1. 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

2. Outline Introduction Our work Pictures and proofs Extensions3
Pictures and proofs 4
Extensions 5
Conclusions
Uncoded transmission in MAC channels

3. Table of contents Introduction Our work Pictures and proofs Extensions3
Pictures and proofs 4
Extensions 5
Conclusions
Uncoded transmission in MAC channels

4. 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

5. 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
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6. 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
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7. Introduction
Warmup question
What is the number of independent streams that can be simultaneously supported by the channel?
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8. 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
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9. 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?
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10. 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!
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11. 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)
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12. 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
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13. 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
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14. 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
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15. 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 → ∞
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16. 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
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17. 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)!
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18. 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
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19. 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
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20. 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
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21. 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 ?
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22. 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
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23. 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

24. Table of contents Introduction Our work Pictures and proofs Extensions3
Pictures and proofs 4
Extensions 5
Conclusions
Uncoded transmission in MAC channels

25. 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
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26. 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)
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27. 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
Regions for transmitted codewords
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28. 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)
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29. Proof outline (ML): Union bound
Proofs
Proof outline (ML): Union bound
Let’s say x0 is transmitted.
Pe ≤ P (xˆ = x)
x/=x0
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30. 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
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31. 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
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32. 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)) ≤ σ2
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33. 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)) ≤ σ2
Depends only on i, i.e. number of columns, not on particular columns considered
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34. 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 +
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35. 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
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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
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37. 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

38. 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

39. 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

40. 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
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41. 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 '?
'<;;
..;
-0.2
100
Uncoded transm1Ss1on m MAC channels

42. 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 '?
'<;;
..;
-0.2
300
Uncoded transm1Ss1on m MAC channels

43. 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

44. 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
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45. 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

46. 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

47. Table of contents Introduction Our work Pictures and proofs Extensions3
Pictures and proofs 4
Extensions 5
Conclusions
Uncoded transmission in MAC channels

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

49. \ 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

50. 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

51. 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

52. 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

53. 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

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