Uncoded transmission in MAC channels achieves arbitrarily small error probability Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Department of Electrical Engineering, Stanford University December 17, 2012 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Outline 5 Conclusions Introduction Our work Proofs Extensions 1 4 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Table of contents 5 Conclusions Introduction Our work Proofs 1 Introduction Our work Proofs 4 Extensions 5 Conclusions M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Motivation: Uplink communication in large networks Introduction 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

System model Hij ∼ N (0, 1), νi ∼ N (0, σ2) y = Hx + ν Introduction System model x1 ν1 + H H y1 y2 yNR H x2 H + ν2 H H N H . R . R H N . . + H xNU R U νNR 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 5 / 28

xˆ = argminx∈{−1,+1}NU ||y − Hx|| Introduction System model x1 ν1 + H H y1 y2 yNR H x2 H + ν2 H H N H . R . R H N . . + H xNU R U νNR NU single antenna users NR antenna receiver Hij ∼ N (0, 1), νi ∼ N (0, σ2) No CSI at transmitters, perfect CSI at receiver Receiver does perfect ML decoding xˆ = argminx∈{−1,+1}NU ||y − Hx|| M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 5 / 28

NU users, each user transmitting at a fixed rate R Given Pe Introduction 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Background For MIMO point-to-point channel For a MAC channel Introduction 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 7 / 28

Background For MIMO point-to-point channel For a MAC channel Introduction 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 7 / 28

Introduction Main question Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ? M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 8 / 28

Introduction 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 8 / 28

Introduction 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 8 / 28

Table of contents 5 Conclusions Introduction Our work Proofs Results Table of contents 1 Introduction Our work Proofs 4 Extensions 5 Conclusions M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 10 / 28

Reliability in communication Results Reliability in communication Theo rem 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 → ∞ M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 10 / 28

Reliability in communication Results Reliability in communication Theo rem 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 10 / 28

Minimum number of required Rx antennas per user Results 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 Every user has average power 1 Sum rate required from the system is NU T Equal rate capacity of the system is 1 log |(I + HH )| 2 σ2 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28

Minimum number of required Rx antennas per user Results 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 Every user has average power 1 Sum rate required from the system is NU Equal rate capacity of the system is . NR log NU + cNR = 1 2 Thus, the lowest NR to support reliable transmissions is 2NU NR ≥ 2c + log N U M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28

Minimum number of required Rx antennas per user Results 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 Every user has average power 1 Sum rate required from the system is NU Equal rate capacity of the system is . NR log NU + cNR = 1 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28

Minimum number of required Rx antennas per user Results 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 Every user has average power 1 Sum rate required from the system is NU Equal rate capacity of the system is . NR log NU + cNR = 1 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 ? M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28

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 c > 0 NU log NU 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 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Table of contents 5 Conclusions Introduction Our work Proofs 1 Introduction Our work Proofs 4 Extensions 5 Conclusions M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Received space Hx2NU −1 View of the 2NU points in the projected space Proofs Received space Hx2NU −1 Hx3 Hx0 Decision region Hx2 Transmitted point Incorrect point Hx1 View of the 2NU points in the projected space M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Proof outline: Union bound Proofs Proof outline: Union bound Let’s say x0 is transmitted. Pe ≤ P (xˆ = x) x/=x0 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 15 / 28

Proof outline: Union bound Proofs Proof outline: Union bound Let’s say x0 is transmitted. Pe ≤ P (xˆ = x) x/=x0 Idea Group wrong codewords by the number of positions in which they differ from x0 Thus NU Pe ≤ P (xˆ = x), i=1 x:d(x,x0)=i where d(·, ·) computes the hamming distance between its two arguments M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 15 / 28

Proof outline: Computing pairwise error probabilities Proofs Proof outline: Computing pairwise error probabilities Let xi be a codeword with i positions different from x0. Then ||H(xi − x0)|| 2σ P (xˆ = xi) = Q || i 2hb(j)|| \ = Q j=1 2σ b(j) is the jth position where xi and x0 differ, hb is the bth column of H 1 || i 2hb(j)||2 \ ≤ 2 exp − j=1 8σ2 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Proof outline: Average pairwise error probability Proofs Proof outline: Average pairwise error probability Idea Average over channel realizations 1 \\ || i j=1 2hb(j)||2 EH (P (xˆ = xi)) ≤ EH 2 exp − 8σ2 This is just the moment generating function of an appropriately scaled chi squared distribution. Closed form expression NR 1 i − 2 EH (P (xˆ = xi)) ≤ 2 1 + σ2 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 17 / 28

Proof outline: Average pairwise error probability Proofs Proof outline: Average pairwise error probability Idea Average over channel realizations 1 \\ || i j=1 2hb(j)||2 EH (P (xˆ = xi)) ≤ EH 2 exp − 8σ2 This is just the moment generating function of an appropriately scaled chi squared distribution. Closed form expression NR 1 i − 2 EH (P (xˆ = xi)) ≤ 2 1 + σ2 Depends only on i, i.e. number of columns, not on particular columns considered M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 17 / 28

Proof outline: Upper bound on total error probability Proofs Proof outline: Upper bound on total error probability Let EH (P (xˆ = xi)) = Pe,i, since it is independent of xi. Then the total error probability is NU NU 1 N Pe ≤ P (xˆ = x) = U Pe,i 2 i i=1 x:d(x,x0)=i i=1 1 NU NU i NR − ≤ 1 + σ2 2 2 i i=1 1 NU i − 2 NR ≤ NU max NU 1 + i∈{1,...,NU } 2 i σ2 NU H2( i \ i − 2 σ2 NR ≤ max 2 2 i∈{1,...,NU } NU 1 + M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Proof outline: Asymptotic limits Proofs Proof outline: Asymptotic limits Define NR NU α = Decoding error probability Pe ≤ 2NU g(NU ) where i α i log n − log 2 σ2 n g(n) � max H2 1≤i≤n − log 1 + + n 2 � max g(i, n) 1≤i≤n M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

" 0.0 •I Behaviour of g(n), constant a Proof outline: "' :· Proofs Proof outline: Behaviour of g(n), constant a a= 0. 3, <72 = 0. 5, n =50 , g(n) = max! i ng( i,n) = 0.26 0.4 0.2 " 0.0 \.',:, "' : :· -0.2 -:-- : \o;; -- •I -0.4 ···-·-·-····-·-·-·· --:- ---------·------- 1 20 30 40 50 M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels 20 1 2s

" 0.0 . '\ ' • Behaviour of g(n), constant a Proof outline: Proofs Proof outline: Behaviour of g(n), constant a a = 0.3,o2 = 0. 5, n = 100 , g(n) = max! i ng( i,n) = 0.08 0.4 0.2 -------- -- ------- --·- " 0.0 . '>, \ -0.2 ' • -\ -- '\ -0.4 --------·---------- --:- ---------·------- 20 40 60 80 100 M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels 20 1 2s

Behaviour of g(n), constant a Proofs Behaviour of g(n), constant a Proof outline: M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels 20 1 2s

Behaviour of g(n), constant a Proofs Behaviour of g(n), constant a Proof outline: M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels 20 1 2s

Proof outline: Behaviour of g(n), constant α Proofs Proof outline: Behaviour of g(n), constant α Message For fixed α > 0 and large enough n, g(n) ≤ c(α) < 0 M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 20 / 28

"'-"- 0.0 • .........., : ' Behaviour of g(n), a= ; Proof outline: Proofs Proof outline: Behaviour of g(n), a= ; 1 £ 0.1, <7 2 0. 5, n 100 . g(n) maxl $i$n9(i,n) 0.03 0.4 ···-·-·-····-·-·-····- 0.2 "'-"- 0.0 "' , .........., -0.2 ···---'-- • \ ·i\ -0.4 ···-·-·-····-·-·-·· ··:- ···-···-·..·-···-· : ' \ 20 40 60 80 100 M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels 21 1 2s

. •... ..._;• """' Behaviour of g(n), a= ; Proof outline: 1 ··-: Proofs Proof outline: Behaviour of g(n), a= ; 1 < O.l,<T2 0. 5,n 300 , g(n) maxl $i$n9 (i,n) -0.01 0.4 0.2 '? """' 0.0 ; . . \ ·.•• ..._;• -0.2 o' . •... \ \ -0.4 ··-: 50 100 150 200 250 300 M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels 21 1 2s

Pe ≤ 2 log NU Proof outline: Behaviour of g(n), α = Message Proofs Proof outline: Behaviour of g(n), α = 2+E log n Message For fixed α > 0 and large enough n, g(n) ≤ c(E) < 0, c(E) goes down as Θ(− ) 1 log n Thus −cNU Pe ≤ 2 log NU M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Table of contents 5 Conclusions Introduction Our work Proofs Extensions Table of contents 1 Introduction Our work Proofs 4 Extensions 5 Conclusions M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Extensions So far discussion focused on BPSK constellations: Each user transmits from {−1, +1} Gaussian channel statistics (Rayleigh fading) M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

\ for the number of receiver antennas per user Extensions Arbitrary finite constellation C Pairwise error probability can be bounded by 1 || i dhb(j)||2 \ P (xˆ = xi) ≤ 2 exp − j=1 , 8σ2 where d is the minimum distance of the constellation i.e. d = min x∈C,y∈C,x=/ y ||x − y|| In general loose Same scaling Θ ( 1 \ for the number of receiver antennas per user log NU M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Arbitrary fading statistics Extensions Arbitrary fading statistics Idea Central limit theorem and rate of convergence of distributions! Specifically Error due to i mismatches depends on the statistics of i j=1 hb(j) Berry Esseen bound guarantees Θ( 1 ) convergence, i.e. √n C˜ sup |Fn(x) − Fg (x)| ≤ √ x n Can be shown that Pe,i ≤ Ci− 2 , NR for all i ≥ i0 Thus Asymptotic behaviour of Pe,i in i same as that for Gaussian fading M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Arbitrary fading statistics Extensions Arbitrary fading statistics Idea Central limit theorem and rate of convergence of distributions! Specifically Error due to i mismatches depends on the statistics of i j=1 hb(j) Berry Esseen bound guarantees Θ( 1 ) convergence, i.e. √n C˜ sup |Fn(x) − Fg (x)| ≤ √ x n Can be shown that Pe,i ≤ Ci− 2 , NR for all i ≥ i0 But Pe = i0 Pe,i + NU Pe,i ? i=1 i=i0+1 Terms with less than i0 mismatches → 0 with large NR M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Table of contents 5 Conclusions Introduction Our work Proofs Summary Table of contents 1 Introduction Our work Proofs 4 Extensions 5 Conclusions M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

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 Ongoing work: Extensions to suboptimal decoders, imperfect CSI, correlated channels M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

Thank you for your attention Questions/Comments? Summary Thank you for your attention Questions/Comments? M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels