to the CF and the other link going back. Let the “cooperation rate” be the total outgoing rate of the CF. This paper demonstrates the existence of a class of MACs where the ratio of the sum-capacity gain to cooperation rate tends to infinity as the cooperation rate tends to zero. For any $k\geq 2$ , examples of channels in this class include the $k$ -user binary adder MAC and the $k$ -user Gaussian MAC.

Rate-Distortion Region of a Gray-Wyner Model with Side Information.

In this work, we establish a full single-letter characterization of the rate-distortion region of an instance of the Gray–Wyner model with side information at the decoders. Specifically, in this model, an encoder observes a pair of memoryless, arbitrarily correlated, sources and communicates with two receivers over an error-free rate-limited link of capacity , as well as error-free rate-limited individual links of capacities to the first receiver and to the second receiver. Both receivers reproduce the source component losslessly; and Receiver 1 also reproduces the source component lossily, to within some prescribed fidelity level . In addition, Receiver 1 and Receiver 2 are equipped, respectively, with memoryless side information sequences and . Important in this setup, the side information sequences are arbitrarily correlated among them, and with the source pair ; and are not assumed to exhibit any particular ordering. Furthermore, by specializing the main result to two Heegard–Berger models with successive refinement and scalable coding, we shed light on the roles of the common and private descriptions that the encoder should produce and the role of each of the common and private links. We develop intuitions by analyzing the developed single-letter rate-distortion regions of these models, and discuss some insightful binary examples.

Multistream Distributed Cophasing

In this paper, we develop a distributed cophasing (DCP) technique for physical layer fusion of multiple data streams in a wireless sensor network with multiple destination nodes (DNs). The DNs can either be connected to a fusion center (referred to as centralized data processing; CDP) or process data independently and communicate with each other via a rate-limited link (referred to as distributed data processing; DDP). In the first stage of this two-stage cophasing scheme, sensors estimate the channel to the DNs using pilot symbols transmitted by the latter; following which they simultaneously transmit multiple streams of data symbols by prerotating them according to the estimated channel phases to the different DNs. The achievable rates for both CDP and DDP are derived to quantify the gains obtainable by the multistream DCP. In order to aid data detection at the receiver, we propose a least-squares-based iterative algorithm for blind channel estimation in CDP–DCP. Following this, we develop a message passing based blind channel estimation algorithm for DDP–DCP. It is found using Monte Carlo simulations that for the CDP system, the proposed blind channel estimation algorithm achieves a probability of error performance very close to that with perfect CSI at the DNs, while using only a moderate number of unknown data symbols for channel estimation. We also derive approximate expressions for the error probability performance of the proposed system for both CDP and DDP and validate their accuracy using Monte Carlo simulations.

Lossless Coding of Correlated Sources With Actions

This paper studies the problem of the distributed compression of correlated sources with an action-dependent joint distribution. This class of problems is, in fact, an extension of the Slepian–Wolf model, but where cost-constrained actions taken by the encoder or the decoder affect the generation of one of the sources. The purpose of this paper is to study the impact of actions on the achievable rates. In particular, two cases where transmission occurs over a rate-limited link are studied; case A for actions taken at the decoder and case B where actions are taken at the encoder. A complete single-letter characterization of the set of achievable rates is given in both cases. Furthermore, a network coding setup for the case where actions are taken at the encoder is investigated. The sources are generated at different nodes of the network and are required at a set of terminal nodes, yet transmission occurs over a general, acyclic, directed network. For this setup, generalized cut-set bounds are derived, and a full characterization of the set of achievable rates using single-letter expressions is provided. For this scenario, random linear network coding is proved to be optimal, even though this is not a classical multicast problem. In addition, two binary examples are investigated and demonstrate how actions taken at different nodes of the system have a significant effect on the achievable rate region, when compared with a naive time-sharing strategy.

GreenInfra: Capacity of Large-Scale Hybrid Networks With Cost-Effective Infrastructure

The cost-effective impact and fundamental limits of infrastructure support with rate-limited wired backhaul links (i.e., GreenInfra support), directly connecting base stations (BSs), are analyzed in a large-scale hybrid network of unit node density, where multiantenna BSs are deployed. We consider a general scenario such that the rate of each BS-to-BS link scales at an arbitrary rate relative to the number of randomly located wireless nodes $n$ . For the operating regimes with respect to the number of BSs and the number of antennas at each BS, we first analyze the minimum rate of each backhaul link $C_{\text{BS}}$ , required to guarantee the same throughput scaling as in the infinite-capacity backhaul link case. We then identify the operating regimes in which the required rate $C_{\text{BS}}$ scales slower than $n^\epsilon$ for an arbitrarily small $\epsilon > 0$ (i.e., the regimes where $C_{\text{BS}}$ does not need to be infinitely large). We also show the case where our network with GreenInfra is fundamentally in the infrastructure-limited regime, in which the performance is limited by the rate of backhaul links. In addition, we derive a generalized throughput scaling law including the case where the rate of each backhaul link scales slower than $C_{\text{BS}}$ . To validate the throughput scaling law for finite values of system parameters, numerical evaluation is also shown via computer simulations.

Multiple Access Channels With Combined Cooperation and Partial Cribbing

In this paper, the multiple access channel (MAC) with combined cooperation and partial cribbing is studied, and its capacity region is characterized. Cooperation means that each of the two encoders sends a message to the other via a rate-limited link prior to transmission, while partial cribbing means that each of the two encoders obtains a deterministic function of the other encoder’s output with or without delay. Prior work in this field dealt separately with cooperation and partial cribbing, but by combining these two methods, we can achieve significantly higher rates. Surprisingly, the capacity region of the MAC with combined cooperation and partial cribbing can be expressed using only one auxiliary random variable (RV) similar to the capacity regions of the MAC with cooperation and with partially cribbing encoders. The reason is that in an optimal coding scheme, the encoders use both cooperation and partial cribbing to generate a common message between the encoders. Furthermore, the Gaussian MAC with combined one-sided cooperation and quantized cribbing is studied. For this model, an achievability scheme is given. This scheme shows how many cooperation or quantization bits are required to practically achieve the capacity region of the Gaussian MAC with full message cooperation or perfect cribbing. To ratify the main results, two additional models are studied. In both models, only one auxiliary RV is needed. The first is a rate distortion dual setting for the MAC with degraded message set and combined cooperation and cribbing. The second is a state-dependent MAC with cooperation, where the state is known at a partially cribbing encoder and at the decoder. However, there are cases where more than one auxiliary RV is needed, e.g., when the cooperation and the cribbing are not used for the same purposes. The MAC with an action-dependent state is presented, where the action is based on the cooperation but not on the cribbing. Therefore, in this case, more than one auxiliary RV is needed. As a result, when the common information shared by the two encoders is used unevenly by the users in the channel, more than one auxiliary RV is needed to express the capacity region.

Joint quantisation levels and power optimisation in uplink network multiple‐input and multiple‐output under constraint backhaul

This article studies the performance of the cooperative system where each base station (BS) is connected to a central processor (CP) via a noiseless and rate-limited backhaul link. The single-user compress-and-forward scheme is deployed at the BSs, and the problem for jointly setting the quantisation noise and the transmit power levels is left open. The authors formulate the weighted sum-rate (WSR) problem of optimising quantisation and transmit power levels for K-users and B-BSs situation. However, the problem is non-convex and difficult to be solved directly by conventional convex methods. In this work, an algorithm is proposed to solve the optimisation problem and achieve its global optimality. More importantly, the authors reveal the reason for the effect of the quantisation levels on the performance of cooperative system. Lower and upper bounds of quantisation levels are derived under the given signal to interference noise ratio (SINR) requirements. The lower and upper bounds are then demonstrated to be rather close to the numerical results. It is also numerically shown that the proposed algorithm can achieve the global optimality and is robust with respect to different multi-user situations.

The Non-Regular CEO Problem

We consider the CEO problem for non-regular source distributions (such as uniform or truncated Gaussian). A group of agents observe independently corrupted versions of data and transmit coded versions over rate-limited links to a CEO. The CEO then estimates the underlying data based on the received coded observations. Agents are not allowed to convene before transmitting their observations. This formulation is motivated by the practical problem of a firm’s CEO estimating (non-regular) beliefs about a sequence of events, before acting on them. Agents’ observations are modeled as jointly distributed with the underlying data through a given conditional probability density function. We study the asymptotic behavior of the minimum achievable mean squared error distortion at the CEO in the limit when the number of agents $L$ and the sum rate $R$ tend to infinity. We establish a $1/R^{2}$ convergence of the distortion, an intermediate regime of performance between the exponential behavior in discrete CEO problems [Berger, Zhang, and Viswanathan (1996)], and the $1/R$ behavior in Gaussian CEO problems [Viswanathan and Berger (1997)]. Achievability is proved by a layered architecture with scalar quantization, distributed entropy coding, and midrange estimation. The converse is proved using the Bayesian Chazan–Zakai–Ziv bound.

The CEO Problem With Secrecy Constraints

We study a lossy source coding problem with secrecy constraints in which a remote information source should be transmitted to a single destination via multiple agents in the presence of a passive eavesdropper. The agents observe noisy versions of the source and independently encode and transmit their observations to the destination via noiseless rate-limited links. The destination should estimate the remote source based on the information received from the agents within a certain mean distortion threshold. The eavesdropper, with access to side information correlated to the source, is able to listen in on one of the links from the agents to the destination in order to obtain as much information as possible about the source. This problem can be viewed as the so-called CEO problem with additional secrecy constraints. We establish inner and outer bounds on the rate-distortion-equivocation region of this problem. We also obtain the region in special cases where the bounds are tight. Furthermore, we study the quadratic Gaussian case and provide the optimal rate-distortion-equivocation region when the eavesdropper has no side information and an achievable region for a more general setup with side information at the eavesdropper.

Distributed and Cascade Lossy Source Coding With a Side Information “Vending Machine”

Source coding with a side information “vending machine” is a recently proposed framework in which the statistical relationship between the side information and the source, instead of being given and fixed as in the classical Wyner-Ziv problem, can be controlled by the decoder. This control action is selected by the decoder based on the message encoded by the source node. Unlike conventional settings, the message can thus carry not only information about the source to be reproduced at the decoder, but also control information aimed at improving the quality of the side information. In this paper, the analysis of the tradeoffs between rate, distortion, and cost associated with the control actions is extended from the previously studied point-to-point setup to two basic multiterminal models. First, a distributed source coding model is studied, in which two encoders communicate over rate-limited links to a decoder, whose side information can be controlled. The control actions are selected by the decoder based on the messages encoded by both source nodes. For this setup, inner bounds are derived on the rate-distortion-cost region for both cases in which the side information is available causally and noncausally at the decoder. These bounds are shown to be tight under specific assumptions, including the scenario in which the sequence observed by one of the nodes is a function of the source observed by the other and the side information is available causally at the decoder. Then, a cascade scenario in which three nodes are connected in a cascade and the last node has controllable side information is also investigated. For this model, the rate-distortion-cost region is derived for general distortion requirements and under the assumption of causal availability of side information at the last node.

Secure Multiterminal Source Coding With Side Information at the Eavesdropper

The problem of secure multiterminal source coding with side information at the eavesdropper is investigated. This scenario consists of a main encoder (referred to as Alice) that wishes to compress a single source but simultaneously satisfying the desired requirements on the distortion level at a legitimate receiver (referred to as Bob) and the equivocation rate—average uncertainty—at an eavesdropper (referred to as Eve). It is further assumed the presence of a (public) rate-limited link between Alice and Bob. In this setting, Eve perfectly observes the information bits sent by Alice to Bob and has also access to a correlated source which can be used as side information. A second encoder (referred to as Charlie) helps Bob in estimating Alice's source by sending a compressed version of its own correlated observation via a (private) rate-limited link, which is only observed by Bob. For instance, the problem at hands can be seen as the unification between the Berger–Tung and the secure source coding setups. Inner and outer bounds on the so-called rate-distortion-equivocation region are derived. The inner region turns to be tight for two cases: 1) uncoded side information at Bob and 2) lossless reconstruction of both sources at Bob—secure distributed lossless compression. Application examples to secure lossy source coding of Gaussian and binary sources in the presence of Gaussian and binary/ternary (respectively) side informations are also considered. Optimal coding schemes are characterized for some cases of interest where the statistical differences between the side information at the decoders and the presence of a nonzero distortion at Bob can be fully exploited to guarantee secrecy.

Secure Source Coding With a Helper

We consider a secure lossless source coding problem with a rate-limited helper. In particular, Alice observes an independent and identically distributed (i.i.d.) source $X^{n}$ and wishes to transmit this source losslessly to Bob over a rate-limited link of capacity not exceeding $R_{x}$ . A helper, say Helen, observes an i.i.d. correlated source $Y^{n}$ and can transmit information to Bob over another link of capacity not exceeding $R_{y}$ . A passive eavesdropper (say Eve) can observe the coded output of Alice, i.e., the link from Alice to Bob is public. The uncertainty about the source $X^{n}$ at Eve (denoted by $\Delta$ ) is measured by the conditional entropy ${{H(X^{n}\vert J_{x})}\over{n}}$ , where $J_{x}$ is the coded output of Alice and $n$ is the block length. We completely characterize the rate-equivocation region for this secure source coding model, where we show that Slepian–Wolf binning of $X^{n}$ with respect to the coded side information received at Bob is optimal. We next consider a modification of this model in which Alice also has access to the coded output of Helen. We call this model as the two-sided helper model. For the two-sided helper model, we characterize the rate-equivocation region. While the availability of side information at Alice does not reduce the rate of transmission from Alice, it significantly enhances the resulting equivocation at Eve. In particular, the resulting equivocation for the two-sided helper case is shown to be $\min (H(X),R_{y})$ , i.e., one bit from the two-sided helper provides one bit of uncertainty at Eve. From this result, we infer that Slepian–Wolf binning of $X$ is suboptimal and one can further decrease the information leakage to the eavesdropper by utilizing the side information at Alice. We, finally, generalize both of these results to the case in which there is additional uncoded side information $W^{n}$ available at Bob and characterize the rate-equivocation regions under the assumption that $Y^{n}\rightarrow X^{n}\rightarrow W^{n}$ forms a Markov chain.

Wiretap Channel with Action-Dependent States and Rate-Limited Feedback

We introduce the wiretap channel with action-dependent states and rate-limited feedback. In the new model, the state sequence is dependent on the action sequence which is selected according to the message, and a secure rate-limited feedback link is shared between the transmitter and the receiver. We obtain the capacity-equivocation region and secrecy capacity of such a channel both for the case where the channel inputs depend noncausally on the state sequence and the case where they are restricted to causal dependence. We construct the capacity-achieving coding schemes utilizing Wyner's random binning, Gel'fand and Pinsker’s coding technique, and rate splitting. Furthermore, we compare our results with the existing approaches without feedback, with noiseless feedback, and without action-dependent states. The simulation results show that the secrecy capacity of our model is bigger than that of the first two existed approaches. Besides, it is also shown that, by taking actions to affect the channel states, we guarantee the data integrity of the message transmitted in the two-stage communication systems although the tolerable overhead of transmission time is brought.