Fundamental Limits of Many-User MAC With Finite Payloads and Fading

Consider a (multiple-access) wireless communication system where users are connected to a unique base station over a shared-spectrum radio links. Each user has a fixed number $k$ of bits to send to the base station, and his signal gets attenuated by a random channel gain (quasi-static fading). In this paper we consider the many-user asymptotics of Chen-Chen-Guo’2017, where the number of users grows linearly with the blocklength. Differently, though, we adopt a per-user probability of error (PUPE) criterion (as opposed to classical joint-error probability criterion). Under PUPE the finite energy-per-bit communication is possible, and we are able to derive bounds on the tradeoff between energy and spectral efficiencies. We reconfirm the curious behaviour (previously observed for non-fading MAC) of the possibility of almost perfect multi-user interference (MUI) cancellation for user densities below a critical threshold. Further, we demonstrate the suboptimality of standard solutions such as orthogonalization (i.e. TDMA/FDMA) and treating interference as noise (i.e. pseudo-random CDMA without multi-user detection). Notably, the problem treated here can be seen as a variant of support recovery in compressed sensing for the unusual definition of sparsity with one non-zero entry per each contiguous section of $2^{k}$ coordinates. This identifies our problem with that of the sparse regression codes (SPARCs) and hence our results can be equivalently understood in the context of SPARCs with sections of length 2100. Finally, we discuss the relation of the almost perfect MUI cancellation property and the replica-method predictions.

[1]  Jean-Francois Chamberland,et al.  Unsourced Random Access With Coded Compressed Sensing: Integrating AMP and Belief Propagation , 2020, IEEE Transactions on Information Theory.

[2]  O. Papaspiliopoulos.  High-Dimensional Probability: An Introduction with Applications in Data Science , 2020 .

[3]  Alexey Frolov,et al.  A Polar Code Based TIN-SIC Scheme for the Unsourced Random Access in the Quasi-Static Fading MAC , 2020, 2020 IEEE International Symposium on Information Theory (ISIT).

[4]  Ramji Venkataramanan,et al.  Capacity-Achieving Spatially Coupled Sparse Superposition Codes With AMP Decoding , 2020, IEEE Transactions on Information Theory.

[5]  Galen Reeves,et al.  All-or-Nothing Phenomena: From Single-Letter to High Dimensions , 2019, 2019 IEEE 8th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP).

[6]  Jean-Francois Chamberland,et al.  Polar Coding and Random Spreading for Unsourced Multiple Access , 2019, ICC 2020 - 2020 IEEE International Conference on Communications (ICC).

[7]  Sekhar Tatikonda,et al.  Sparse Regression Codes , 2019, Found. Trends Commun. Inf. Theory.

[8]  Yury Polyanskiy,et al.  Energy Efficient Coded Random Access for the Wireless Uplink , 2019, IEEE Transactions on Communications.

[9]  Yury Polyanskiy,et al.  Energy efficient random access for the quasi-static fading MAC , 2019, 2019 IEEE International Symposium on Information Theory (ISIT).

[10]  Christos Thrampoulidis,et al.  Improved bounds on Gaussian MAC and sparse regression via Gaussian inequalities , 2019, 2019 IEEE International Symposium on Information Theory (ISIT).

[11]  Yury Polyanskiy,et al.  Quasi-static fading MAC with many users and finite payload , 2019, 2019 IEEE International Symposium on Information Theory (ISIT).

[12]  Galen Reeves,et al.  The All-or-Nothing Phenomenon in Sparse Linear Regression , 2019, COLT.

[13]  Giuseppe Caire,et al.  SPARCs for Unsourced Random Access , 2019, IEEE Transactions on Information Theory.

[14]  Galen Reeves,et al.  The Replica-Symmetric Prediction for Random Linear Estimation With Gaussian Matrices Is Exact , 2019, IEEE Transactions on Information Theory.

[15]  Giuseppe Caire,et al.  Massive MIMO Unsourced Random Access , 2019, ArXiv.

[16]  Robert Calderbank,et al.  CHIRRUP: a practical algorithm for unsourced multiple access , 2018, Information and Inference: A Journal of the IMA.

[17]  Shi Jin,et al.  Concise Derivation for Generalized Approximate Message Passing Using Expectation Propagation , 2018, IEEE Signal Processing Letters.

[18]  Krishna R. Narayanan,et al.  A Coupled Compressive Sensing Scheme for Uncoordinated Multiple Access , 2018, ArXiv.

[19]  Giuseppe Caire,et al.  On the Fundamental Limits of MIMO Massive Multiple Access Channels , 2018, ICC 2019 - 2019 IEEE International Conference on Communications (ICC).

[20]  Pavel Rybin,et al.  On LDPC Code Based Massive Random-Access Scheme for the Gaussian Multiple Access Channel , 2018, NEW2AN.

[21]  A. Robert Calderbank,et al.  Compressed Neighbour Discovery using Sparse Kerdock Matrices , 2018, 2018 IEEE International Symposium on Information Theory (ISIT).

[22]  Ramji Venkataramanan,et al.  The Error Probability of Sparse Superposition Codes With Approximate Message Passing Decoding , 2017, IEEE Transactions on Information Theory.

[23]  Jun Cheng,et al.  A user-independent serial interference cancellation based coding scheme for the unsourced random access Gaussian channel , 2017, 2017 IEEE Information Theory Workshop (ITW).

[24]  Yury Polyanskiy,et al.  A perspective on massive random-access , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[25]  Yury Polyanskiy,et al.  Low complexity schemes for the random access Gaussian channel , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[26]  Nicolas Macris,et al.  The mutual information in random linear estimation , 2016, 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[27]  Xu Chen,et al.  Capacity of Gaussian Many-Access Channels , 2016, IEEE Transactions on Information Theory.

[28]  Jianhua Lu,et al.  Concise Derivation of Complex Bayesian Approximate Message Passing via Expectation Propagation , 2015, ArXiv.

[29]  Florent Krzakala,et al.  Approximate Message-Passing Decoder and Capacity Achieving Sparse Superposition Codes , 2015, IEEE Transactions on Information Theory.

[30]  Sergio Verdú,et al.  Total variation distance and the distribution of relative information , 2014, 2014 Information Theory and Applications Workshop (ITA).

[31]  Giuseppe Durisi,et al.  Quasi-Static Multiple-Antenna Fading Channels at Finite Blocklength , 2013, IEEE Transactions on Information Theory.

[32]  S. Boucheron,et al.  Concentration inequalities : a non asymptotic theory of independence , 2013 .

[33]  Giuseppe Durisi,et al.  Quasi-static SIMO fading channels at finite blocklength , 2013, 2013 IEEE International Symposium on Information Theory.

[34]  Florent Krzakala,et al.  Compressed sensing of approximately-sparse signals: Phase transitions and optimal reconstruction , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[35]  Wen Li,et al.  Some new two-sided bounds for determinants of diagonally dominant matrices , 2012 .

[36]  Bhaskar D. Rao,et al.  Limits on Support Recovery of Sparse Signals via Multiple-Access Communication Techniques , 2011, IEEE Transactions on Information Theory.

[37]  Richard G. Baraniuk,et al.  Asymptotic Analysis of Complex LASSO via Complex Approximate Message Passing (CAMP) , 2011, IEEE Transactions on Information Theory.

[38]  H. Vincent Poor,et al.  Minimum Energy to Send $k$ Bits Through the Gaussian Channel With and Without Feedback , 2011, IEEE Transactions on Information Theory.

[39]  Galen Reeves,et al.  The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing , 2010, IEEE Transactions on Information Theory.

[40]  H. Vincent Poor,et al.  Minimum energy to send k bits with and without feedback , 2010, 2010 IEEE International Symposium on Information Theory.

[41]  Andrew R. Barron,et al.  Least squares superposition codes of moderate dictionary size, reliable at rates up to capacity , 2010, 2010 IEEE International Symposium on Information Theory.

[42]  H. Vincent Poor,et al.  Channel Coding Rate in the Finite Blocklength Regime , 2010, IEEE Transactions on Information Theory.

[43]  Galen Reeves,et al.  Approximate Sparsity Pattern Recovery: Information-Theoretic Lower Bounds , 2010, IEEE Transactions on Information Theory.

[44]  Andrea Montanari,et al.  The dynamics of message passing on dense graphs, with applications to compressed sensing , 2010, 2010 IEEE International Symposium on Information Theory.

[45]  G. Reeves,et al.  A note on optimal support recovery in compressed sensing , 2009, 2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers.

[46]  Dongning Guo,et al.  A single-letter characterization of optimal noisy compressed sensing , 2009, 2009 47th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[47]  Andrea Montanari,et al.  Message-passing algorithms for compressed sensing , 2009, Proceedings of the National Academy of Sciences.

[48]  Venkatesh Saligrama,et al.  Information Theoretic Bounds for Compressed Sensing , 2008, IEEE Transactions on Information Theory.

[49]  Martin J. Wainwright,et al.  Information-Theoretic Limits on Sparsity Recovery in the High-Dimensional and Noisy Setting , 2007, IEEE Transactions on Information Theory.

[50]  V. Koltchinskii,et al.  High Dimensional Probability , 2006, math/0612726.

[51]  Jari Salo,et al.  On the Asymptotic Geometric Mean of MIMO Channel Eigenvalues , 2006, 2006 IEEE International Symposium on Information Theory.

[52]  Sergio Verdú,et al.  Randomly spread CDMA: asymptotics via statistical physics , 2005, IEEE Transactions on Information Theory.

[53]  Giuseppe Caire,et al.  The throughput of some wireless multiaccess systems , 2002, IEEE Trans. Inf. Theory.

[54]  Sergio Verdú,et al.  Spectral efficiency in the wideband regime , 2002, IEEE Trans. Inf. Theory.

[55]  J. Rosenthal.  A First Look at Rigorous Probability Theory , 2000 .

[56]  Shlomo Shamai,et al.  Spectral Efficiency of CDMA with Random Spreading , 1999, IEEE Trans. Inf. Theory.

[57]  Te Sun Han,et al.  An Information-Spectrum Approach to Capacity Theorems for the General Multiple-Access Channel , 1998, IEEE Trans. Inf. Theory.

[58]  Shlomo Shamai,et al.  Fading Channels: Information-Theoretic and Communication Aspects , 1998, IEEE Trans. Inf. Theory.

[59]  Charles Knessl,et al.  Integral representations and asymptotic expansions for Shannon and Renyi entropies , 1998 .

[60]  G. H. Khachatrian,et al.  A new approach to the design of codes for synchronous-CDMA systems , 1995, IEEE Trans. Inf. Theory.

[61]  S Kirkpatrick,et al.  Critical Behavior in the Satisfiability of Random Boolean Expressions , 1994, Science.

[62]  M. Feder,et al.  A generalization of the entropy power inequality with applications , 1993, IEEE Trans. Inf. Theory.

[63]  Charles R. Johnson,et al.  Improving Hadamard's inequality * , 1984 .

[64]  W. R. van Zwet,et al.  A Strong Law for Linear Functions of Order Statistics , 1980 .

[65]  S. G. Choi,et al.  On a Combinatorial Problem in Number Theory , 1971 .

[66]  R. R. Bahadur.  A Note on Quantiles in Large Samples , 1966 .

[67]  Suhas S. Kowshik,et al.  Coding for random access in wireless networks , 2018 .

[68]  H. Vincent Poor,et al.  Channel coding: non-asymptotic fundamental limits , 2010 .

[69]  Yoshiyuki Kabashima,et al.  Low-density parity-check codes—A statistical physics perspective , 2002 .

[70]  L. Birge,et al.  An alternative point of view on Lepski's method , 2001 .

[71]  M. Ledoux.  The concentration of measure phenomenon , 2001 .

[72]  Giuseppe Caire,et al.  The optimal received power distribution for IC-based iterative multiuser joint decoders , 2001 .

[73]  F. Alajaji,et al.  Lectures Notes in Information Theory , 2000 .

[74]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[75]  I. Olkin,et al.  Entropy of the Sum of Independent Bernoulli Random Variables and of the Multinomial Distribution , 1981 .

[76]  D. A. Bell,et al.  Information Theory and Reliable Communication , 1969 .

[77]  W. H. Mills,et al.  Determination of a Subset from Certain Combinatorial Properties , 1966, Canadian Journal of Mathematics.