Introduction Results Proofs Summary Uncoded transmission in MAC channels achieves arbitrarily small error probability Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Department of Electrical Engineering, Stanford University Allerton, 2012 Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Motivation: Uplink communication in large networks Introduction Results Proofs Summary Motivation: Uplink communication in large networks Often limited by transmitter side constraints Power/energy requirements Delay constraints Processing capability Examples Cellular uplink Sensor networks Low power radio Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

System model Hij ∼ N (0, 1), νi ∼ N (0, σ2), y = Hx + ν Introduction Results Proofs Summary System model x1 ν1 H H + y1 H H + ν2 y2 H H n 2 H . . 1 H n . . + νm ym H xn NU single antenna users NR antenna receiver Hij ∼ N (0, 1), νi ∼ N (0, σ2), y = Hx + ν NU users, each with 1 antenna, NR receive antennas No CSI at transmitters, perfect CSI at receiver ML decoding at receiver Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

NU users, each user transmitting at a fixed rate R Given Pe Introduction Results Proofs Summary We ask NU users, each user transmitting at a fixed rate R Given Pe How many receive antennas are required ? We show As NU → ∞, Pe → 0 for any ratio NR NU Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Background For MIMO point-to-point channel For a MAC channel Introduction Results Proofs Summary Background For MIMO point-to-point channel Multiplexing gain 1 min(NR , NU ) 2 Achieved by coding across time Pe → 0 with larger blocklengths For a MAC channel NU transmitters each with 1 antenna NR receiver antennas Total transmit power grows like NU Multiplexing gain 1 min(NR , NU ) log NU 2 Achieved by coding over large blocklengths Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Background For MIMO point-to-point channel For a MAC channel Introduction Results Proofs Summary Background For MIMO point-to-point channel Multiplexing gain 1 min(NR , NU ) 2 Achieved by coding across time Pe → 0 with larger blocklengths For a MAC channel NU transmitters each with 1 antenna NR receiver antennas Total transmit power grows like NU Multiplexing gain 1 min(NR , NU ) log NU 2 Achieved by coding over large blocklengths Both results rely on coding across time to get low Pe Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Introduction Results Proofs Summary Main question Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ? Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Introduction Results Proofs Summary Main question Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ? Intuition Large systems already have sufficient degrees of freedom to achieve low error probability Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Introduction Results Proofs Summary Main question Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ? Intuition Large systems already have sufficient degrees of freedom to achieve low error probability To investigate this, from now on, we consider BPSK transmissions without coding Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Reliability in communication Introduction Results Proofs Summary Reliability in communication Theorem For NR = αNU for any α > 0 and a large enough n0, there exists a c > 0 such that the probability of block error goes to zero exponentially fast with NU , i.e. P(xˆ /= x ) ≤ 2−cNU for all NU > n0 In other words Equal rate transmission Any NR ratio NU Error probability Pe → 0 with NU → ∞ Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Reliability in communication Introduction Results Proofs Summary Reliability in communication Theorem For NR = αNU for any α > 0 and a large enough n0, there exists a c > 0 such that the probability of block error goes to zero exponentially fast with NU , i.e. P(xˆ /= x ) ≤ 2−cNU for all NU > n0 In other words Equal rate transmission Any NR ratio NU Error probability Pe → 0 with NU → ∞ This is due to a combination of receiver diversity and spatial multiplexing Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Minimum number of required Rx antennas per user Introduction Results Proofs Summary Minimum number of required Rx antennas per user Let us consider coding across time Every user needs a rate of 1 bit per channel use on average Sum rate required from the system is NU Equal rate capacity of the system is 1 NR log NU + cNR 2 Thus, the lowest NR to support reliable transmissions is 2NU NR ≥ 2c + log N U Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Minimum number of required Rx antennas per user Introduction Results Proofs Summary Minimum number of required Rx antennas per user Let us consider coding across time Every user needs a rate of 1 bit per channel use on average Sum rate required from the system is NU Equal rate capacity of the system is 1 NR log NU + cNR 2 Thus, the lowest NR to support reliable transmissions is 2NU NR ≥ 2c + log N U Smallest NR ratio achievable with coding is NU NR 2 NU ≥ log NU + 2c Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Minimum number of required Rx antennas per user Introduction Results Proofs Summary Minimum number of required Rx antennas per user Let us consider coding across time Every user needs a rate of 1 bit per channel use on average Sum rate required from the system is NU Equal rate capacity of the system is 1 NR log NU + cNR 2 Thus, the lowest NR to support reliable transmissions is 2NU NR ≥ 2c + log N U Smallest NR ratio achievable with coding is NU NR 2 NU ≥ log NU + 2c Can we achieve similar ratios for reliable uncoded systems also ? Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Minimum number of required Rx antennas per user: Results Introduction Results Proofs Summary Minimum number of required Rx antennas per user: Results Theorem For NR = 2+E for any E > 0 and a large enough n0, there exists a NU log NU c > 0 such that the probability of block error goes to zero exponentially fast with NU , i.e. −cNU P(xˆ /= x ) ≤ 2 log NU for all NU > n0 In other words Coding does not reduce the number of required antennas per user for 1 bit per channel use Scaling behaviour of minimum NR is Θ( 1 ) NU log NU Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Received space View of the 2NU points in the projected space Hx2NU −1 Introduction Results Proofs Summary Received space Hx2NU −1 Hx3 Hx0 Decision region Hx2 Transmitted point Incorrect point Hx1 View of the 2NU points in the projected space Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Proof outline: Union bound Introduction Results Proofs Summary Proof outline: Union bound Let m = NR , n = NU Expected decoding error probability with i mistakes 1 ( i \−m/2 E(Pe,i ) ≤ 2 1 + σ2 Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Proof outline: Union bound Introduction Results Proofs Summary Proof outline: Union bound Let m = NR , n = NU Expected decoding error probability with i mistakes 1 ( i \−m/2 E(Pe,i ) ≤ 2 1 + σ2 Expected decoding error probability 1 (n\ ( n i \−m/2 σ2 E(Pe ) ≤ \ 1 + 2 i i =1 Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Proof outline: Asymptotic limits Introduction Results Proofs Summary Proof outline: Asymptotic limits Define m n α = Expected decoding error probability ( i \−αn/2 σ2 1 2nH2( n ) E(Pe ) ≤ n max i 1 + 1≤i ≤n 2 = 2ng (n) where ( i \ ( α i \ log n − log 2 σ2 n g (n) c,. max H2 1≤i ≤n − log 1 + + n 2 Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Proof outline: Behaviour of g (n) Introduction Results Proofs Summary Proof outline: Behaviour of g (n) For α > 0, g (n) becomes Θ(−1) Thus Pe ≤ 2−d1n for some d1 > 0 0.5 0.4 0.3 For α = 2+E , E > 0, g (n) becomes g(n) 0.2 log n Θ(− log n ) Eventually 1 0.1 0.0 −0.1 0 20 40 60 n 80 100 Pe ≤ 2− log n for some d2 > 0 d2n g (n) versus n for m = 2.7n log n Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Introduction Results Proofs Summary Key insights Sufficient degrees of freedom already present in large systems Receiver diversity allows reliable communication Positive rate possible for every user without coding Results carry over to general discrete constellations and fading statistics Ongoing work: Extensions to suboptimal decoders, imperfect CSI Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels

Thank you for your attention Questions/Comments? Introduction Results Proofs Summary Thank you for your attention Questions/Comments? Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels