Binary-input Channels Research Articles

Optimal 1-bit Error Exponent for 2-Hop Relaying With Binary-Input Channels

Optimal 1-bit Error Exponent for 2-Hop Relaying With Binary-Input Channels

Upper Bounds on the Mismatched Reliability Function and Capacity Using a Genie Receiver

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 inherent symmetric requirement from the codewords, leading to these new upper bounds. Since the computations of most of the known bounds on the mismatch capacity are rather complicated, we further present a method to obtain relaxed bounds that are easier to compute. As an example, we analyze the obtained bounds in the binary-input channels case. We conclude by presenting simpler bounds on the reliability function, and provide sufficient conditions for their tightness in certain ranges of rates.

Binary Modeling and Capacity-Approaching Coding for the IM/DD Channel

The paper provides a new perspective on peak- and average-constrained Gaussian channels. Such channels model optical wireless communication (OWC) systems which employ intensity-modulation with direct detection (IM/DD). First, the paper proposes a new, capacity-preserving vector binary channel (VBC) model, consisting of dependent binary noisy bit-pipes. Then, to simplify coding over this VBC, the paper proposes coding schemes with varying levels of complexity, building on the capacity of binary-symmetric channels (BSC) and channels with state. The achievable rates are compared to capacity and capacity bounds, showing that coding for the BSC with state over the VBC achieves rates close to capacity at moderate to high signal-to-noise ratio (SNR), whereas simpler schemes achieve lower rates at lower complexity. The presented coding schemes are realizable using capacity-achieving codes for binary-input channels, such as polar codes. Numerical results are provided to validate the theoretical results and demonstrate the applicability of the proposed schemes.

Sequential Transmission Over Binary Asymmetric Channels With Feedback

In this paper, we consider variable-length coding over the memoryless binary asymmetric channel (BAC) with full noiseless feedback, including the binary symmetric channel (BSC) as a special case. In 2012, Naghshvar et al. introduced a coding scheme, which we refer to as the small-enough-difference (SED) coding scheme. For symmetric binary-input channels, the deterministic variable-length feedback (VLF) code constructed with the SED coding scheme asymptotically achieves both capacity and Burnashev’s optimal error exponent. Building on the work of Naghshvar et al., this paper extends the SED coding scheme to the BAC and develops a non-asymptotic VLF achievability bound that is shown to achieve both capacity and the optimal error exponent. For the specific case of the BSC, we develop an additional non-asymptotic VLF achievability bound using a two-phase analysis that leverages both a submartingale synthesis and a Markov chain time of first passage analysis. Numerical evaluations show that both new VLF achievability bounds outperform Polyanskiy’s achievability bound for variable-length stop-feedback codes.

Systematic Polar Coded Modulation for Informed Receivers

The problem of coding for informed receivers (IR) is considered, where a transmitter multicasts a bunch of messages to its associated receivers, each of which already has a subset of messages as side information. Which receiver has what side information is unknown to the transmitter. A family of polar codes for IR was proposed for binary-input channels in (Huang and Shieh, 2018), which was shown to provide excellent side information gain regardless of side information configuration and content. A trial extension of this scheme to high-order modulations, however, was unsatisfactory. In this paper, two families of systematic polar coded modulations for IR are proposed. In the first family, by building a connection between the message bits and modulated symbols under systematic polar codes, we leverage the bit assignments and labeling techniques such that receiver side information can be directly translated to a minimum distance gain. In the second family, a permutation operation is added before modulation such that the message bits can be properly reordered to further advance and balance the gains among different side information configurations. Simulations show that the proposed schemes provide large and balanced side information gains and significantly outperform the system directly adapted from (Huang and Shieh, 2018) onto high-order modulations.

Channel Polarization Through the Lens of Blackwell Measures

Each memoryless binary-input channel (BIC) can be uniquely described by its Blackwell measure, which is a probability distribution on the unit interval $[0,1]$ with mean $1/2$. Conversely, any such probability distribution defines a BIC. The evolution of the Blackwell measure under Arikan's polar transform is derived for general BICs, and is analogous to density evolution as cited in the literature. The present analysis emphasizes functional equations. Consequently, the evolution of a variety of channel functionals is characterized, including the symmetric capacity, Bhattacharyya parameter, moments of information density, Hellinger affinity, Gallager's reliability function, the Hirschfeld-Gebelein-Renyi maximal correlation, and the Bayesian information gain. The evolution of measure is specialized for symmetric BICs according to their decomposition into binary symmetric (sub)-channels (BSCs), which simplifies iterative computations and the construction of polar codes. It is verified that, as a consequence of the Blackwell--Sherman--Stein theorem, all channel functionals $\mathrm{I}_f$ that can be expressed as an expectation of a convex function $f$ with respect to the Blackwell measure of a channel polarize in each iteration due to the polar transformation on the class of symmetric BICs. Moreover, for $f$ either convex or non-convex, a necessary and sufficient condition is established to determine whether the random process associated with each $\mathrm{I}_f$ is a martingale, submartingale, or supermartingale. Represented via functional inequalities in terms of $f$, this condition is numerically verifiable for all $\mathrm{I}_f$, and can generate analytical proofs. To exhibit one such proof, it is shown that the random process associated with the squared maximal correlation parameter is a supermartingale, and converges almost surely on the unit interval $[0,1]$.

Dual Capacity Upper Bounds for Binary-Input Single-Tap ISI Channels

The capacity of noisy channels with memory and finite input has been difficult to characterize explicitly in many cases. In this paper, we consider single-tap binary-input Gaussian channels with inter-symbol interference (ISI). The dual capacity method is used to obtain upper bounds using Markov test distributions. The bound, expressed as a single-letter optimization problem, is solved numerically. For higher memory, the notion of cycle basis is used to provide an exponential reduction in the computational complexity of the optimization problem. Bounds were obtained for the important case of the dicode channel, and these are better than the previously known upper bounds and are close to achievable rates over a wide range of SNRs.

Achievable Rate and Capacity Analysis for Ambient Backscatter Communications

In this paper, we analyze the achievable rate for ambient backscatter communications under three different channels: the binary input and binary output (BIBO) channel, the binary input and signal output (BISO) channel, and the binary input and energy output (BIEO) channel. Instead of assuming Gaussian input distribution, the proposed study matches the practical ambient backscatter scenarios, where the input of the tag can only be binary. We derive the closed-form capacity expression as well as the capacity-achieving input distribution for the BIBO channel. To show the influence of the signal-to-noise ratio (SNR) on the capacity, a closed-form tight ceiling is also derived when SNR turns relatively large. For BISO and BIEO channel, we obtain the closed-form mutual information, while the semi-closed-form capacity value can be obtained via one dimensional searching. Simulations are provided to corroborate the theoretical studies. Interestingly, the simulations show that: (i) the detection threshold maximizing the capacity of BIBO channel is the same as the one from the maximum likelihood signal detection; (ii) the maximal of the mutual information of all channels is achieved almost by a uniform input distribution; and (iii) the mutual information of the BIEO channel is larger than that of the BIBO channel, but is smaller than that of the BISO channel.

On the Throughput of Channels That Wear Out

This paper investigates the fundamental limits of communication over a noisy discrete memoryless channel that wears out, in the sense of signal-dependent catastrophic failure. In particular, we consider a channel that starts as a memoryless binary-input channel and when the number of transmitted ones causes a sufficient amount of damage, the channel ceases to convey signals. Constant composition codes are adopted to obtain an achievability bound, and the left-concave right-convex inequality is then refined to obtain a converse bound on the log-volume throughput for channels that wear out. Since infinite blocklength codes will always wear out the channel for any finite threshold of failure, and therefore cannot convey information at positive rates, we analyze the performance of finite blocklength codes to determine the maximum expected transmission volume at a given level of average error probability. We show that this maximization problem has a recursive form and can be solved by dynamic programming. Numerical results demonstrate that a sequence of block codes is preferred to a single block code for streaming sources.

Progressive Matrix Growth Algorithm for Constructing Rate-Compatible Length-Scalable Raptor-Like Quasi-Cyclic LDPC Codes

With the worldwide deployment and rapid development of digital television terrestrial broadcasting (DTTB) systems, the demands of diversified multimedia services from the fixed/mobile receptions increase significantly. To satisfy the various requirements of quality of service, the forward error correction codes with multiple code rates and lengths become a trend for DTTB standards. In this paper, rate-compatible length-scalable raptor-like quasi-cyclic low density parity check (RL-QC-LDPC) codes are proposed for future DTTB systems, where a wide range of code rates and information/parity block lengths can be obtained with low implementation complexity. A low-complexity construction method, called progressive matrix growth (PMG), is also proposed for such kind of RL-QC-LDPC codes. Unlike conventional construction methods, PMG performs 2-D progressive extension of both information variable nodes and check nodes with limited computational complexity and memory resources. Based on the performance prediction provided by the tool of multi-edge type density evolution, several code families are constructed with PMG as examples, and careful discussions on design parameters are given. The numerical results show that the average gaps of SNR threshold to Shannon limit of binary-input additive white Gaussian noise channel are only 0.41~0.44 dB for the code families with information-bit puncturing.

LLR-Based SC Decoding of Polar Codes for Two-User Binary-Input MAC

This letter considers a hardware-friendly log-likelihood ratio (LLR)-based successive cancellation (SC) decoding of polar codes for two-user binary-input multiple access channels. Based on the known recursive equations in the likelihood domain, we obtain LLR-based recursive equations for the SC decoding. An approximate LLR-based decoding is also introduced, which shows small performance loss at high error rate, but has significantly low complexity compared with the previous likelihood-based decoding.

Construction of Polar Codes for Arbitrary Discrete Memoryless Channels

It is known that polar codes can be efficiently constructed for binary-input channels. At the same time, existing algorithms for general input alphabets are less practical because of high complexity. We address the construction problem for the general case, and analyze an algorithm that is based on successive reduction of the output alphabet size of the subchannels in each recursion step. For this procedure, we estimate the approximation error as $O(\mu ^{-1/(q-1)})$ , where $q$ is the input alphabet size and $\mu$ is the “quantization parameter,” i.e., the maximum size of the subchannel output alphabet allowed by the algorithm. The complexity of the code construction scales as $O(N\mu ^{2} \log \mu)$ , where $N$ is the length of the code. We also show that if the polarizing operation relies on modulo- $q$ addition, it is possible to merge subsets of output symbols without any loss in subchannel capacity. Performing this procedure before each approximation step results in a further speed-up of the code construction, and the resulting codes have smaller gap to capacity. We also show that a similar acceleration can be attained for polar codes over finite field alphabets. Experimentation shows that the suggested construction algorithms can be used to construct long polar codes for alphabets of size $q=16$ and more with acceptable loss of the code rate for a variety of polarizing transforms.

A Reused-Public-Path Successive Cancellation List Decoding for Polar Codes With CRC

Since the decoding of polar codes goes wrong in sometime, several decoding methods have been proposed to overcome this issue. Unfortunately, the several schemes are still suffering from high decoding complexity. To this end, we propose a reused-public-path successive-cancellation list decoder, which can re-do the decoding from ${E}$ -th level when the current decoding process fails. In addition, a reused public-path scheme is proposed. By properly designing, simulation results in binary-input additive white gaussian noise channel show that this decoding approach provides significant computational complexity reduction in the moderate and high signal-to-noise ratio regime without performance loss.

Dual Capacity Upper Bounds for Noisy Runlength Constrained Channels

Binary-input memoryless channels with a run length constrained input are considered. Upper bounds to the capacity of such noisy run length constrained channels are derived using the dual capacity method with Markov test distributions satisfying the Karush-Kuhn-Tucker conditions for the capacity-achieving output distribution. Simplified algebraic characterizations of the bounds are presented for the binary erasure channel and the binary symmetric channel. These upper bounds are very close to achievable rates, and improve upon previously known feedback-based bounds for a large range of channel parameters. For the binary-input additive white Gaussian noise channel, the upper bound is simplified to a small-scale numerical optimization problem. These results provide some of the simplest upper bounds for an open capacity problem that has theoretical and practical relevance.

Intrinsic Capacity

Every channel can be expressed as a convex combination of deterministic channels with each deterministic channel corresponding to one particular intrinsic state. Such convex combinations are in general not unique, each giving rise to a specific intrinsic-state distribution. In this paper we study the maximum and the minimum capacities of a channel when the realization of its intrinsic state is causally available at the encoder and/or the decoder. Several conclusive results are obtained for binary-input channels and binary-output channels. Byproducts of our investigation include a generalization of the Birkhoff-von Neumann theorem and a condition on the uselessness of causal state information at the encoder.

Improved Belief Propagation Decoding Algorithm for Short Polar Codes

In this paper, we discuss the belief propagation (BP) decoding of polar codes. The performance of polar codes for short lengths is not satisfactory. Therefore, motivated by this we propose a novel technique to improve the performance of polar codes under BP decoding. In order to enhance the reliability of the variable nodes’ propagated messages in BP decoding, the messages are multiplied by previous messages of variable nodes. It is also shown that the BP decoding can be performed on polar codes if the factor graph is constructed according to the parity check matrix of polar codes. Simulation results in binary-input additive white Gaussian noise channel show that using the proposed method a gain of 1–2 dB can be achieved. We also show that the complexity of proposed decoder is same as the complexity of standard BP decoder. Furthermore, we show that the proposed decoder requires about 10–25 iterations less than the standard BP decoder.

On the Binary Input Gaussian Wiretap Channel with/without Output Quantization

In this paper, we investigate the effect of output quantization on the secrecy capacity of the binary-input Gaussian wiretap channel. As a result, a closed-form expression with infinite summation terms of the secrecy capacity of the binary-input Gaussian wiretap channel is derived for the case when both the legitimate receiver and the eavesdropper have unquantized outputs. In particular, computable tight upper and lower bounds on the secrecy capacity are obtained. Theoretically, we prove that when the legitimate receiver has unquantized outputs while the eavesdropper has binary quantized outputs, the secrecy capacity is larger than that when both the legitimate receiver and the eavesdropper have unquantized outputs or both have binary quantized outputs. Further, numerical results show that in the low signal-to-noise ratio (SNR) (of the main channel) region, the secrecy capacity of the binary input Gaussian wiretap channel when both the legitimate receiver and the eavesdropper have unquantized outputs is larger than the capacity when both the legitimate receiver and the eavesdropper have binary quantized outputs; as the SNR increases, the secrecy capacity when both the legitimate receiver and the eavesdropper have binary quantized outputs tends to overtake.

When is Noisy State Information at the Encoder as Useless as No Information or as Good as Noise-Free State?

For any binary-input channel with perfect state information at the decoder, if the mutual information between the noisy state observation at the encoder and the true channel state is below a positive threshold determined solely by the state distribution, then the capacity is the same as that with no encoder side information. A complementary phenomenon is revealed for the generalized probing capacity. Extensions beyond binary-input channels are developed.

Embedded physical‐layer authentication in cognitive radio requires efficient low‐rate channel coding schemes

In this study, the author investigates the practical limitation of the recently proposed embedded cryptographic signature authentication scheme at the physical layer. By employing the log-likelihood ratio of a tag bit and its approximation, the author shows that the equivalent authentication channel observed by the secondary receiver can be viewed as a binary-input additive white Gaussian noise channel. Then, the sphere-packing lower bound can be employed to show the transmission capability for practical finite-length authentication tags. To achieve the same effective coverage area for both the primary and secondary receivers, it essentially requires efficient low-rate channel coding schemes with near sphere-packing-bound performance at the secondary receiver, which contrasts sharply with the pessimistic conclusion of Jiang et al.

Design of superposition polar coding for binary-input less-noisy broadcast channels

Abstract In this study, we design a superposition polar code that can achieve the capacity region of binary-input less-noisy broadcast channels asymptotically. Simulation results show that a better rate region is achievable by superposition polar coding than by time sharing between two point-to-point channel polar codes.