1. 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
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2. 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
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3. 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
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4. Introduction
Warmup question
What is the number of independent users that can be reliably supported by the channel?
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5. 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
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6. 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 ) ?
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7. 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!
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8. 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)
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9. 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,
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10. Table of contents 1 Introduction 4 Conclusions Overview of results2
Overview of results 3
Proof sketch
4 Conclusions
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11. 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
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12. 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 → ∞
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13. 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
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14. 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
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15. 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
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16. 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
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17. 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
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18. 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
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19. 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
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20. 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
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21. A randomization scheme (demonstration)
Proofs
A randomization scheme (demonstration)
rISQ decoder: Variation of PS (xr) (blue point) with xr (orange point)
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22. 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
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23. 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
E 2 σ2
Probability of kNU symbol errors
Pe,k,E,δ ≤ 2−cNU log NU valid for any k, E, σ2, α > 0
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24. 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
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
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25. Averaging over randomized grids
Proofs
Averaging over randomized grids
Shifting the grid: By varying δ
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26. 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?
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27. 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 !
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28. 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 )
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29. 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)
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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!
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31. 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
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32. 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
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33. “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
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34. 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
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35. 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
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36. 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)
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37. Thank you for your attention Questions/Comments?
Summary
Thank you for your attention Questions/Comments?
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