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Introduction Results Proofs Summary
Uncoded transmission in MAC channels achievesarbitrarily 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
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
Introduction Results Proofs Summary
System model
x1
x2
...
xn
+
Hm1
Hm2
Hmn
...
+
H21
H22
H2n
+H11
H12
H1n
ν1
ν2
νm
y1
y2
ym
NR antenna receiverNU single antenna users
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
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 NRNU
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Background
For MIMO point-to-point channel
Multiplexing gain 12 min(NR ,NU)
Achieved by coding across timePe → 0 with larger blocklengths
For a MAC channel
NU transmitters each with 1 antennaNR receiver antennasTotal transmit power grows like NU
Multiplexing gain 12 min(NR ,NU) logNU
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
Introduction Results Proofs Summary
Background
For MIMO point-to-point channel
Multiplexing gain 12 min(NR ,NU)
Achieved by coding across timePe → 0 with larger blocklengths
For a MAC channel
NU transmitters each with 1 antennaNR receiver antennasTotal transmit power grows like NU
Multiplexing gain 12 min(NR ,NU) logNU
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
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 achievelow error probability
To investigate this, from now on, we consider BPSK transmissionswithout coding
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
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 achievelow error probability
To investigate this, from now on, we consider BPSK transmissionswithout coding
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
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 achievelow error probability
To investigate this, from now on, we consider BPSK transmissionswithout coding
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Reliability in communication
Theorem
For NR = αNU for any α > 0 and a large enough n0, there exists ac > 0 such that the probability of block error goes to zeroexponentially fast with NU , i.e.
P(x̂ 6= x) ≤ 2−cNU for all NU > n0
In other words
Equal rate transmission
Any NRNU
ratio
Error probability Pe → 0 with NU →∞
This is due to a combination of receiver diversity and spatialmultiplexing
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Reliability in communication
Theorem
For NR = αNU for any α > 0 and a large enough n0, there exists ac > 0 such that the probability of block error goes to zeroexponentially fast with NU , i.e.
P(x̂ 6= x) ≤ 2−cNU for all NU > n0
In other words
Equal rate transmission
Any NRNU
ratio
Error probability Pe → 0 with NU →∞
This is due to a combination of receiver diversity and spatialmultiplexing
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
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 12NR logNU + cNR
Thus, the lowest NR to support reliable transmissions is
NR ≥2NU
2c + logNU
Smallest NRNU
ratio achievable with coding is
NR
NU≥ 2
logNU + 2c
Can we achieve similar ratios for reliable uncoded systems also ?
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
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 12NR logNU + cNR
Thus, the lowest NR to support reliable transmissions is
NR ≥2NU
2c + logNU
Smallest NRNU
ratio achievable with coding is
NR
NU≥ 2
logNU + 2c
Can we achieve similar ratios for reliable uncoded systems also ?
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
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 12NR logNU + cNR
Thus, the lowest NR to support reliable transmissions is
NR ≥2NU
2c + logNU
Smallest NRNU
ratio achievable with coding is
NR
NU≥ 2
logNU + 2c
Can we achieve similar ratios for reliable uncoded systems also ?
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Minimum number of required Rx antennas per user:Results
Theorem
For NRNU
= 2+εlogNU
for any ε > 0 and a large enough n0, there exists ac > 0 such that the probability of block error goes to zeroexponentially fast with NU , i.e.
P(x̂ 6= x) ≤ 2−cNUlog NU for all NU > n0
In other words
Coding does not reduce the number of required antennas peruser for 1 bit per channel use
Scaling behaviour of minimum NRNU
is Θ( 1logNU
)
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Received space
Hx0
Hx1
Hx3
Hx2
Hx2NU−1
Transmitted point
Decision region
Incorrect point
View of the 2NU points in the projected space
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Proof outline: Union bound
Let m = NR , n = NU
Expected decoding error probability with i mistakes
E(Pe,i ) ≤1
2
(1 +
i
σ2
)−m/2
Expected decoding error probability
E(Pe) ≤n∑
i=1
1
2
(n
i
)(1 +
i
σ2
)−m/2
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Proof outline: Union bound
Let m = NR , n = NU
Expected decoding error probability with i mistakes
E(Pe,i ) ≤1
2
(1 +
i
σ2
)−m/2
Expected decoding error probability
E(Pe) ≤n∑
i=1
1
2
(n
i
)(1 +
i
σ2
)−m/2
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Proof outline: Asymptotic limits
Defineα =
m
n
Expected decoding error probability
E(Pe) ≤ n max1≤i≤n
1
22nH2(
in)
(1 +
i
σ2
)−αn/2
= 2ng(n)
where
g(n) , max1≤i≤n
H2
(i
n
)− α
2log
(1 +
i
σ2
)+
log n − log 2
n
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Proof outline: Behaviour of g(n)
0 20 40 60 80 100n
−0.1
0.0
0.1
0.2
0.3
0.4
0.5
g(n
)
g(n) versus n for m = 2.7nlog n
For α > 0, g(n) becomes Θ(−1)
ThusPe ≤ 2−d1n
for some d1 > 0
For α = 2+εlog n , ε > 0, g(n) becomes
Θ(− 1log n )
Eventually
Pe ≤ 2−d2nlog n
for some d2 > 0
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
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 fadingstatistics
Ongoing work: Extensions to suboptimal decoders, imperfectCSI
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels
Introduction Results Proofs Summary
Thank you for your attentionQuestions/Comments?
Mainak Chowdhury, Andrea Goldsmith, Tsachy Weissman Uncoded transmission in MAC channels