Reliable Uncoded Communication in the SIMO MAC with Efficient Decoding Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Department of Electrical Engineering, Stanford University July 1, 2013, Broadcom Corp., Sunnyvale Uncoded transmission in MAC channels

Outline Introduction Our work Pictures and proofs Extensions 3 Pictures and proofs 4 Extensions 5 Conclusions Uncoded transmission in MAC channels

Table of contents Introduction Our work Pictures and proofs Extensions 3 Pictures and proofs 4 Extensions 5 Conclusions 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 Uncoded transmission in MAC channels

NU single antenna users Introduction System model x1 x2 ν1 + ν2 H H y1 y2 yNR H H H H H N . . R R H N . . + νNR NR antenna receiver H xNU NU single antenna users Hij ∼ N (0, 1), νi ∼ N (0, σ2) y = Hx + ν NU users, each with 1 antenna NR receive antennas at base station R U Uncoded transmission in MAC channels 5 / 36

System model No CSI at transmitters, perfect CSI at receiver x1 ν1 y1 Introduction System model x1 ν1 + H H y1 y2 yNR H x2 H + ν2 N H H H . R . R H N . . + H xNU R U νNR NU single antenna users NR antenna receiver No CSI at transmitters, perfect CSI at receiver Uncoded transmission in MAC channels 5 / 36

Introduction Warmup question What is the number of independent streams that can be simultaneously supported by the channel? Uncoded transmission in MAC channels 6 / 36

Introduction Warmup question What is the number of independent streams that can be simultaneously supported by the channel? Asymptotic regime of high SNR min(NR, NU ) streams Theoretically achieved using coding across time Transmitter channel knowledge usually important Uncoded transmission in MAC channels 6 / 36

Introduction Warmup question What is the number of independent streams that can be simultaneously supported by the channel? Asymptotic regime of high SNR min(NR, NU ) streams Theoretically achieved using coding across time Transmitter channel knowledge usually important A different (practical) operating regime Low SNR, one-shot communication, symbols from constellation set Does the maximum number of streams still stay the same? Uncoded transmission in MAC channels 6 / 36

Introduction Warmup question What is the number of independent streams that can be simultaneously supported by the channel? Asymptotic regime of high SNR min(NR, NU ) streams Theoretically achieved using coding across time Transmitter channel knowledge usually important A different (practical) operating regime Low SNR, one-shot communication, symbols from constellation set Does the maximum number of streams still stay the same? Not necessarily Different operating regimes can yield totally different intuition/insights! Uncoded transmission in MAC channels 6 / 36

Introduction Our setting We consider the regime of large NU , fixed rate R and QoS requirement Pe We ask How many receive antennas are required ? What kind of decoders can give reliable decoding ? How useful is coding in large systems ? We show As NU → ∞, Pe → 0 for any fixed ratio NR , using either of the following: NU Maximum likelihood decoder Interval Search and Quantize (ISQ) decoder (deterministic and randomized) Uncoded transmission in MAC channels 7 / 36

Introduction Our setting We consider the regime of large NU , fixed rate R and QoS requirement Pe We ask How many receive antennas are required ? What kind of decoders can give reliable decoding ? How useful is coding in large systems ? We show As NU → ∞, Pe → 0 for any fixed ratio NR , using either of the following: NU Maximum likelihood decoder Interval Search and Quantize (ISQ) decoder (deterministic and randomized) Number of data streams supported can be much larger than min(NR, NU )! Uncoded transmission in MAC channels 7 / 36

Table of contents Introduction Our work Pictures and proofs Extensions Results Table of contents Introduction Our work 3 Pictures and proofs 4 Extensions 5 Conclusions Uncoded transmission in MAC channels 10 / 36

Results Decoders Decoder 1: Maximum likelihood (ML) decoder Compute xˆ = argminx∈{−1,+1}NU ||y − Hx|| Complexity exponential in NU Decoder 2: Interval Search and Quantize (ISQ) decoder Compute xˆ = sign(argminx∈[−1,+1]NU ||y − Hx||) sign(x) is elementwise application of sign(·) If solution is non unique (for NR < NU ), return randomized projection onto solution space Complexity of decoding and random projection is polynomial in NU Uncoded transmission in MAC channels 10 / 36

Reliability in communication: (ML decoder) Results Reliability in communication: (ML decoder) Let x0 = −1 be transmitted 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 =/ x0) ≤ 2−cNU for all NU > n0 In other words Equal rate transmission Fixed NR ratio NU Error probability Pe → 0 with NU → ∞ Uncoded transmission in MAC channels 10 / 36

Reliability in communication: (ML decoder) Results Reliability in communication: (ML decoder) Let x0 = −1 be transmitted 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 =/ x0) ≤ 2−cNU for all NU > n0 In other words Equal rate transmission Fixed NR ratio NU Error probability Pe → 0 with NU → ∞ This is a combination of receiver diversity and spatial multiplexing Uncoded transmission in MAC channels 10 / 36

Reliability in communication: (ISQ decoder) Results Reliability in communication: (ISQ decoder) Theorem For NR = αNU for any α, k > 0 and a large enough n0, there exists a c > 0 such that the probability of kNU symbol errors goes to zero super-exponentially fast with NU , i.e. P (|xˆ − x0|0 > kNU ) ≤ 2−cNU log NU for all NU > n0 In other words Per user symbol error goes to zero Net rate of decay of error probability is at best polynomial Error even without noise (self interference)! Uncoded transmission in MAC channels

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 Uncoded transmission in MAC channels 12 / 36

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 Uncoded transmission in MAC channels 12 / 36

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 Uncoded transmission in MAC channels 12 / 36

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 ? Uncoded transmission in MAC channels 12 / 36

Results Minimum number of required Rx antennas per user: Results for ML decoder 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 with the maximum likelihood decoder 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 Uncoded transmission in MAC channels

Pk /= x) ≤ 2−cNU log NU for all NU > n0 Results Minimum number of required Rx antennas per user: Results for ISQ decoder Theorem For NR = α for any E > 0, k > 0 and a large enough n0, there exists a NU c > 0 such that the probability of kNU symbol errors with the randomized ISQ decoder goes to zero exponentially fast with NU , i.e. Pk /= x) ≤ 2−cNU log NU for all NU > n0 In other words Per-user reliability still holds with NU → ∞ (0) Decay of per-user error probability is at most polynomial (not exponential Q) Uncoded transmission in MAC channels

Table of contents Introduction Our work Pictures and proofs Extensions 3 Pictures and proofs 4 Extensions 5 Conclusions Uncoded transmission in MAC channels

Decision region (transmitter space): ML decoder, Proofs Decision region (transmitter space): ML decoder, NR = 2, NU = 3 1 0.5 −0.5 −1 0.5 1 0.5 −0.5 −0.5 −1 −1 Possible transmitted codewords Uncoded transmission in MAC channels

Decision region (receiver space): ML decoder, Proofs Decision region (receiver space): ML decoder, NR = 2, NU = 3 3 2 1 −1 −2 −3 −3 −2 −1 1 2 3 Possible transmitted codewords (as seen by the receiver) Uncoded transmission in MAC channels

NR=2,Nu=3 Decision region (transmitter space): ISQ decoder, " Proofs Decision region (transmitter space): ISQ decoder, NR=2,Nu=3 " 0 5 _, 1 0.5 1 -1 Regions for transmitted codewords Uncoded transm1Ss1on m MAC channels 1s 1 36

Decision region (receiver space): Proofs Decision region (receiver space): ISQ decoder, NR=2,Nu=3 _, :, -- ,-----_, -- ---- ---- -- Regions for some selected transmitted codewords (as seen by the receiver) Uncoded transm1Ss1on m MAC channels 19 1 36

Proof outline (ML): Union bound Proofs Proof outline (ML): Union bound Let’s say x0 is transmitted. Pe ≤ P (xˆ = x) x/=x0 Uncoded transmission in MAC channels 20 / 36

Proof outline (ML): Union bound Proofs Proof outline (ML): 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 Uncoded transmission in MAC channels 20 / 36

Proof outline (ML): Computing pairwise error probabilities Proofs Proof outline (ML): 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 Uncoded transmission in MAC channels

Proof outline (ML): Average pairwise error probability Proofs Proof outline (ML): 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 Uncoded transmission in MAC channels 22 / 36

Proof outline (ML): Average pairwise error probability Proofs Proof outline (ML): 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 Uncoded transmission in MAC channels 22 / 36

Proof outline (ML): Upper bound on total error probability Proofs Proof outline (ML): 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 + Uncoded transmission in MAC channels

Proof outline (ML): Asymptotic limits Proofs Proof outline (ML): 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 Uncoded transmission in MAC channels

Proof outline (ML): Behaviour of g(n), constant a Proofs Proof outline (ML): Behaviour of g(n), constant a Uncoded transm1Ss1on m MAC channels 2s 1 36

Proof outline (ML): Behaviour of g(n), constant a Proofs Proof outline (ML): Behaviour of g(n), constant a Uncoded transm1Ss1on m MAC channels 2s 1 36

Proof outline (ML): Behaviour of g(n), constant a Proofs Proof outline (ML): Behaviour of g(n), constant a Uncoded transm1Ss1on m MAC channels 2s 1 36

Proof outline (ML): Behaviour of g(n), constant a Proofs Proof outline (ML): Behaviour of g(n), constant a Uncoded transm1Ss1on m MAC channels 2s 1 36

Proof outline (ML): Behaviour of g(n), constant α Proofs Proof outline (ML): Behaviour of g(n), constant α Message For fixed α > 0 and large enough n, g(n) ≤ c(α) < 0 Uncoded transmission in MAC channels 25 / 36

Proof outline (ML): Behaviour of g(n), a= ; Proofs Proof outline (ML): Behaviour of g(n), a= ; 1 £ O.l,o2 0. 5, n 100 , g (n) max!$i$n9 (i,n) 0.03 '? '<;; ..; 0.0 -0.2 100 Uncoded transm1Ss1on m MAC channels 26 1 36

Proof outline (ML): Behaviour of g(n), a= ; Proofs Proof outline (ML): Behaviour of g(n), a= ; 1 £ O.l,o2 0. 5, n 300 , g(n) max!$i$n9 (i,n) -0.01 '? '<;; ..; 0.0 -0.2 300 Uncoded transm1Ss1on m MAC channels 26 1 36

Pe ≤ 2 log NU Proof outline (ML): Behaviour of g(n), α = Message Proofs Proof outline (ML): 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 Uncoded transmission in MAC channels

Proof outline (ISQ): (E, δ) grid Proofs Proof outline (ISQ): (E, δ) grid We look at Pe,k , which is the probability of error in kNU symbols Definition For any E, the (E, δ) grid GE,δ is the following set {x : xi mod E = δi, |xi| < 1}. Idea Look at decoder xˆ = sign(argminx∈GE,δ ||y − Hx||) Using techniques as in the previous proof, 1 NU E 2−c˜NU log NU Pe,k,E,δ ≤ Relate the grid error probability to the ISQ decoder error probability Uncoded transmission in MAC channels

Proof outline (ISQ): Grid versus interval performances Proofs Proof outline (ISQ): Grid versus interval performances Lemma The minimizer over GE,δ will w.p. 1 be “close” to at least one solution of the ISQ decoder, in the following sense ||xˆE,δ − xˆISQ||∞ ≤ E. Plausibility arguments and implications Easier to visualize for NR ≥ NU For NR < NU , the same results continue to hold on average for any distribution f (δ) on the offset δ Computationally feasible to sample a vector “close” to xˆE,δ ∼ f (δ) xˆE,δ has “close” to zero entries in at most sublinear positions Uncoded transmission in MAC channels

Proof outline (ISQ) (contd.) Proofs Proof outline (ISQ) (contd.) Conclusion Pe,k ≤ 2−CNU log NU , i.e. ISQ estimate differs from x0 in at most a sublinear number of symbols ˜ Uncoded transmission in MAC channels

Table of contents Introduction Our work Pictures and proofs Extensions 3 Pictures and proofs 4 Extensions 5 Conclusions Uncoded transmission in MAC channels

Extensions So far discussion focused on BPSK constellation: Each user transmits from {−1, +1} Gaussian channel statistics (Rayleigh fading) 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 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 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 Uncoded transmission in MAC channels

Table of contents Introduction Our work Pictures and proofs Extensions Summary Table of contents Introduction Our work 3 Pictures and proofs 4 Extensions 5 Conclusions Uncoded transmission in MAC channels

Summary Key insights Sufficient degrees of freedom already present in large systems Receiver diversity allows reliable communication with efficient decoders Positive rate possible for every user without coding Ongoing work: Imperfect CSI, correlation patterns (in channel or symbols) Uncoded transmission in MAC channels

Thank you for your attention Questions/Comments? Summary Thank you for your attention Questions/Comments? Uncoded transmission in MAC channels