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Yury Polyanskiy - One of the best experts on this subject based on the ideXlab platform.
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beta beta bounds finite Blocklength analog of the golden formula
IEEE Transactions on Information Theory, 2018Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, Vincent H PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: 1) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu, which involves the ratio of two Neyman–Pearson $\beta $ functions (beta–beta converse bound); and 2) a novel beta–beta channel-coding achievability bound, expressed again as the ratio of two Neyman–Pearson $\beta $ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta–beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu’s wideband-slope approximation. The proof parallels the derivation of the latter, with the beta–beta bounds used in place of the golden formula. The beta–beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula
IEEE Transactions on Information Theory, 2018Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, H. Vincent PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: 1) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu, which involves the ratio of two Neyman–Pearson $\beta $ functions (beta–beta converse bound); and 2) a novel beta–beta channel-coding achievability bound, expressed again as the ratio of two Neyman–Pearson $\beta $ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta–beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu’s wideband-slope approximation. The proof parallels the derivation of the latter, with the beta–beta bounds used in place of the golden formula. The beta–beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Beta-Beta Bounds: Finite-Blocklength Analog of the Golden Formula
arXiv: Information Theory, 2017Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, H. Vincent PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: (i) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu (2014), which involves the ratio between two Neyman-Pearson $\beta$ functions (beta-beta converse bound), and (ii) a novel beta-beta channel-coding achievability bound, expressed again as the ratio between two Neyman-Pearson $\beta$ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu's (2002) wideband-slope approximation. The proof parallels the elegant derivation in Verdu (2002), with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Peak-to-Average Power Ratio of Good Codes for Gaussian Channel
IEEE Transactions on Information Theory, 2014Co-Authors: Yury Polyanskiy, Yihong WuAbstract:Consider a problem of forward error-correction for the additive white Gaussian noise (AWGN) channel. For finite Blocklength codes, the backoff from the channel capacity is inversely proportional to the square root of the Blocklength. In this paper, it is shown that the codes achieving this tradeoff must necessarily have peak-to-average power ratio (PAPR) proportional to logarithm of the Blocklength. This is extended to codes approaching capacity slower, and to PAPR measured at the output of an orthogonal frequency division multiplexing modulator. As a by-product, the convergence of (Smith's) amplitude-constrained AWGN capacity to Shannon's classical formula is characterized in the regime of large amplitudes. This converse-type result builds upon recent contributions in the study of empirical output distributions of good channel codes.
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Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength
arXiv: Information Theory, 2013Co-Authors: Wei Yang, Giuseppe Durisi, Tobias Koch, Yury PolyanskiyAbstract:This paper investigates the maximal achievable rate for a given Blocklength and error probability over quasi-static multiple-input multiple-output (MIMO) fading channels, with and without channel state information (CSI) at the transmitter and/or the receiver. The principal finding is that outage capacity, despite being an asymptotic quantity, is a sharp proxy for the finite-Blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion is shown to be zero regardless of whether the fading realizations are available at both transmitter and receiver, at only one of them, or at neither of them. These results follow from analytically tractable converse and achievability bounds. Numerical evaluation of these bounds verifies that zero dispersion may indeed imply fast convergence to the outage capacity as the Blocklength increases. In the example of a particular $1 \times 2$ single-input multiple-output (SIMO) Rician fading channel, the Blocklength required to achieve $90\%$ of capacity is about an order of magnitude smaller compared to the Blocklength required for an AWGN channel with the same capacity. For this specific scenario, the coding/decoding schemes adopted in the LTE-Advanced standard are benchmarked against the finite-Blocklength achievability and converse bounds.
Ankur A. Kulkarni - One of the best experts on this subject based on the ideXlab platform.
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shannon meets von neumann a minimax theorem for channel coding in the presence of a jammer
IEEE Transactions on Information Theory, 2020Co-Authors: Sharu Theresa Jose, Ankur A. KulkarniAbstract:We study the setting of channel coding over a family of channels whose state is controlled by an adversarial jammer by viewing it as a zero-sum game between a finite Blocklength encoder-decoder team, and the jammer. The encoder-decoder team choose stochastic encoding and decoding strategies to minimize the average probability of error in transmission, while the jammer chooses a distribution on the state-space to maximize this probability. The min-max value of the game is equivalent to channel coding for a compound channel - we call this the Shannon solution of the problem. The max-min value corresponds to finding a mixed channel with the largest value of the minimum achievable probability of error. When the min-max and max-min values are equal, the problem is said to admit a saddle-point or von Neumann solution. While a Shannon solution always exists, the communicating team’s problem is nonconvex for finite Blocklengths, whereby a von Neumann solution may not exist. Despite this, we show that the min-max and max-min values become equal asymptotically in the large Blocklength limit, for all but finitely many rates. We explicitly characterize this limiting value as a function of the rate and obtain tight finite Blocklength bounds on the min-max and max-min value. As a corollary we get an explicit expression for the $\epsilon $ -capacity of a compound channel under stochastic codes - the first such result, to the best of our knowledge. Our results demonstrate a deeper relation between the compound channel and mixed channel than was previously known. They also show that the conventional information-theoretic viewpoint, articulated via the Shannon solution, coincides asymptotically with the game-theoretic one articulated via the von Neumann solution. Key to our results is the derivation of new finite Blocklength upper bounds on the min-max value of the game via a novel achievability scheme, and lower bounds on the max-min value obtained via the linear programming relaxation based approach we introduced in [2] .
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Improved Finite Blocklength Converses for Slepian–Wolf Coding via Linear Programming
IEEE Transactions on Information Theory, 2019Co-Authors: Sharu Theresa Jose, Ankur A. KulkarniAbstract:A new finite Blocklength converse for the Slepian–Wolf coding problem, which significantly improves on the best-known converse due to Miyake and Kanaya, is presented. To obtain this converse, an extension of the linear programming (LP)-based framework for finite Blocklength point-to-point coding problems is employed. However, a direct application of this framework demands a complicated analysis for the Slepian–Wolf problem. An analytically simpler approach is presented, wherein LP-based finite Blocklength converses for this problem are synthesized from point-to-point lossless source coding problems with perfect side-information at the decoder. New finite Blocklength converses for these point-to-point problems are derived by employing the LP-based framework, and the new converse for Slepian–Wolf coding is obtained by an appropriate combination of these converses.
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Improved Finite Blocklength Converses for Slepian-Wolf Coding via Linear Programming
arXiv: Information Theory, 2018Co-Authors: Sharu Theresa Jose, Ankur A. KulkarniAbstract:A new finite Blocklength converse for the Slepian- Wolf coding problem is presented which significantly improves on the best known converse for this problem, due to Miyake and Kanaya [2]. To obtain this converse, an extension of the linear programming (LP) based framework for finite Blocklength point- to-point coding problems from [3] is employed. However, a direct application of this framework demands a complicated analysis for the Slepian-Wolf problem. An analytically simpler approach is presented wherein LP-based finite Blocklength converses for this problem are synthesized from point-to-point lossless source coding problems with perfect side-information at the decoder. New finite Blocklength metaconverses for these point-to-point problems are derived by employing the LP-based framework, and the new converse for Slepian-Wolf coding is obtained by an appropriate combination of these converses.
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ITA - Linear Programming Based Finite Blocklength Converses in Information Theory
2018 Information Theory and Applications Workshop (ITA), 2018Co-Authors: Ankur A. Kulkarni, Sharu Theresa JoseAbstract:A linear programming based framework is presented to derive finite Blocklength converses for coding problems in information theory which is also extendable to network settings. In the point-to-point setting, the LP based framework recovers and in fact improves on almost all well-known finite Blocklength converses for lossy joint source-channel coding, lossy source coding and channel coding. Moreover, the LP based framework is shown to be asymptotically tight for the averaged and compound channels under the maximum probability of error criterion. Further, for multiterminal Slepian- Wolf source coding problem, a systematic approach to synthesize new converses from considering point-to-point lossless source coding (with side-information at decoder) sub-problems is introduced. The method derives new finite Blocklength converse for Slepian- Wolf coding which significantly improves on the converse of Miyake and Kanaya.
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ITW - Linear programming based finite Blocklength converses for some network-like problems
2017 IEEE Information Theory Workshop (ITW), 2017Co-Authors: Sharu Theresa Jose, Ankur A. KulkarniAbstract:The linear programming (LP) based approach we introduced in [1] for finding finite Blocklength converses for joint source-channel coding is extended to some network-like settings. Finite Blocklength channel coding of compound and averaged channels under the maximum probability error criterion is considered. Through the LP approach new converses are obtained which imply a weak converse for both channels and a strong converse for the compound channel. The LP approach is also extended to the networked setting and a new finite Blocklength converse for Slepian-Wolf coding which improves on the converse in Han [2, Lemma 7.2.2] is derived.
H. Vincent Poor - One of the best experts on this subject based on the ideXlab platform.
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ICASSP - On Polar Coding For Finite Blocklength Secret Key Generation Over Wireless Channels
ICASSP 2020 - 2020 IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2020Co-Authors: Henri Hentila, Yanina Y. Shkel, Visa Koivunen, H. Vincent PoorAbstract:We consider the problem of secret key generation from correlated Gaussian random variables in the finite Blocklength regime. Such keys could be used to encrypt communication in IoT networks, and have provable secrecy guarantees in contrast to classic cryptographic approaches. We investigate the performance of polar coding schemes for generating the secret key over short Blocklengths. Our simulation results show that the proposed scheme achieves close to theoretical upper bounds at short Blocklengths.
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GLOBECOM - mmWave-MIMO Based 5G Wireless Ad-Hoc Networks in the Finite Blocklength Regime
2019 IEEE Global Communications Conference (GLOBECOM), 2019Co-Authors: Xi Zhang, Jingqing Wang, H. Vincent PoorAbstract:The integration of millimeter wave (mmWave) and multiple-input and multiple-output (MIMO) techniques has been designed to provide reliable communications with large degrees of freedom while supporting the explosively growing number of mobile users. Under stringent requirements in terms of latency and reliability, due to the infinite Blocklength requirement of the Shannon’s theorem, researchers have investigated new methods to characterize the wireless data transmissions considering the block error probability. The finite Blocklength coding (FBC) technique has been developed to model the finite Blocklength coding rate in the non-asymptotic regime while supporting short-packet communications over 5G wireless ad- hoc networks. However, because of the design complexity when characterizing the second-order coding rate over mmWave MIMO based wireless channels while being integrated with FBC, how to accurately derive the finite Blocklength coding rate over the mmWave MIMO wireless fading channels is still an open problem over 5G wireless ad-hoc networks. To tackle the above- mentioned challenges, we propose and develop the system model which can efficiently integrate the mmWave-MIMO techniques with the finite Blocklength coding over 5G wireless ad-hoc networks. In particular, we derive the system equations which characterize the foundational information-theoretical relationship between the finite Blocklength channel capacity and the coding rate measures over our proposed mmWave MIMO based 5G wireless ad-hoc networks in the finite Blocklength regime. Also conducted is the MATLAB-based performance evaluation results, which validate and analyze our proposed schemes over mmWave MIMO based 5G wireless ad-hoc networks in the finite Blocklength regime.
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Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula
IEEE Transactions on Information Theory, 2018Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, H. Vincent PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: 1) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu, which involves the ratio of two Neyman–Pearson $\beta $ functions (beta–beta converse bound); and 2) a novel beta–beta channel-coding achievability bound, expressed again as the ratio of two Neyman–Pearson $\beta $ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta–beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu’s wideband-slope approximation. The proof parallels the derivation of the latter, with the beta–beta bounds used in place of the golden formula. The beta–beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Beta-Beta Bounds: Finite-Blocklength Analog of the Golden Formula
arXiv: Information Theory, 2017Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, H. Vincent PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: (i) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu (2014), which involves the ratio between two Neyman-Pearson $\beta$ functions (beta-beta converse bound), and (ii) a novel beta-beta channel-coding achievability bound, expressed again as the ratio between two Neyman-Pearson $\beta$ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu's (2002) wideband-slope approximation. The proof parallels the elegant derivation in Verdu (2002), with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Channel Coding Rate in the Finite Blocklength Regime
IEEE Transactions on Information Theory, 2010Co-Authors: Yury Polyanskiy, H. Vincent Poor, Sergio VerduAbstract:This paper investigates the maximal channel coding rate achievable at a given Blocklength and error probability. For general classes of channels new achievability and converse bounds are given, which are tighter than existing bounds for wide ranges of parameters of interest, and lead to tight approximations of the maximal achievable rate for Blocklengths n as short as 100. It is also shown analytically that the maximal rate achievable with error probability ? isclosely approximated by C - ?(V/n) Q-1(?) where C is the capacity, V is a characteristic of the channel referred to as channel dispersion , and Q is the complementary Gaussian cumulative distribution function.
Vincent H Poor - One of the best experts on this subject based on the ideXlab platform.
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on polar coding for finite Blocklength secret key generation over wireless channels
International Conference on Acoustics Speech and Signal Processing, 2020Co-Authors: Henri Hentila, Yanina Y. Shkel, Visa Koivunen, Vincent H PoorAbstract:We consider the problem of secret key generation from correlated Gaussian random variables in the finite Blocklength regime. Such keys could be used to encrypt communication in IoT networks, and have provable secrecy guarantees in contrast to classic cryptographic approaches. We investigate the performance of polar coding schemes for generating the secret key over short Blocklengths. Our simulation results show that the proposed scheme achieves close to theoretical upper bounds at short Blocklengths.
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mmwave mimo based 5g wireless ad hoc networks in the finite Blocklength regime
Global Communications Conference, 2019Co-Authors: Xi Zhang, Jingqing Wang, Vincent H PoorAbstract:The integration of millimeter wave (mmWave) and multiple-input and multiple-output (MIMO) techniques has been designed to provide reliable communications with large degrees of freedom while supporting the explosively growing number of mobile users. Under stringent requirements in terms of latency and reliability, due to the infinite Blocklength requirement of the Shannon’s theorem, researchers have investigated new methods to characterize the wireless data transmissions considering the block error probability. The finite Blocklength coding (FBC) technique has been developed to model the finite Blocklength coding rate in the non-asymptotic regime while supporting short-packet communications over 5G wireless ad- hoc networks. However, because of the design complexity when characterizing the second-order coding rate over mmWave MIMO based wireless channels while being integrated with FBC, how to accurately derive the finite Blocklength coding rate over the mmWave MIMO wireless fading channels is still an open problem over 5G wireless ad-hoc networks. To tackle the above- mentioned challenges, we propose and develop the system model which can efficiently integrate the mmWave-MIMO techniques with the finite Blocklength coding over 5G wireless ad-hoc networks. In particular, we derive the system equations which characterize the foundational information-theoretical relationship between the finite Blocklength channel capacity and the coding rate measures over our proposed mmWave MIMO based 5G wireless ad-hoc networks in the finite Blocklength regime. Also conducted is the MATLAB-based performance evaluation results, which validate and analyze our proposed schemes over mmWave MIMO based 5G wireless ad-hoc networks in the finite Blocklength regime.
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sum capacity of the mimo many access gaussian noise channel
IEEE Transactions on Communications, 2019Co-Authors: Alex Dytso, Yanina Y. Shkel, Gang Feng, Vincent H PoorAbstract:Providing massive connectivity is one of the key challenges for the next generation of wireless communication networks, and hence the capacity limits of massive connectivity need to be thoroughly studied. The uplink in the regime of massive connectivity is captured by the many-access channel (MnAC) model, assuming the number of users to be extremely large and comparable to the Blocklength. This work investigates a generalized MnAC, in which the transmitters and/or the receiver can be equipped with multiple antennas, and the channel gain of each user is allowed to be different. This model is referred to as the multiple-input and multiple-output (MIMO) MnAC model. In the MnAC paradigm, the message length (i.e., the number of bits communicated) is not necessarily linear in the Blocklength. Therefore, instead of the conventional code rate, the message length is studied and defined as a function of the Blocklength. This work characterizes the sum-message-length capacity (SMC) of the MIMO Gaussian MnAC in the regime where the number of users increases sub-linearly in the Blocklength (i.e., $K_{n} = o(n)$ ). The SMC is numerically compared to lower bounds on achievable rate at finite Blocklengths and is shown to be a good approximation for system performance. The impact of the number of antennas per user on SMC is also investigated. While in the single antenna MnAC model the conventional code rate is always zero, it is shown that in the MIMO MnAC it is possible to achieve positive rate by increasing the number of antennas per user. Furthermore, the antenna-user index is defined and the SMC is characterized for different antenna-user joint regimes. This provides useful insights for future MIMO MnAC system design.
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beta beta bounds finite Blocklength analog of the golden formula
IEEE Transactions on Information Theory, 2018Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, Vincent H PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: 1) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu, which involves the ratio of two Neyman–Pearson $\beta $ functions (beta–beta converse bound); and 2) a novel beta–beta channel-coding achievability bound, expressed again as the ratio of two Neyman–Pearson $\beta $ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta–beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu’s wideband-slope approximation. The proof parallels the derivation of the latter, with the beta–beta bounds used in place of the golden formula. The beta–beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
Wei Yang - One of the best experts on this subject based on the ideXlab platform.
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beta beta bounds finite Blocklength analog of the golden formula
IEEE Transactions on Information Theory, 2018Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, Vincent H PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: 1) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu, which involves the ratio of two Neyman–Pearson $\beta $ functions (beta–beta converse bound); and 2) a novel beta–beta channel-coding achievability bound, expressed again as the ratio of two Neyman–Pearson $\beta $ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta–beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu’s wideband-slope approximation. The proof parallels the derivation of the latter, with the beta–beta bounds used in place of the golden formula. The beta–beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula
IEEE Transactions on Information Theory, 2018Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, H. Vincent PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: 1) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu, which involves the ratio of two Neyman–Pearson $\beta $ functions (beta–beta converse bound); and 2) a novel beta–beta channel-coding achievability bound, expressed again as the ratio of two Neyman–Pearson $\beta $ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta–beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu’s wideband-slope approximation. The proof parallels the derivation of the latter, with the beta–beta bounds used in place of the golden formula. The beta–beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Beta-Beta Bounds: Finite-Blocklength Analog of the Golden Formula
arXiv: Information Theory, 2017Co-Authors: Wei Yang, Yury Polyanskiy, Giuseppe Durisi, Austin Collins, H. Vincent PoorAbstract:It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-Blocklength extension of this relation. This extension consists of two elements: (i) a finite-Blocklength channel-coding converse bound by Polyanskiy and Verdu (2014), which involves the ratio between two Neyman-Pearson $\beta$ functions (beta-beta converse bound), and (ii) a novel beta-beta channel-coding achievability bound, expressed again as the ratio between two Neyman-Pearson $\beta$ functions. To demonstrate the usefulness of this finite-Blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-Blocklength extension of Verdu's (2002) wideband-slope approximation. The proof parallels the elegant derivation in Verdu (2002), with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-Blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.
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Fading Channels: Capacity and Channel Coding Rate in the Finite-Blocklength Regime
2015Co-Authors: Wei YangAbstract:Information-theoretic studies on the fundamental limits of communication over wireless fading channels typically rely on simplifying assumptions, such as perfect channel state information (CSI), infinite Blocklength, and vanishing probability of error. Although these assumptions are reasonable for most of the current wireless communication systems, they may be inaccurate for next-generation wireless systems. Indeed, next-generation wireless systems will need to support a much wider range of features, such as ultra-high data rate, extremely low latency, and low energy consumption, for which the assumptions listed above may not be valid. This thesis investigates the fundamental limits of fading channels under a set of assumptions that are more relevant for future wireless systems. First, we characterize the capacity of Rayleigh block-fading multiple-input multiple-output (MIMO) channels with no a priori CSI at the transmitter and the receiver in the high signal-to-noise ratio regime. We show that unitary space time modulation, which is capacity-achieving for MIMO systems with a small number of antennas, is not capacity-achieving when the total number of antennas exceeds the coherence time of the fading channel, a situation that is relevant for large-MIMO systems. We also provide the input distribution that achieves the capacity of large-MIMO fading channels. Second, we study the maximal achievable rate for a given Blocklength and error probability over MIMO quasi-static fading channels, subject to different power constraints on the transmitted codewords: the short-term (i.e., per-codeword) power constraint and the long-term (i.e., average-over-all-codeword) power constraint. For channels subject to a short-term power constraint, we prove that outage capacity---despite being an asymptotic quantity---is a sharp proxy for the finite-Blocklength fundamental limits of slow-fading channels. Specifically, the channel dispersion---a quantity that measures the backoff from capacity in the finite-Blocklength regime---is shown to be zero regardless of whether the fading realizations are available at the transmitter and/or the receiver. The situation is drastically different when a long-term power constraint is present. In this case, if the transmitter has perfect CSI, then the outage capacity is higher than in the short-term power constraint case. Approaching the outage capacity, however, requires codes with much longer Blocklengths. In both cases, we develop easy-to-evaluate approximations for the maximal achievable rate and demonstrate their accuracy by comparison to nonasymptotic achievability and converse bounds. Finally, we investigate the minimum energy required to transmit $k$ information bits with a given reliability over a MIMO Rayleigh block-fading channel, with and without CSI at the receiver. It is well known that the ratio between the minimum energy per bit and the noise level converges to $-1.59$ dB as $k$ goes to infinity, regardless of whether CSI is available at the receiver or not. We show that lack of CSI at the receiver causes a slowdown in the speed of convergence to $-1.59$ dB as $k\to\infty$ compared to the case of perfect receiver CSI. Specifically, in the no-CSI case, the gap to $-1.59$ dB is proportional to $((\log k) /k)^{1/3}$, whereas when perfect CSI is available at the receiver, this gap is proportional to $1/\sqrt{k}$.
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diversity versus multiplexing at finite Blocklength
International Symposium on Wireless Communication Systems, 2014Co-Authors: Johan Ostman, Wei Yang, Giuseppe Durisi, Tobias KochAbstract:A finite blocklenth analysis of the diversity-multiplexing tradeoff is presented, based on nonasymptotic bounds on the maximum channel coding rate of multiple-antenna block-memoryless Rayleigh-fading channels. The bounds in this paper allow one to numerically assess for which packet size, number of antennas, and degree of channel selectivity, diversity-exploiting schemes are close to optimal, and when instead the available spatial degrees of freedom should be used to provide spatial multiplexing. This finite Blocklength view on the diversity-multiplexing tradeoff provides insights on the design of delay-sensitive ultra-reliable communication links.