Erdal Arikan, Emre Telatar
It is shown that for any binary-input discrete memoryless channel $W$ with symmetric capacity $I(W)$ and any rate $R <I(W)$, the probability of block decoding error for polar coding under successive cancellation decoding satisfies $P_e \le 2^{-N^β}$ for any $β<\frac12$ when the block-length $N$ is large enough.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
A method of channel polarization, proposed by Arikan, allows us to construct efficient capacity-achieving channel codes. In the original work, binary input discrete memoryless channels are considered. A special case of $q$-ary channel polarization is considered by Sasoglu, Telatar, and Arikan. In this paper, we consider more general channel polarization on $q$-ary channels. We further show explicit constructions using Reed-Solomon codes, on which asymptotically fast channel polarization is induced.
In this letter, we consider the proof of non-stationary channel polarization theory. First we construct a multi-channel stochastic process for the non-stationary channel polarization operation. Then based on this stochastic process, we extend Arıkan's standard martingale proof method on the average channel capacity and average channel Bhattacharyya parameter, by which we have proved the non-stationary channel polarization theory.
It is known that if an Abelian group operation is used in an Arıkan-style construction, we have multilevel polarization where synthetic channels can approach intermediate channels that are neither almost perfect nor almost useless. An open problem in polarization theory is to determine the polarization levels of a given channel. In this paper, we discuss the polarization levels of a family of channels that we call automorphic-symmetric channels. We show that the polarization levels of an automorphic-symmetric channel are determined by characteristic subgroups. In particular, if the group that …
We give a unified treatment of some inequalities that are used in the proofs of channel polarization theorems involving a binary-input discrete memoryless channel.
This study examines multilevel channel polarization for a certain class of erasure channels that the input alphabet size is an arbitrary composite number. We derive limiting proportions of partially noiseless channels for such a class. The results of this study are proved by an argument of convergent sequences, inspired by Alsan and Telatar's simple proof of polarization, and without martingale convergence theorems for polarization process.
Complex Analysis of Channel Polarization on Discrete BMS Channels
本文提出了一种基于复函数理论的组件演化框架，用于有限块长下离散二进制输入记忆less对称(BMS)信道的信道极化分析，推导了Bhattacharyya参数的解析表达式，并揭示了二进制擦除信道和二进制对称信道的极性极端性。
Channel polarization, originally proposed for binary-input channels, is generalized to arbitrary discrete memoryless channels. Specifically, it is shown that when the input alphabet size is a prime number, a similar construction to that for the binary case leads to polarization. This method can be extended to channels of composite input alphabet sizes by decomposing such channels into a set of channels with prime input alphabet sizes. It is also shown that all discrete memoryless channels can be polarized by randomized constructions. The introduction of randomness does not change the order of …
In this paper, we present the concept of non-uniform channel polarization and a scheme for rate-compatible polar code construction for any codeword length using additive white Gaussian noise (AWGN) channels and the successive cancellation (SC) decoder. A Non-Uniform Polarization technique based on the Gaussian Approximation (NUPGA) is developed and an efficient rate-compatible design technique is devised to choose the best channels for transmission by a process of re-polarization of the codeword with the desired length. Simulations illustrate the proposed NUPGA design against existing rate-com…
A rate-dependent upper bound of the best achievable block error probability of polar codes with successive-cancellation decoding is derived.
It is shown that given two copies of a q-ary input channel $W$, where q is prime, it is possible to create two channels $W^-$ and $W^+$ whose symmetric capacities satisfy $I(W^-)\le I(W)\le I(W^+)$, where the inequalities are strict except in trivial cases. This leads to a simple proof of channel polarization in the q-ary case.
A method to polarize channels universally is introduced. The method is based on combining two distinct channels in each polarization step, as opposed to Arikan's original method of combining identical channels. This creates an equal number of only two types of channels, one of which becomes progressively better as the other becomes worse. The locations of the good polarized channels are independent of the underlying channel, guaranteeing universality. Polarizing the good channels further with Arikan's method results in universal polar codes of rate 1/2. The method is generalized to construct c…
This study proposes \emph{modular arithmetic erasure channels} (MAECs), a novel class of erasure-like channels with an input alphabet that need not be binary. This class contains the binary erasure channel (BEC) and some other known erasure-like channels as special cases. For MAECs, we provide recursive formulas of Arıkan-like polar transform to simulate channel polarization. In other words, we show that the synthetic channels of MAECs are equivalent to other MAECs. This is a generalization of well-known recursive formulas of the polar transform for BECs. Using our recursive formulas, we also …
Let $W$ be a channel where the input alphabet is endowed with an Abelian group operation, and let $(W_n)_{n\geq 0}$ be Arıkan's channel-valued polarization process that is obtained from $W$ using this operation. We prove that the process $(W_n)_{n\geq 0}$ converges almost surely to deterministic homomorphism channels in the noisiness/weak-$\ast$ topology. This provides a simple proof of multilevel polarization for a large family of channels, containing among others, discrete memoryless channels (DMC), and channels with continuous output alphabets. This also shows that any continuous channel fu…
Regular perturbation is applied to the Manakov equation and motivates a generalized correlated phase-and-additive noise model for wavelength-division multiplexing over dual-polarization optical fiber channels. The model includes three hidden Gauss-Markov processes: phase noise, polarization rotation, and additive noise. Particle filtering is used to compute lower bounds on the capacity of multi-carrier communication with frequency-dependent powers and delays. A gain of 0.17 bits/s/Hz/pol in spectral efficiency or 0.8 dB in power efficiency is achieved with respect to existing models at their p…
We demonstrate, for a widely used model of channels with polarization dependent loss (PDL), that channel capacity is achieved by a simple interference cancellation scheme in conjunction with a universal precoder. Crucially, the proposed scheme is not only information-theoretically optimal, but it is also exceptionally simple and concrete. It transforms the PDL channel into separate scalar additive white Gaussian noise channels, allowing off-the-shelf coding and modulation schemes designed for such channels to approach capacity. The signal-to-noise ratio (SNR) penalty incurred under 6 dB of PDL…
This note is a written and extended version of the Shannon Lecture I gave at 2019 International Symposium on Information Theory. It gives an account of the original ideas that motivated the development of polar coding and discusses some new ideas for exploiting channel polarization more effectively in order to improve the performance of polar codes.
The polar transformation of a binary erasure channel (BEC) can be exactly approximated by other BECs. Arıkan proposed that polar codes for a BEC can be efficiently constructed by using its useful property. This study proposes a new class of arbitrary input generalized erasure channels, which can be exactly approximated the polar transformation by other same channel models, as with the BEC. One of the main results is the recursive formulas of the polar transformation of the proposed channel. In the study, we evaluate the polar transformation by using the $α$-mutual information. Particularly, wh…
A Low-Complexity Detector for Memoryless Polarization-Multiplexed Fiber-Optical Channels
针对非线性相位噪声下的偏振复用 M-PSK 光纤信道，提出利用双接收幅度相位补偿和直线判决边界的低复杂度检测器，推广了单偏振结果。
This paper presents the first proof of polarization for the deletion channel with a constant deletion rate and a regular hidden-Markov input distribution. A key part of this work involves representing the deletion channel using a trellis and describing the plus and minus polar-decoding operations on that trellis. In particular, the plus and minus operations can be seen as combining adjacent trellis stages to yield a new trellis with half as many stages. Using this viewpoint, we prove a weak polarization theorem for standard polar codes on the deletion channel. To achieve strong polarization, w…
In this letter, we present a correlated channel model for a dual-polarization antenna to omnidirectional antennas in indoor small-cell multiple-input multiple-output (MIMO) systems. In an indoor environment, we confirm that the cross-polarization discrimination (XPD) in the direction of angle-of-departure can be represented as the spatial correlation of the MIMO channel. We also evaluate a dual-polarization antenna-based MIMO channel model and a spatially correlated channel model using a three-dimensional (3D) ray-tracing simulator. Furthermore, we provide the equivalent distance between adjac…
We propose a novel scheme for rate-compatible arbitrary-length polar code construction for the additive white Gaussian noise (AWGN) channel. The proposed scheme is based on the concept of non-uniform channel polarization. The original polar codes can only be designed with code lengths that are powers of two. Puncturing, shortening and extension are three strategies to obtain arbitrary code lengths and code rates for polar codes. There are other ways to design codes with arbitrary length but which have encoding and decoding with higher complexity such as multi-kernel, concatenated codes and spe…
In this paper we investigate, starting with a symmetric B-DMC, the evolution of various probabilities of the likelihood ratios of the synthetic channels created by the recursive application of the basic polarization transformations. The analysis provides a new perspective into the theory of channel polarization initiated by Arıkan and helps us to address a problem related to approximating the computations of the likelihood ratios of the synthetic channels.
Polarization phenomenon over any finite field $\mathbb{F}_{q}$ with size $q$ being a power of a prime is considered. This problem is a generalization of the original proposal of channel polarization by Arikan for the binary field, as well as its extension to a prime field by Sasoglu, Telatar, and Arikan. In this paper, a necessary and sufficient condition of a matrix over a finite field $\mathbb{F}_q$ is shown under which any source and channel are polarized. Furthermore, the result of the speed of polarization for the binary alphabet obtained by Arikan and Telatar is generalized to arbitrary …
The geometry of dual-polarization four-dimensional constellations is optimized in the optical fiber channel using a recent nonlinear interference model. A 0.27 bit/4D rate gain and 13% reach increase are attained compared to polarization-multiplexed formats.
It is shown that polar codes achieve the symmetric capacity of discrete memoryless channels with arbitrary input alphabet sizes. It is shown that in general, channel polarization happens in several, rather than only two levels so that the synthesized channels are either useless, perfect or "partially perfect". Any subset of the channel input alphabet which is closed under addition, induces a coset partition of the alphabet through its shifts. For any such partition of the input alphabet, there exists a corresponding partially perfect channel whose outputs uniquely determine the coset to which …
Channel Polarization under Channel Noise with Memory
本文研究具有记忆性噪声下的信道极化，证明标准极化框架下子信道仍收敛，量化了忽略记忆的互信息差距，并指出极化速率慢于无记忆信道，Bhattacharyya参数有特定渐近界。
We show that the mismatched capacity of binary discrete memoryless channels can be improved by channel combining and splitting via Arıkan's polar transformations. We also show that the improvement is possible even if the transformed channels are decoded with a mismatched polar decoder.
A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity $I(W)$ of any given binary-input discrete memoryless channel (B-DMC) $W$. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of $N$ independent copies of a given B-DMC $W$, a second set of $N$ binary-input channels $\{W_N^{(i)}:1\le i\le N\}$ such that, as $N$ becomes large, the fraction of indices $i$ for which $I(W_N^{(i)})$ …
A hybrid ARQ (HARQ) scheme for polar code, which is called active-bit relocation under masks (ARUM), is proposed. In each transmission, the data bits are encoded and bit-wisely XOR-masked using a binary vector before being transmitted through the channel. The masking process combines multiple transmissions together which forms another step of inter-transmission channel transform. The reliabilities are updated after every transmission, and the less reliable bits in earlier ones are relocated to the more reliable positions at the latest transmitted block. ARUM is a very flexible HARQ scheme whic…
Stronger Polarization for the Deletion Channel
本文提出一种删除信道极化编码方案，通过在几乎所有极化层级放置保护带并对整个接收字进行网格译码，将错误概率衰减从立方根提升至平方根，达到与无记忆信道极化码相同的速率，同时保持容量可达和多项式译码复杂度。
We propose a scheme that universally achieves the smallest possible compression rate for a class of sources with side information, and develop an application of this result for a joint source channel coding problem over a broadcast channel.
We consider the multiplexing gain (MUXG) of the fully connected K-user line-of-sight (LOS) interference channels (ICs). A polarimetric antenna composed of 3 orthogonal electric dipoles and 3 orthogonal magnetic dipoles is considered where all 6 dipoles are co-located. In case of K-user IC with single polarization, the maximum achievable MUXG is K regardless of the number of transmit and receive antennas because of the key-hole effect. With polarization, a trivial upper bound on the MUXG is 2K. We propose a zero forcing (ZF) scheme for the K-user LOS IC, where each user uses one or more polarim…
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…
A New Metric Function for SC-based Polar Decoders: Polarization, Pruning, and Fast Decoders
本文提出度量函数极化方法以获取极化树各层最优度量，推导相关理论性质，并提出基于变熵的列表剪枝策略，有效降低FSCL等译码算法的计算复杂度。
For a binary-input memoryless symmetric channel $W$, we consider the asymptotic behavior of the polarization process in the large block-length regime when transmission takes place over $W$. In particular, we study the asymptotics of the cumulative distribution $\mathbb{P}(Z_n \leq z)$, where $\{Z_n\}$ is the Bhattacharyya process defined from $W$, and its dependence on the rate of transmission. On the basis of this result, we characterize the asymptotic behavior, as well as its dependence on the rate, of the block error probability of polar codes using the successive cancellation decoder. This…
The polar codes are proven to be capacity-achieving and are shown to have equivalent or even better finite-length performance than the turbo/LDPC codes under some improved decoding algorithms over the additive white Gaussian noise (AWGN) channels. Polar coding is based on the so-called channel polarization phenomenon induced by a transform over the underlying binary-input channel. The channel polarization is found to be universal in many signal processing problems and has been applied to the coded modulation schemes. In this paper, the channel polarization is further extended to the multiple a…
Polar codes are the first proven capacity-achieving codes. Recently, they are adopted as the channel coding scheme for 5G due to their superior performance.A polar code for encoding length-K information bits in length-N codeword could be specified by the polar code construction method. Most construction methods define a polar code related to channel parameter set, e.g. designed signal-to-noise ratio. Polarization weight (PW) is a channel-independent approximation method, which estimates the subchannel reliability as a function of its index. In this paper, we generalize the PW method by includi…
We introduce explicit schemes based on the polarization phenomenon for the tasks of one-way secret key agreement from common randomness and private channel coding. For the former task, we show how to use common randomness and insecure one-way communication to obtain a strongly secure key such that the key construction has a complexity essentially linear in the blocklength and the rate at which the key is produced is optimal, i.e., equal to the one-way secret-key rate. For the latter task, we present a private channel coding scheme that achieves the secrecy capacity using the condition of stron…
The general subject considered in this thesis is a recently discovered coding technique, polar coding, which is used to construct a class of error correction codes with unique properties. In his ground-breaking work, Arıkan proved that this class of codes, called polar codes, achieve the symmetric capacity --- the mutual information evaluated at the uniform input distribution ---of any stationary binary discrete memoryless channel with low complexity encoders and decoders requiring in the order of $O(N\log N)$ operations in the block-length $N$. This discovery settled the long standing open pr…
Polarization Drift Channel Model for Coherent Fibre-Optic Systems
本文提出相干光纤系统中偏振态动态变化的理论信道模型，将一维相位噪声随机游走推广为 Poincaré 球面上的随机偏振漂移，并通过实验验证。
It has been shown that an extension of the basic binary polar transformation also polarizes over finite fields. With it the direct encoding of q-ary sources and channels is a process that can be implemented with simple and efficient algorithms. However, direct polar decoding of q-ary sources and channels is more involved. In this paper we obtain a recursive equation for the likelihood ratio expressed as a LR vector. With it successive cancellation (SC) decoding is applied in a straightforward way. The complexity is quadratic in the order of the field, but the use of the LR vector introduces fa…
A polar coding scheme for fading channels is proposed in this paper. More specifically, the focus is Gaussian fading channel with a BPSK modulation technique, where the equivalent channel could be modeled as a binary symmetric channel with varying cross-over probabilities. To deal with variable channel states, a coding scheme of hierarchically utilizing polar codes is proposed. In particular, by observing the polarization of different binary symmetric channels over different fading blocks, each channel use corresponding to a different polarization is modeled as a binary erasure channel such th…
We prove that the relation $E_0(ρ, W^{-}) + E_0(ρ, W^{+}) \geq 2 E_0(ρ, W)$ holds for any binary input discrete memoryless channel $W$, and $ρ\geq 0$.
The notion of source polarization is introduced and investigated. This complements the earlier work on channel polarization. An application to Slepian-Wolf coding is also considered. The paper is restricted to the case of binary alphabets. Extension of results to non-binary alphabets is discussed briefly.
We study polarization for nonbinary channels with input alphabet of size q=2^r,r=2,3,... Using Arikan's polarizing kernel H_2, we prove that the virtual channels that arise in the process of polarization converge to q-ary channels with capacity 1,2,...,r bits, and that the total transmission rate approaches the symmetric capacity of the channel. This leads to an explicit transmission scheme for q-ary channels. The error probability of decoding using successive cancellation behaves as exp(-N^α), where N is the code length and α is any constant less than 0.5.
Deep CSI Compression for Dual-Polarized Massive MIMO Channels with Disentangled Representation Learning
本文提出DiReNet，通过将双极化CSI解耦为极化共享、垂直极化特定和水平极化特定信息，减少极化相关性带来的冗余，从而提升CSI压缩与恢复性能，并降低近三分之一参数量。
We analyze successive cancellation (SC) decoder by using two random functions. The first function is related to the likelihoods of 0 and 1 in each code position, while the second gives the difference between their posterior probabilities. We then study the second power moments of both functions. We show that these moments are being squared in channel transformations, while their product tends to 0 for growing lengths $n$. This gives an elementary proof of polarization properties of SC decoding. We also derive a simple ordering of decoding channels with construction complexity of order $n\log n…
Antenna Selection in Polarization Reconfigurable MIMO (PR-MIMO) Communication Systems
本文研究极化可重构MIMO系统中的天线选择问题，提出联合极化预编码-后编码及PR-HS/MRT方案，实现3至5 dB信噪比增益并显著提升信道容量和符号错误率性能。
Capacity Bounds under Imperfect Polarization Tracking
本文首次刻画了不完全信道知识下偏振漂移信道的容量，并提出基于Kabsch算法的数据辅助信道估计器，仿真表明其相比最小二乘误差算法可将可实现信息速率提升0.2至0.35比特/符号。
A Novel Channel Model for Reconfigurable Intelligent Surfaces with Consideration of Polarization and Switch Impairments
针对现有RIS信道模型忽略开关损伤与极化效率的问题，本文提出纳入反射场、散射场、天线谐振及移相器等效模型的新模型，并通过3.5 GHz的4×4 RIS阵列与盲控制算法验证其实用性。
Polar codes were introduced by Arikan in 2008 and are the first family of error-correcting codes achieving the symmetric capacity of an arbitrary binary-input discrete memoryless channel under low complexity encoding and using an efficient successive cancellation decoding strategy. Recently, non-binary polar codes have been studied, in which one can use different algebraic geometry codes to achieve better error decoding probability. In this paper, we study the performance of binary polar codes that are obtained from non-binary algebraic geometry codes using concatenation. For binary polar code…
This paper studies the exploitation of triple polarization (TP) for multi-user (MU) holographic multiple-input multiple-output surface (HMIMOS) wireless communication systems, aiming at capacity boosting without enlarging the antenna array size. We specifically consider that both the transmitter and receiver are equipped with an HMIMOS comprising compact sub-wavelength TP patch antennas. To characterize TP MUHMIMOS systems, a TP near-field channel model is proposed using the dyadic Green's function, whose characteristics are leveraged to design a user-cluster-based precoding scheme for mitigat…
Holevo, Schumacher, and Westmoreland's coding theorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channel with classical inputs and quantum outputs. Although they demonstrated the existence of such codes, their proof does not provide an explicit construction of codes for this task. The aim of the present paper is to fill this gap by constructing near-explicit "polar" codes that are capacity-achieving. The codes exploit the channel polarization phenomenon observed by Arikan for the case of classical channels. Channel polarization i…
We introduce an error resilient distributed computing method based on an extension of the channel polarization phenomenon to distributed algorithms. The method leverages an algorithmic split operation that transforms two identical compute nodes to slow and fast workers, which parallels the channel split operation in Polar Codes. This operation preserves the average runtime, analogous to the conservation of Shannon capacity in channel polarization. By leveraging a recursive construction in a similar spirit to the Fast Fourier Transform, this method synthesizes virtual compute nodes with dispers…
Real global-scale quantum communications and quantum key distribution systems cannot be implemented by the current fiber and free-space links. These links have high attenuation, low polarization-preserving capability or extreme sensitivity to the environment. A potential solution to the problem is the space-earth quantum channels. These channels have no absorption since the signal states are propagated in empty space, however a small fraction of these channels is in the atmosphere, which causes slight depolarizing effect. Furthermore, the relative motion of the ground station and the satellite…
We prove that, for all binary-input symmetric memoryless channels, polar codes enable reliable communication at rates within $ε> 0$ of the Shannon capacity with a block length, construction complexity, and decoding complexity all bounded by a {\em polynomial} in $1/ε$. Polar coding gives the {\em first known explicit construction} with rigorous proofs of all these properties; previous constructions were not known to achieve capacity with less than $\exp(1/ε)$ decoding complexity except for erasure channels. We establish the capacity-achieving property of polar codes via a direct analysis of th…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。