1. Introduction Results Proofs Summary
Uncoded transmission in MAC channels achieves arbitrarily small error probability
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Department of Electrical Engineering, Stanford University
Allerton, 2012
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

2. Motivation: Uplink communication in large networks
Introduction Results Proofs Summary
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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

3. System model Hij ∼ N (0, 1), νi ∼ N (0, σ2), y = Hx + ν
Introduction Results Proofs Summary
System model x1 ν1 H H + y1 H H + ν2 y2 H H n 2 H . . 1 H n . . + νm ym H xn
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 No CSI at transmitters, perfect CSI at receiver
ML decoding at receiver
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

4. NU users, each user transmitting at a fixed rate R Given Pe
Introduction Results Proofs Summary
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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

5. Background For MIMO point-to-point channel For a MAC channel
Introduction Results Proofs Summary
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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

6. Background For MIMO point-to-point channel For a MAC channel
Introduction Results Proofs Summary
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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

7. Introduction Results Proofs Summary
Main question
Is coding necessary in large systems to achieve Reliability in communication (Pe → 0) ? Optimal number of receiver antennas per user ?
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

8. Introduction Results Proofs Summary
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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

9. Introduction Results Proofs Summary
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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

10. Reliability in communication
Introduction Results Proofs Summary
Reliability in communication
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 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 → ∞
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

11. Reliability in communication
Introduction Results Proofs Summary
Reliability in communication
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 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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

12. Minimum number of required Rx antennas per user
Introduction Results Proofs Summary
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 Sum rate required from the system is NU
Equal rate capacity of the system is 1 NR log NU + cNR 2
Thus, the lowest NR to support reliable transmissions is
2NU
NR ≥ 2c + log N U
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

13. Minimum number of required Rx antennas per user
Introduction Results Proofs Summary
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 Sum rate required from the system is NU
Equal rate capacity of the system is 1 NR log NU + cNR 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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

14. Minimum number of required Rx antennas per user
Introduction Results Proofs Summary
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 Sum rate required from the system is NU
Equal rate capacity of the system is 1 NR log NU + cNR 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 ?
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

15. Minimum number of required Rx antennas per user: Results
Introduction Results Proofs Summary
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
NU log NU
c > 0 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
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

16. Received space View of the 2NU points in the projected space Hx2NU −1
Introduction Results Proofs Summary
Received space
Hx2NU −1
Hx3
Hx0
Decision region
Hx2
Transmitted point
Incorrect point
Hx1
View of the 2NU points in the projected space
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

17. Proof outline: Union bound
Introduction Results Proofs Summary
Proof outline: Union bound
Let m = NR , n = NU
Expected decoding error probability with i mistakes
1 (
i \−m/2
E(Pe,i ) ≤ σ2
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

18. Proof outline: Union bound
Introduction Results Proofs Summary
Proof outline: Union bound
Let m = NR , n = NU
Expected decoding error probability with i mistakes
1 (
i \−m/2
E(Pe,i ) ≤ σ2
Expected decoding error probability
1 (n\ ( n
i \−m/2 σ2
E(Pe ) ≤ \
1 +
2 i
i =1
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

19. Proof outline: Asymptotic limits
Introduction Results Proofs Summary
Proof outline: Asymptotic limits
Define
m n
α =
Expected decoding error probability (
i \−αn/2 σ2 1
2nH2( n )
E(Pe ) ≤ n max i
1 +
1≤i ≤n 2
= 2ng (n)
where
( i \ ( α
i \ log n − log 2
σ2 n
g (n) c,. max H2
1≤i ≤n
− log 1 + +
n 2
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

20. Proof outline: Behaviour of g (n)
Introduction Results Proofs Summary
Proof outline: Behaviour of g (n)
For α > 0, g (n) becomes Θ(−1) Thus
Pe ≤ 2−d1n for some d1 > 0
0.5
0.4
0.3
For α = 2+E , E > 0, g (n) becomes
g(n)
0.2
log n
Θ(− log n ) Eventually 1
0.1
0.0
−0.1 0 20
40 60 n
80 100
Pe ≤ 2− log n
for some d2 > 0
d2n
g (n) versus n for m = 2.7n
log n
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

21. Introduction Results Proofs 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
Results carry over to general discrete constellations and fading statistics
Ongoing work: Extensions to suboptimal decoders, imperfect CSI
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels

22. Thank you for your attention Questions/Comments?
Introduction Results Proofs Summary
Thank you for your attention Questions/Comments?
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman
Uncoded transmission in MAC channels