1. Reliable Uncoded Communication in the SIMO MAC with Low-Complexity Decoding
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Department of Electrical Engineering, Stanford University
July 10, 2013, ISIT
Uncoded transmission in MAC channels

2. Outline 1 Introduction 4 Conclusions Overview of results Proof sketch2
Overview of results 3
Proof sketch
4 Conclusions
Uncoded transmission in MAC channels

3. Table of contents 1 Introduction 4 Conclusions Overview of results2
Overview of results 3
Proof sketch
4 Conclusions
Uncoded transmission in MAC channels

4. 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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5. 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

6. Introduction
Warmup question
What is the number of independent users that can be reliably supported by the channel?
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7. 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 Achieveable using coding across time
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8. 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?
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9. 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?
Not necessarily
Different operating regimes can yield totally different intuition/insights!
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10. 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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11. Introduction
Our work
Question
Can we achieve similar results with lower-complexity decoders?
One answer
In the regime of α ≥ 1, with a particular relaxation of the ML decoder,
Pe → 0
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12. Table of contents 1 Introduction 4 Conclusions Overview of results2
Overview of results 3
Proof sketch
4 Conclusions
Uncoded transmission in MAC channels

13. 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
Proposed decoder: Interval Search and Quantize (ISQ) Compute xˆISQ = sign(argminx∈[−1,+1]NU ||y − Hx||) sign(x) is elementwise application of sign(·)
Solution is unique w.p. 1
Complexity of decoding is polynomial in NU
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14. 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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15. 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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16. Asymptotic reliability (ISQ decoder)
Results
Asymptotic reliability (ISQ decoder)
Theorem
For NR = αNU for any α ≥ 1, 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 (||xˆISQ − 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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17. Asymptotic reliability (ISQ decoder)
Results
Asymptotic reliability (ISQ decoder)
Theorem
For NR = αNU for any α ≥ 1, 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 (||xˆISQ − 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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18. 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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19. Decision region (x space): ML decoder, NR = 2, NU = 3,
Proofs
Decision region (x space): ML decoder, NR = 2, NU = 3,
α = 2/3
0.0
Possible transmitted n-user codewords
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20. Decision region (Hx space): ML decoder,
Proofs
Decision region (Hx space): ML decoder,
NR = 2,Nu = 3,a = 2/3 2 -1 -2
- '
-- o
Possible received codewords
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21. Decision region (x space): ISQ decoder,
Proofs
Decision region (x space): ISQ decoder,
NR = 2, NU = 3, α = 2/3
0.0
Regions for transmitted n-user codewords
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22. Decision region (Hx space): ISQ decoder, NR = 2,Nu = 3,a = 2/3
Proofs
Decision region (Hx space): ISQ decoder,
NR = 2,Nu = 3,a = 2/3
Mapping of regions under multiplication by H. Note overlap of decision regions
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23. Proofs
Decision region (Hx space) with one more receiver antenna: NR = 3, NU = 3, α = 1
0.0
Mapping of regions under multiplication by H
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24. Proof outline (ISQ) Grid search approximation of interval search
Proofs
Proof outline (ISQ)
Grid search approximation of interval search
Block error probability versus symbol error probability Grid search versus interval search
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25. Proof outline (ISQ): Grid approximation - (E, δ) grid
Proofs
Proof outline (ISQ): Grid approximation - (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}.
Idea
Look at decoder xˆE,δ = sign(argminx∈GE,δ ||y − Hx||)
Use union bounding techniques for ML decoders
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26. Proof outline (ISQ): Block error versus symbol error
Proofs
Proof outline (ISQ): Block error versus symbol error
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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27. Proof outline (ISQ): Block error versus symbol error
Proofs
Proof outline (ISQ): Block error versus symbol error
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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28. Proof outline (ISQ): Grid search versus interval search
Proofs
Proof outline (ISQ): Grid search versus interval search
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
For NR ≥ NU , the channel matrix will be full rank w.p. 1
||y − Hx||2 will be strictly convex
xˆE,δ will be a vertex of the hypercube containing xˆISQ
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29. Proof outline (ISQ): Grid search versus interval search
Proofs
Proof outline (ISQ): Grid search versus interval search
An illustration
Interval minimum versus grid minimum
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30. Proof outline (ISQ): Grid search versus interval search
Proofs
Proof outline (ISQ): Grid search versus interval search
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
For NR ≥ NU , the channel matrix will be full rank w.p. 1
||y − Hx||2 will be strictly convex
xˆE,δ will be a vertex of the hypercube containing xˆISQ
xˆE,δ has sublinear number of entries close to 0 w.h.p.
Conclusion
sign(xˆE,δ ) differs from sign(xˆISQ) in at most sublinear symbols w.h.p.
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31. 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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32. 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: ISQ decoder with larger multiplexing ratio, imperfect
CSI, correlation patterns (in channel or symbols)
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33. Thank you for your attention Questions/Comments?
Summary
Thank you for your attention Questions/Comments?
Uncoded transmission in MAC channels