1. Uncoded transmission in MAC channels achieves arbitrarily small error probability
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Department of Electrical Engineering, Stanford University
December 17, 2012
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

2. Outline 5 Conclusions Introduction Our work Proofs Extensions 1 4
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

3. Table of contents 5 Conclusions Introduction Our work Proofs1
Introduction
Our work
Proofs 4
Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

5. System model Hij ∼ N (0, 1), νi ∼ N (0, σ2) y = Hx + ν
Introduction
System model x1 ν1 + H H y1 y2
yNR H x2 H + ν2 H H N 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
NU users, each with 1 antenna, NR receive antennas
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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6. xˆ = argminx∈{−1,+1}NU ||y − Hx||
Introduction
System model x1 ν1 + H H y1 y2
yNR H x2 H + ν2 H H N 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)
No CSI at transmitters, perfect CSI at receiver Receiver does perfect ML decoding
xˆ = argminx∈{−1,+1}NU ||y − Hx||
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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7. NU users, each user transmitting at a fixed rate R Given Pe
Introduction
We ask
NU users, each user transmitting at a fixed rate R Given Pe
How many receive antennas are required ?
We show
As NU → ∞, Pe → 0 for any ratio NR NU
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

8. Background For MIMO point-to-point channel For a MAC channel
Introduction
Background
For MIMO point-to-point channel
Multiplexing gain 1 min(NR, NU ) 2
Achieved by coding across time
Pe → 0 with larger blocklengths
For a MAC channel
NU transmitters each with 1 antenna
NR receiver antennas
Total transmit power grows like NU
Multiplexing gain 1 min(NR, NU ) log NU 2
Achieved by coding over large blocklengths
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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9. Background For MIMO point-to-point channel For a MAC channel
Introduction
Background
For MIMO point-to-point channel
Multiplexing gain 1 min(NR, NU ) 2
Achieved by coding across time
Pe → 0 with larger blocklengths
For a MAC channel
NU transmitters each with 1 antenna
NR receiver antennas
Total transmit power grows like NU
Multiplexing gain 1 min(NR, NU ) log NU 2
Achieved by coding over large blocklengths
Both results rely on coding across time to get low Pe
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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10. Introduction
Main question
Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ?
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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11. Introduction
Main question
Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ?
Intuition
Large systems already have sufficient degrees of freedom to achieve low error probability
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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12. Introduction
Main question
Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ?
Intuition
Large systems already have sufficient degrees of freedom to achieve low error probability
To investigate this, from now on, we consider BPSK transmissions without coding
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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13. Table of contents 5 Conclusions Introduction Our work Proofs
Results
Table of contents 1
Introduction
Our work
Proofs 4
Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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14. Reliability in communication
Results
Reliability in communication
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 =/ x) ≤ 2−cNU for all NU > n0
In other words
Equal rate transmission
Any NR ratio NU
Error probability Pe → 0 with NU → ∞
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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15. Reliability in communication
Results
Reliability in communication
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 =/ x) ≤ 2−cNU for all NU > n0
In other words
Equal rate transmission
Any NR ratio NU
Error probability Pe → 0 with NU → ∞
This is due to a combination of receiver diversity and spatial multiplexing
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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16. 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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17. 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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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
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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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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
Smallest NR ratio achievable with coding is NU
NR 2
NU ≥ log NU + 2c
Can we achieve similar ratios for reliable uncoded systems also ?
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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20. Minimum number of required Rx antennas per user: Results
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 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

21. Table of contents 5 Conclusions Introduction Our work Proofs1
Introduction
Our work
Proofs 4
Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

22. Received space Hx2NU −1 View of the 2NU points in the projected space
Proofs
Received space
Hx2NU −1
Hx3
Hx0
Decision region
Hx2
Transmitted point
Incorrect point
Hx1
View of the 2NU points in the projected space
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

23. Proof outline: Union bound
Proofs
Proof outline: Union bound
Let’s say x0 is transmitted.
Pe ≤ P (xˆ = x)
x/=x0
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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24. Proof outline: Union bound
Proofs
Proof outline: 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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25. Proof outline: Computing pairwise error probabilities
Proofs
Proof outline: 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

26. Proof outline: Average pairwise error probability
Proofs
Proof outline: 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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27. Proof outline: Average pairwise error probability
Proofs
Proof outline: 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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28. Proof outline: Upper bound on total error probability
Proofs
Proof outline: 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 +
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

29. Proof outline: Asymptotic limits
Proofs
Proof outline: 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

30. " 0.0 •I Behaviour of g(n), constant a Proof outline: "' :·
Proofs
Proof outline:
Behaviour of g(n), constant a
a= 0. 3, <72 = 0. 5, n =50 , g(n) = max! i ng( i,n) = 0.26
0.4
0.2
" 0.0
\.',:,
"' : :·
-0.2
-:--
: \o;; -- •I
···-·-·-····-·-·-··
--: · 1 20 30 40 50
M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels s

31. " 0.0 . '\ ' • Behaviour of g(n), constant a Proof outline:
Proofs
Proof outline:
Behaviour of g(n), constant a
a = 0.3,o2 = 0. 5, n = 100 , g(n) = max! i ng( i,n) = 0.08
0.4 ·-
" 0.0 .
'>, \
-0.2 ' •
-\ -- '\ ·
--: · 20 40 60 80
100
M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels s

32. Behaviour of g(n), constant a
Proofs
Behaviour of g(n), constant a
Proof outline:
M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels s

33. Behaviour of g(n), constant a
Proofs
Behaviour of g(n), constant a
Proof outline:
M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels s

34. Proof outline: Behaviour of g(n), constant α
Proofs
Proof outline: Behaviour of g(n), constant α
Message
For fixed α > 0 and large enough n,
g(n) ≤ c(α) < 0
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels
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35. "'-"- 0.0 • .........., : ' Behaviour of g(n), a= ; Proof outline:
Proofs
Proof outline:
Behaviour of g(n), a= ; 1
£ , <7 2
0. 5, n
100 . g(n)
maxl $i$n9(i,n)
0.03
···-·-·-····-·-·-····-
0.2
"'-"
"' , ,
-0.2
···---'-- • \
·i\
···-·-·-····-·-·-··
··:- ···-···-·..·-···-·
: ' \ 20 40 60 80
100
M Chowdhury A Goldsm1th T We1ssm3n Uncoded trdnsmiSSIOn m MAC chdnnels
21 1 2s

36. . •... ..._;• """' Behaviour of g(n), a= ; Proof outline: 1 ··-:
Proofs
Proof outline:
Behaviour of g(n), a= ; 1
< O.l,<T2
0. 5,n
300 , g(n)
maxl $i$n9 (i,n)
-0.01
0.4
0.2 '?
"""'
0.0
; . .
\ ·.••
..._;•
-0.2
o' .
•... \ \
-0.4
··-: 50
100
150
200
250
300
M Chowdhury A Goldsm1th T We1ssm3n
Uncoded trdnsmiSSIOn m MAC chdnnels
21 1 2s

37. Pe ≤ 2 log NU Proof outline: Behaviour of g(n), α = Message
Proofs
Proof outline: 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

38. Table of contents 5 Conclusions Introduction Our work Proofs
Extensions
Table of contents 1
Introduction
Our work
Proofs 4
Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

39. Extensions So far discussion focused on
BPSK constellations: Each user transmits from {−1, +1}
Gaussian channel statistics (Rayleigh fading)
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

40. \ 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

41. 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

42. 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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

43. Table of contents 5 Conclusions Introduction Our work Proofs
Summary
Table of contents 1
Introduction
Our work
Proofs 4
Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

44. Summary
Key insights
Sufficient degrees of freedom already present in large systems
Receiver diversity allows reliable communication Positive rate possible for every user without coding
Ongoing work: Extensions to suboptimal decoders, imperfect CSI, correlated channels
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels

45. Thank you for your attention Questions/Comments?
Summary
Thank you for your attention Questions/Comments?
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels