Reliable Uncoded Communication in the Underdetermined SIMO MAC with Low-Complexity Decoding Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Department of Electrical Engineering, Stanford University October 3, 2013, Allerton Uncoded transmission in MAC channels

Motivation: Uplink communication Introduction Motivation: Uplink communication 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

System model Hij ∼ N (0, 1), νi ∼ N (0, σ2) Introduction System model x1 ν1 + H H y1 y2 yNR x2 H H + ν2 H N H 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, x is n-user codeword NU single-antenna users with BPSK signalling NR = αNU receive antennas at base station No CSI at transmitters, perfect CSI at receiver Uncoded transmission in MAC channels

Introduction Warmup question What is the number of independent users that can be reliably supported by the channel? Uncoded transmission in MAC channels 4 / 30

Introduction Warmup question What is the number of independent users that can be reliably supported by the channel? Capacity of point to point MIMO “Superuser” with total power NU min(NR, NU ) log(NU SN R) in high SNR regime Suggests that min(NR, NU ) users can be simultaneously supported Achievable using coding across time Uncoded transmission in MAC channels 4 / 30

Practical regime of operation Introduction Practical regime of operation In low SNR, finite blocklength, symbols from constellation set, does the maximum number of users supported stay the same i.e. min(NR, NU ) ? Uncoded transmission in MAC channels 5 / 30

Practical regime of operation Introduction Practical regime of operation In low SNR, finite blocklength, symbols from constellation set, does the maximum number of users supported stay the same i.e. min(NR, NU ) ? Not necessarily Different operating regimes can yield totally different intuition/insights! Uncoded transmission in MAC channels 5 / 30

Result with ML decoding [Allerton ’12] Introduction Result with ML decoding [Allerton ’12] We consider the regime of large NU , fixed rate R and QoS requirement Pe We show As NU → ∞, for any fixed ratio NR , using the maximum likelihood (ML) NU decoder Pe → 0 May be anticipated from capacity of SIMO MAC with transmitter pooling min(NR, NU ) log(NU SN R) Uncoded transmission in MAC channels 10 / 30

Introduction Our work Question Can we achieve similar results with lower-complexity decoders? One answer [ISIT’ 2013] In the regime of α ≥ 1, with a convex relaxation of the ML decoder, Pe → 0 This work Even in the regime of α > 0, with a more general relaxation of the ML decoder, Uncoded transmission in MAC channels 10 / 30

Table of contents 1 Introduction 4 Conclusions Overview of results 2 Overview of results 3 Proof sketch 4 Conclusions Uncoded transmission in MAC channels 10 / 30

Compute xˆ = argminx∈{−1,+1}NU ||y − Hx|| Results Decoders considered Maximum likelihood (ML) decoder Compute xˆ = argminx∈{−1,+1}NU ||y − Hx|| Complexity exponential in NU Interval Search and Quantize (ISQ) decoder Compute xˆISQ = sign(argminx∈[−1,+1]NU ||y − Hx||) Complexity of decoding is polynomial in NU Randomized Interval Search and Quantize (rISQ) decoder Compute xˆrISQ = sign(PS (xr)) where xr ∼ Unif([−1, +1]NU ), PS (y) = argminx∈S ||x − y||∞, S = argminx∈[−1,1]NU ||y − Hx|| Same as ISQ w.h.p. for α ≥ 1 Uncoded transmission in MAC channels 10 / 30

Asymptotic reliability (ML decoder) Results Asymptotic reliability (ML decoder) Let x0 = −1 be transmitted 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 with NU , i.e. P (xˆ =j x0) ≤ 2−cNU for all NU > n0 In other words Equal rate uncoded transmission Fixed NR ratio NU Error probability Pe → 0 with NU → ∞ Uncoded transmission in MAC channels 10 / 30

Asymptotic reliability (ML decoder) Results Asymptotic reliability (ML decoder) Let x0 = −1 be transmitted 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 with NU , i.e. P (xˆ =j x0) ≤ 2−cNU for all NU > n0 In other words Equal rate uncoded 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 / 30

Asymptotic reliability (rISQ decoder) Results Asymptotic reliability (rISQ 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 with NU , i.e. P (||xˆrISQ − x0||0 > kNU ) ≤ 2−cNU log NU for all NU > n0 In other words Per user symbol error probability goes to zero Q Net rate of decay of symbol error probability is polynomial Q Uncoded transmission in MAC channels 11 / 30

Asymptotic reliability (rISQ decoder) Results Asymptotic reliability (rISQ 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 with NU , i.e. P (||xˆrISQ − x0||0 > kNU ) ≤ 2−cNU log NU for all NU > n0 In other words Per user symbol error probability goes to zero Q Net rate of decay of symbol error probability is polynomial Q Thus the price of a lower complexity decoder is a polynomial rate of decay of symbol error probability Uncoded transmission in MAC channels 11 / 30

Remainder of talk (proof outline) Results Remainder of talk (proof outline) Motivation for rISQ decoder Randomized (E, δ)-grid decoder Properties and connections betweeen the above two decoders Uncoded transmission in MAC channels

Table of contents 1 Introduction 4 Conclusions Overview of results Proofs Table of contents 1 Introduction 2 Overview of results 3 Proof sketch 4 Conclusions Uncoded transmission in MAC channels

Need for randomization Proofs Need for randomization Solution for ISQ decoder not unique for 0 < α < 1 In general, it is a set S May contain “bad” points for α < 0.5 [Donoho, Tanner ’10] Definition of “bad” An estimate xˆ is bad if the number of differences in the symbol positions is asymptotically linear in NU , i.e. || sign(xˆ) − x0||0 = Θ(NU ). Idea: Study behaviour of random point from S Uncoded transmission in MAC channels

Need for randomization Proofs Need for randomization S x0 ISQ decoder: S (blue line) represents the solution set of the ISQ decoder for a random 2 × 3 H and no noise, the green point x0 is the transmitted n-user codeword Uncoded transmission in MAC channels

A randomization scheme Proofs A randomization scheme Idea Generate point from random distribution over hypercube and project (R∞ norm) it onto S Efficient to compute solution set, sample and project R∞ norm projection allows us to provably lie within hypercube Uncoded transmission in MAC channels

A randomization scheme (demonstration) Proofs A randomization scheme (demonstration) rISQ decoder: Variation of PS (xr) (blue point) with xr (orange point) Uncoded transmission in MAC channels

Grid approximation of ISQ - (E, δ) grid Proofs Grid approximation of ISQ - (E, δ) grid Definition For any E, δ, the (E, δ) grid GE,δ is the following set {x : xi mod E = δi, |xi| < 1}. Performance guarantees Look at decoder xˆE,δ = argminx∈GE,δ ||y − Hx|| Use same union bounding techniques as for ML decoders Uncoded transmission in MAC channels

Analysing the grid approximation ... Proofs Analysing the grid approximation ... Main steps Group ( 1 )NU (2NU − 1) wrong codewords by the number of errors E Probability of block error NU H2( i '\ ( ( ) U } E max ∈{ 1 NU NR 2 Pe,E,δ ≤ NU i 1,...,N 2 NU 1 + σ2 ) i − 2 2 ( )− 2U αN Bound evaluates to ( 1 )NU NU 1 + 1 E 2 σ2 Probability of kNU symbol errors Pe,k,E,δ ≤ 2−cNU log NU valid for any k, E, σ2, α > 0 Uncoded transmission in MAC channels 19 / 30

Analysing the grid approximation ... Proofs Analysing the grid approximation ... Main steps Group ( 1 )NU (2NU − 1) wrong codewords by the number of errors E Probability of block error NU H2( i '\ ( ( ) U } E max ∈{ 1 NU NR 2 Pe,E,δ ≤ NU i 1,...,N 2 NU 1 + σ2 ) i − 2 2 ( )− 2U αN Bound evaluates to ( 1 )NU NU 1 + 1 E 2 σ2 Probability of kNU symbol errors Pe,k,E,δ ≤ 2−cNU log NU valid for any k, E, σ2, α > 0 In subsequent slides, w.h.p. ≡ probability greater than 1 − 2−cNU log NU Uncoded transmission in MAC channels 19 / 30

Averaging over randomized grids Proofs Averaging over randomized grids Shifting the grid: By varying δ Uncoded transmission in MAC channels

Averaging over randomized grids Proofs Averaging over randomized grids Idea Consider a distribution f (δ) on δ Induces distribution on xˆE,δ Samples drawn from this induced distribution are “good” Is PS (xˆE,δ ) also good? Uncoded transmission in MAC channels

Connection to rISQ: a fact about projection Proofs Connection to rISQ: a fact about projection Lemma There is at least one point of the solution set S of the ISQ decoder within the R∞ ball around xˆE,δ , i.e. ||xˆE,δ − x0||∞ ≤ E Proof Idea By contradiction! If false, quadratic form ||y − Hx||2 is strictly convex inside the hypercube {x : ||x − xˆE,δ ||∞ ≤ E} Grid minimizer can’t be xˆE,δ ! Uncoded transmission in MAC channels

Connection to rISQ: why efficient sampling suffices Proofs Connection to rISQ: why efficient sampling suffices Recall Distribution f (δ) on δ induces distribution on points of solution set S Such points are “good” for any f (δ) For every s ∈ S, there exists δ, such that s = PS (xˆE,δ ) Uncoded transmission in MAC channels 23 / 30

Connection to rISQ: why efficient sampling suffices Proofs Connection to rISQ: why efficient sampling suffices Recall Distribution f (δ) on δ induces distribution on points of solution set S Such points are “good” for any f (δ) For every s ∈ S, there exists δ, such that s = PS (xˆE,δ ) rISQ This decoder also induces a distribution on S This belongs to the family of distributions on S induced by f (δ) Uncoded transmission in MAC channels 23 / 30

Detour: Hardness of multivariate sampling Proofs Detour: Hardness of multivariate sampling Multivariate sampling in general hard Example [Huang, Kannan ’11] If X1, X2, . . . , Xn are sampled from a prior distribution, and satisfy linear constraints in addition, the constrained sampling problem is hard in general For arbitrary priors, the sampling problem may be efficient! We show There exists efficiently sampled points from the ISQ solution set with “good” asymptotic properties! Uncoded transmission in MAC channels 24 / 30

Detour: Hardness of multivariate sampling Proofs Detour: Hardness of multivariate sampling Multivariate sampling in general hard Example [Huang, Kannan ’11] If X1, X2, . . . , Xn are sampled from a prior distribution, and satisfy linear constraints in addition, the constrained sampling problem is hard in general For arbitrary priors, the sampling problem may be efficient! We show There exists efficiently sampled points from the ISQ solution set with “good” asymptotic properties! Even for arbitrarily underdetermined systems, uncoded communication is reliable with efficient decoders Uncoded transmission in MAC channels 24 / 30

Proofs Thus... Result The rISQ decoder returns “good” points i.e. for any k > 0, ||xˆrISQ − x0|| ≤ kNU w.h.p. Reliability of uncoded symbols with rISQ decoder As NU → ∞, Prob(symbol error) → 0 Uncoded transmission in MAC channels

“Good”ness: Can we do better ? Proofs “Good”ness: Can we do better ? An improved result Points returned by the rISQ decoder also satisfy i.e. ||xˆrISQ − x0|| ≤ N E w.h.p. U Improved rate of decay of symbol error probability Still does not achieve the exponential rates of the ML decoder Uncoded transmission in MAC channels

Extensions Similar results extend to: Still smaller α, e.g., for α = Proofs Extensions Similar results extend to: Still smaller α, e.g., for α = 1 (log n)E Non gaussian fading distributions General non-BPSK constellations Finite blocklength transmissions Uncoded transmission in MAC channels

Table of contents 1 Introduction 4 Conclusions Overview of results Summary Table of contents 1 Introduction 2 Overview of results 3 Proof sketch 4 Conclusions Uncoded transmission in MAC channels

Summary Key insights Sufficient degrees of freedom already present in large systems Positive rate possible for every user without coding Diversity allows reliable communication with low-complexity decoders Extensions: Arbitrary constellation, general fading coefficients 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