Discrete Memoryless Broadcast Channel Research Articles

Identification Over Compound Multiple-Input Multiple-Output Broadcast Channels

The identification capacity region of the compound broadcast channel is determined under an average error criterion, where the sender has no channel state information. We give single-letter identification capacity formulas for discrete channels and multiple-input multiple-output Gaussian channels under an average input constraint. The capacity theorems apply to general discrete memoryless broadcast channels. This is in contrast to the transmission setting, where the capacity is only known for special cases, notably the degraded broadcast channel and the multiple-input multiple-output broadcast channel with private messages. Furthermore, the identification capacity region of the compound multiple-input multiple-output broadcast channel can be larger than the transmission capacity region. This is a departure from the single-user behavior of identification, since the identification capacity of a single-user channel equals the transmission capacity.

Diamond Message Set Groupcasting: From an Inner Bound for the DM Broadcast Channel to the Capacity Region of the Combination Network

Multiple groupcasting over the broadcast channel (BC) is studied in a special setting. In particular, an inner bound is obtained for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -receiver discrete memoryless (DM) BC for the diamond message set which consists of four groupcast messages: one desired by all receivers, one by all but two receivers, and two more desired by all but each one of those two receivers. The inner bound is based on rate-splitting and superposition coding and is given in explicit form herein as a union over coding distributions of four-dimensional polytopes. This inner bound is then shown to be the capacity for a certain class of partially ordered DM BCs with order defined via the less noisy condition. When specialized to the so- called combination network, which is a class of three-layer (two-hop) broadcast networks parameterized by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2^{K}{-}1$ </tex-math></inline-formula> finite-and-arbitrary-capacity noiseless links from the source node in the first layer to as many nodes of the second layer, our top-down approach from the DM BC to the combination network yields an explicit inner bound as a single polytope via the identification of a single coding distribution. This inner bound consists of inequalities which are then identified to be within the class of generalized cut-set outer bounds recently obtained by Salimi et al for broadcast networks. We hence establish the capacity region of the general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -user combination network for the diamond message set, and do so in explicit and structured form. Such a result implies a certain strength of our inner bound for the DM BC in that it (a) produces a hitherto unknown capacity region when specialized to the combination network and (b) may capture many combinatorial aspects of the capacity region of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -receiver DM BC itself for the diamond message set. Moreover, we extend that inner bound by adding binning to it. As in the no-binning case, we provide the more general inner bound in explicit form.

The -User DM Broadcast Channel With Two Groupcast Messages: Achievable Rate Regions and the Combination Network as a Case Study

A novel class of achievable rate regions is obtained for the general <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K$ </tex-math></inline-formula> -receiver discrete memoryless broadcast channel over which two groupcast messages are to be transmitted, with each message required by an arbitrary group of receivers. The associated achievability schemes are parameterized by an expansion of the message set which then determines how random coding techniques are employed. These techniques include generalized versions of up-set message-splitting, the generation of possibly multiple auxiliary codebooks for certain compositions of split messages using superposition coding with subset inclusion order, partial interference decoding at all receivers in general, joint unique decoding at receivers that desire both messages, and non-unique or indirect decoding at receivers that desire only one of the two messages. The generality of the proposed class of schemes implies new achievable rate regions for problems previously not considered as well as those that were studied before, with specific members of that class having rate regions that coincide with previously found capacity regions for special classes of broadcast channels with two private or two nested groupcast messages, wherein the group of receivers desiring one message is contained in that desiring the other. Moreover, new capacity results are established for certain partially ordered classes of broadcast channels for a class of two non-nested groupcast messages. To further show the strength of the proposed achievable rate regions we consider the so-called combination network as a test case. When specialized to the combination network, some members of the class of inner bounds are shown, via converse results, to result in the capacity region when the two messages are (a) intended for two distinct sets of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$K{-}1$ </tex-math></inline-formula> receivers each and (b) nested, in which one message is intended for one or two (common) receivers and both messages are intended for all other (private) receivers. In the latter two nested messages cases, we hence recover, in a top-down manner, previous results by Bidokhti, Prabhakaran, and Diggavi, obtained therein using lower complexity network coding schemes based on rate-splitting and linear superposition coding but tailored to the combination network, while in the first case we obtain a new capacity result for a non-nested message set, which was hitherto unknown. Furthermore, we show the achievability of rate pairs in two interesting examples of combination networks, with three and four common receivers each. These examples were proposed in the previous literature to show the sub-optimality of the aforementioned rate-splitting and linear superposition coding scheme, and hence to motivate the additional consideration of a pre-encoding technique and a block-Markov linear superposition coding for the combination network, with the latter then lifted to the general broadcast channel. Our results suggest that the proposed framework here for the general broadcast channel when specialized to the combination network is strong enough to incorporate the enhancements afforded by those two latter techniques, thereby implying that perhaps block-Markov superposition coding is not necessary in the general broadcast channel. Moreover, there is a trade-off between the complexity of the coding scheme within the class of schemes we propose when applied to the combination network and that of the determination of the distribution of the auxiliary random variables and the encoding function that achieve the capacity region. This may have interesting implications for the general broadcast channel as well.

A Secure Communication Scheme Based on Equivalent Interference Channel Assisted by Physical Layer Secret Keys

Due to the channel estimation error, most of the physical layer secret key generation schemes need information reconciliation to correct error key bits, resulting in reduced efficiency. To solve the problem, this work proposes a novel secure communication scheme based on a equivalent interference channel. Different keys generated from imperfect channel state information are directly applied to signal scrambling and descrambling, which is equivalent to the process of a signal passing through an interference channel. Legitimate communication parties can reduce interference with the help of similar keys and channel coding without sending additional signals, while the eavesdropper channel is deteriorated due to the spatial decorrelation. For this kind of schemes, we first establish a discrete memoryless broadcast channel model to derive the expressions of bit error rate (BER), channel capacity, and security capacity for performance analysis. Simulation results verify the derivations that the proposed scheme achieves secure communication with a correlated eavesdropping channel and has a higher upper bound of transmission rate. Furthermore, we design a new metric to evaluate the efficiency and the result shows that the proposed scheme has superior performance on error reconciliation efficiency, despite its slight increase in BER.

Embedding Covert Information in Broadcast Communications

We analyze a two-receiver binary-input discrete memoryless broadcast channel, in which the transmitter communicates a common message simultaneously to both receivers and a covert message to only one of them. The unintended recipient of the covert message is treated as an adversary who attempts to detect the covert transmission. This model captures the problem of embedding covert messages in an innocent codebook and generalizes previous covert communication models in which innocent behavior corresponds to the absence of communication between legitimate users. We identify the exact asymptotic behavior of the number of covert bits that can be transmitted when the rate of the innocent codebook is close to the capacity of the channel to the adversary. Our results also identify the dependence of the number of covert bits on the channel parameters and the characteristics of the innocent codebook.

Capacity Results for Classes of Partially Ordered $K$ -User Broadcast Channels With Two Nested Multicast Messages

The $K$ -user discrete memoryless (DM) broadcast channel (BC) with two nested multicast messages is studied in which one common message is to be multicast to all receivers and the second private message to a subset of receivers. The receivers that must decode both messages are referred to as private receivers and the others that must decode only the common message as common receivers. For two nested multicast messages, we establish the capacity region for several classes of partially ordered DM BCs characterized by the respective associated sets of pair-wise relationships between and among the common and private receivers, each described by the well-known pair-wise more capable or less noisy condition. For three classes of partially ordered DM BCs, the capacity region is shown to be simply achieved by two-level superposition coding and the proofs of the converses rely on a recently found information inequality. The rate region achievable by two-level superposition coding is then enhanced through a multi-level superposition coding scheme after splitting the private message into as many parts as there are common receivers and indirect decoding. A closed-form two-dimensional polyhedral (polygonal) description is obtained for it for a given coding distribution in spite of the indeterminate number of split rates via a structured form of Fourier-Motzkin elimination. Through a converse result that relies on the Csiszar sum lemma and that information inequality, a specialization of this region that corresponds to splitting the private message into just two sub-messages is proved to be the capacity region for several classes of partially ordered DM BCs beyond those for which two-level superposition coding is capacity optimal, thereby underscoring the benefit of rate-splitting. All previously known capacity results for partially ordered DM BCs with two nested multicast messages for the two and three-receiver DM BCs as well as DM BCs with one private or one common receiver are subsumed in the general results obtained in this work.

On the Evaluation of Marton’s Inner Bound for Two-Receiver Broadcast Channels

Marton’s inner bound is the best known achievable rate region for a general two-receiver discrete memoryless broadcast channel. In this paper, we establish improved bounds on the cardinalities of the auxiliary random variables appearing in this inner bound to the true rate region. We combine a perturbation technique, along with a representation using concave envelopes of information-theoretic functions that involve the use of auxiliary random variables, to achieve this improvement. The new cardinality bounds lead to a proof that a randomized-time-division strategy achieves every rate triple in Marton’s region for binary input broadcast channels. This extends the result by Hajek and Pursley which showed that the Cover-van der Muelen region was exhausted by the randomized-time-division strategy.

Common-Message Broadcast Channels With Feedback in the Nonasymptotic Regime: Stop Feedback

We investigate the maximum coding rate for a given average blocklength and error probability over a K-user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiy et al. (2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e., zero dispersion. In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message K-user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., the binary symmetric broadcast channel). Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.

Common-Message Broadcast Channels With Feedback in the Nonasymptotic Regime: Full Feedback

We investigate the maximum coding rate for a given average blocklength and error probability over a $K$ -user discrete memoryless broadcast channel for the scenario where a common message is transmitted using variable-length stop-feedback codes. For the point-to-point case, Polyanskiy et al. (2011) demonstrated that variable-length coding combined with stop-feedback significantly increases the speed of convergence of the maximum coding rate to capacity. This speed-up manifests itself in the absence of a square-root penalty in the asymptotic expansion of the maximum coding rate for large blocklengths, i.e., zero dispersion . In this paper, we present nonasymptotic achievability and converse bounds on the maximum coding rate of the common-message $K$ -user discrete memoryless broadcast channel, which strengthen and generalize the ones reported in Trillingsgaard et al. (2015) for the two-user case. An asymptotic analysis of these bounds reveals that zero dispersion cannot be achieved for certain common-message broadcast channels (e.g., the binary symmetric broadcast channel). Furthermore, we identify conditions under which our converse and achievability bounds are tight up to the second order. Through numerical evaluations, we illustrate that our second-order expansions approximate accurately the maximum coding rate and that the speed of convergence to capacity is indeed slower than for the point-to-point case.

Achievable Rate Region for Three User Discrete Broadcast Channel Based on Coset Codes

We consider the problem of developing coding techniques and deriving achievable rate regions for discrete memoryless broadcast channels with three receivers (3-DBC). We begin by identifying a novel vector additive 3-DBC for which we characterize an upper bound on the the largest achievable rate region based on unstructured codes, henceforth referred to as $\pmb {\mathscr {U}}\mathcal {M}$ -region. We propose a coding technique based on coset codes that yield an achievable rate triple not contained within $\pmb {\mathscr {U}}\mathcal {M}$ -region. We generalize the proposed coding technique using a new ensemble of codes— partitioned coset codes (PCC)—containing both empirical and algebraic properties, and evaluate its performance to derive an achievable rate region for the general 3-DBC. The new elements in this derivation are binning and joint typicality encoding and decoding of statistically correlated PCCs. We validate the utility of this technique by identifying non-additive instances of 3-DBC for which the proposed coding techniques based on PCC yield strictly larger rates.

Feedback and Partial Message Side-Information on the Semideterministic Broadcast Channel

The capacity of the semideterministic discrete memoryless broadcast channel (SD-BC) with partial message side-information (P-MSI) at the receivers is established. In the setting without a common message, it is shown that P-MSI to the stochastic receiver alone can increase capacity, whereas P-MSI to the deterministic receiver can only increase capacity if also the stochastic receiver has P-MSI. The latter holds only for the setting without a common message: if the encoder also conveys a common message, then P-MSI to the deterministic receiver alone can increase capacity. These capacity results are used to show that feedback from the stochastic receiver can increase the capacity of the SD-BC without P-MSI and the sum-rate capacity of the SD-BC with P-MSI at the deterministic receiver. The link between P-MSI and feedback is a feedback code, which—roughly speaking—turns feedback into P-MSI at the stochastic receiver, and hence helps the stochastic receiver mitigate experienced interference. For the case, where the stochastic receiver has full MSI (F-MSI) and can thus fully mitigate experienced interference also in the absence of feedback, it is shown that feedback cannot increase capacity.

Leveraging Receiver Message Side Information in Two-Receiver Broadcast Channels: A General Approach †

We consider two-receiver broadcast channels where each receiver may know a priori some of the messages requested by the other receiver as receiver message side information (RMSI). We devise a general approach to leverage RMSI in these channels. To this end, we first propose a pre-coding scheme considering the general message setup where each receiver requests both common and private messages and knows a priori part of the private message requested by the other receiver as RMSI. We then construct the transmission scheme of a two-receiver channel with RMSI by applying the proposed pre-coding scheme to the best transmission scheme for the channel without RMSI. To demonstrate the effectiveness of our approach, we apply our pre-coding scheme to three categories of the two-receiver discrete memoryless broadcast channel: (i) channel without state; (ii) channel with states known causally to the transmitter; and (iii) channel with states known non-causally to the transmitter. We then derive a unified inner bound for all three categories. We show that our inner bound is tight for some new cases in each of the three categories, as well as all cases whose capacity region was known previously.

Polar Code Design for Noisy Blackwell Channels

In this letter, we introduce a class of discrete memoryless broadcast channels, called noisy Blackwell channels, which generalize the Blackwell channel to include noise and to more than two users. We design the polar code for the noisy Blackwell channel based on Marton’s coding scheme. We choose auxiliary random variables and the mapping from auxiliary random variables to the channel input, such that the sum-rate capacity can be achieved very closely. Furthermore, we show that the designed polar code can be applied to multi-level cell flash memories for improving random I/O performance. The simulation results demonstrate that our designed polar codes achieve good performance even for a large number of users.

Polar Coding for the Broadcast Channel With Confidential Messages: A Random Binning Analogy

We develop a low-complexity polar coding scheme for the discrete memoryless broadcast channel with confidential messages under strong secrecy and randomness constraints. Our scheme extends previous work by using an optimal rate of uniform randomness in the stochastic encoder, and avoiding assumptions regarding the symmetry or degraded nature of the channels. The price paid for these extensions is that the encoder and the decoders are required to share a secret seed of negligible size and to increase the block length through chaining. We also highlight a close conceptual connection between the proposed polar coding scheme and a random binning proof of the secrecy capacity region.

Coding Schemes With Rate-Limited Feedback That Improve Over the No Feedback Capacity for a Large Class of Broadcast Channels

We propose two coding schemes for the two-receiver discrete memoryless broadcast channel (BC) with rate-limited feedback from one or both receivers. They improve over the no feedback capacity region for a large class of channels, including the class of strictly essentially less-noisy BCs that we introduce in this paper. Examples of strictly essentially less-noisy BCs are the binary symmetric BC or the binary erasure BC with unequal crossover or erasure probabilities at the two receivers. When the feedback rates are sufficiently large, our schemes recover all previously known capacity results for discrete memoryless BCs with feedback. In both our schemes, we let the receivers feedback quantization messages about their receive signals. In the first scheme, the transmitter simply relays the quantization information obtained from Receiver 1 to Receiver 2, and vice versa. This provides each receiver with a second observation of the input signal and can thus improve its decoding performance unless the BC is physically degraded. Moreover, each receiver uses its knowledge of the quantization message describing its own outputs so as to attain the same performance as if this message had not been transmitted at all. In our second scheme, the transmitter first reconstructs and processes the quantized output signals, and then sends the outcome as a common update information to both receivers. A special case of our second scheme also applies to memoryless BCs without feedback but with strictly causal state-information at the transmitter and causal state-information at the receivers. It recovers all previous achievable regions also for this setup with state-information.

On capacity region of certain classes of three‐receiver broadcast channels with side information

The fact that the results for 2-receiver broadcast channels (BCs) are not generalised to the 3-receiver ones is of information theoretical importance. In this study, the authors study two classes of discrete memoryless BCs with non-causal side information, that is, multilevel BC (MBC) and 3-receiver less noisy BC. First, they obtain an achievable rate region and a capacity outer bound for the MBC. Second, they provide the capacity region for a special class of 3-receiver less noisy BC. Third, the obtained special capacity region for the 3-receiver less noisy BC is evaluated for the continuous alphabet fading Gaussian version. It is worth mentioning that the previous works are special cases of the authors works.

Achieving Marton’s Region for Broadcast Channels Using Polar Codes

This paper presents polar coding schemes for the two-user discrete memoryless broadcast channel (DM-BC) which achieve Marton’s region with both common and private messages. This is the best achievable rate region known to date, and it is tight for all classes of two-user DM-BCs whose capacity regions are known. To accomplish this task, we first construct polar codes for both the superposition as well as binning strategy. By combining these two schemes, we obtain Marton’s region with private messages only. Finally, we show how to handle the case of common information. The proposed coding schemes possess the usual advantages of polar codes, i.e., they have low encoding and decoding complexity and a superpolynomial decay rate of the error probability. We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar codes emulating the superposition and binning schemes. To align the polar indices, for both schemes, their solution involves some degradedness constraints that are assumed to hold between the auxiliary random variables and channel outputs. To remove these constraints, we consider the transmission of $k$ blocks and employ a chaining construction that guarantees the proper alignment of the polarized indices. The techniques described in this paper are quite general, and they can be adopted to many other multiterminal scenarios whenever there polar indices need to be aligned.

An Achievable Rate Region for the Broadcast Channel With Feedback

A single-letter achievable rate region is proposed for the two-receiver discrete memoryless broadcast channel with generalized feedback. The coding strategy involves block-Markov superposition coding, using Marton's coding scheme for the broadcast channel without feedback as the starting point. If the message rates in the Marton scheme are too high to be decoded at the end of a block, each receiver is left with a list of messages compatible with its output. Resolution information is sent in the following block to enable each receiver to resolve its list. The key observation is that the resolution information of the first receiver is correlated with that of the second. This correlated information is efficiently transmitted via joint source-channel coding, using ideas similar to the Han-Costa coding scheme. Using the result, we obtain an achievable rate region for the stochastically degraded AWGN broadcast channel with noisy feedback from only one receiver. It is shown that this region is strictly larger than the no-feedback capacity region.

On the Capacity of the Discrete Memoryless Broadcast Channel With Feedback

A coding scheme for the discrete memoryless broadcast channel with {noiseless, noisy, generalized} feedback is proposed, and the associated achievable region derived. The scheme is based on a block-Markov strategy combining the Marton scheme and a lossy version of the Gray–Wyner scheme with side information. In each block, the transmitter sends fresh data and update information that allows the receivers to improve the channel outputs observed in the previous block. For a generalization of Dueck's broadcast channel, our scheme achieves the noiseless-feedback capacity, which is strictly larger than the no-feedback capacity. For a generalization of Blackwell's channel and when the feedback is noiseless, our new scheme achieves rate points that are outside the no-feedback capacity region. It follows by a simple continuity argument that for both these channels and when the feedback noise is sufficiently low, our scheme improves on the no-feedback capacity even when the feedback is noisy.

Polar Codes for Broadcast Channels

Polar codes are introduced for discrete memoryless broadcast channels. For $m$-user deterministic broadcast channels, polarization is applied to map uniformly random message bits from $m$ independent messages to one codeword while satisfying broadcast constraints. The polarization-based codes achieve rates on the boundary of the private-message capacity region. For two-user noisy broadcast channels, polar implementations are presented for two information-theoretic schemes: i) Cover's superposition codes; ii) Marton's codes. Due to the structure of polarization, constraints on the auxiliary and channel-input distributions are identified to ensure proper alignment of polarization indices in the multi-user setting. The codes achieve rates on the capacity boundary of a few classes of broadcast channels (e.g., binary-input stochastically degraded). The complexity of encoding and decoding is $O(n*log n)$ where $n$ is the block length. In addition, polar code sequences obtain a stretched-exponential decay of $O(2^{-n^{\beta}})$ of the average block error probability where $0 < \beta < 0.5$.