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Uncoded transmission in MAC channels achievesarbitrarily 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 1 / 28
Outline
1 Introduction
2 Our work
3 Proofs
4 Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 2 / 28
Introduction
Table of contents
1 Introduction
2 Our work
3 Proofs
4 Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 3 / 28
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 4 / 28
Introduction
System model
x1
x2
...
xNU
+
HN
R1
HN
R2
HNRNU
...
+
H21
H22
H2NU
+H11
H12
H1NU
ν1
ν2
νNR
y1
y2
yNR
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
x̂ = argminx∈{−1,+1}NU ||y −Hx||
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 5 / 28
Introduction
System model
x1
x2
...
xNU
+
HN
R1
HN
R2
HNRNU
...
+
H21
H22
H2NU
+H11
H12
H1NU
ν1
ν2
νNR
y1
y2
yNR
NR antenna receiverNU single antenna users
Hij ∼ N (0, 1), νi ∼ N (0, σ2)
No CSI at transmitters, perfect CSI at receiver
Receiver does perfect ML decoding
x̂ = argminx∈{−1,+1}NU ||y −Hx||M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 5 / 28
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 NRNU
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 6 / 28
Introduction
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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 7 / 28
Introduction
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
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 7 / 28
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 lowerror probability
To investigate this, from now on, we consider BPSK transmissions withoutcoding
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 8 / 28
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 lowerror probability
To investigate this, from now on, we consider BPSK transmissions withoutcoding
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 8 / 28
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 lowerror probability
To investigate this, from now on, we consider BPSK transmissions withoutcoding
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 8 / 28
Results
Table of contents
1 Introduction
2 Our work
3 Proofs
4 Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 9 / 28
Results
Reliability in communication
Theorem
For NR = αNU for any α > 0 and a large enough n0, there exists a c > 0such that the probability of block error goes to zero exponentially fast withNU , 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 spatial multiplexing
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 10 / 28
Results
Reliability in communication
Theorem
For NR = αNU for any α > 0 and a large enough n0, there exists a c > 0such that the probability of block error goes to zero exponentially fast withNU , 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 spatial multiplexing
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 10 / 28
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 12 log |(I + HHT
σ2 )|
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 ?
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28
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.= 1
2NR 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 ?
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28
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.= 1
2NR 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 ?
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28
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.= 1
2NR 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 ?M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 11 / 28
Results
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 a c > 0such that the probability of block error goes to zero exponentially fast withNU , i.e.
P (x̂ 6= x) ≤ 2−cNUlogNU for all NU > n0
In other words
Coding does not reduce the number of required antennas per user for1 bit per channel use
Scaling behaviour of minimum NRNU
is Θ( 1logNU
)
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 12 / 28
Proofs
Table of contents
1 Introduction
2 Our work
3 Proofs
4 Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 13 / 28
Proofs
Received space
Hx0
Hx1
Hx3
Hx2
Hx2NU−1
Transmitted point
Decision region
Incorrect point
View of the 2NU points in the projected space
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 14 / 28
Proofs
Proof outline: Union bound
Let’s say x0 is transmitted.
Pe ≤∑x 6=x0
P (x̂ = x)
Idea
Group wrong codewords by the number of positions in which they differfrom x0
Thus
Pe ≤NU∑i=1
∑x:d(x,x0)=i
P (x̂ = x),
where d(·, ·) computes the hamming distance between its two arguments
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 15 / 28
Proofs
Proof outline: Union bound
Let’s say x0 is transmitted.
Pe ≤∑x 6=x0
P (x̂ = x)
Idea
Group wrong codewords by the number of positions in which they differfrom x0
Thus
Pe ≤NU∑i=1
∑x:d(x,x0)=i
P (x̂ = x),
where d(·, ·) computes the hamming distance between its two arguments
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 15 / 28
Proofs
Proof outline: Computing pairwise error probabilities
Let xi be a codeword with i positions different from x0. Then
P (x̂ = xi) = Q
( ||H(xi − x0)||2σ
)= Q
(||∑i
j=1 2hb(j)||2σ
)b(j) is the jth position where xi and x0 differ,
hb is the bth column of H
≤ 1
2exp
(−||∑i
j=1 2hb(j)||28σ2
)
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 16 / 28
Proofs
Proof outline: Average pairwise error probability
Idea
Average over channel realizations
EH(P (x̂ = xi)) ≤ EH
(1
2exp
(−||∑i
j=1 2hb(j)||28σ2
))This is just the moment generating function of an appropriately scaled chisquared distribution.
Closed form expression
EH(P (x̂ = xi)) ≤1
2
(1 +
i
σ2
)−NR2
Depends only on i, i.e. number of columns, not on particular columnsconsidered
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 17 / 28
Proofs
Proof outline: Average pairwise error probability
Idea
Average over channel realizations
EH(P (x̂ = xi)) ≤ EH
(1
2exp
(−||∑i
j=1 2hb(j)||28σ2
))This is just the moment generating function of an appropriately scaled chisquared distribution.
Closed form expression
EH(P (x̂ = xi)) ≤1
2
(1 +
i
σ2
)−NR2
Depends only on i, i.e. number of columns, not on particular columnsconsidered
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 17 / 28
Proofs
Proof outline: Upper bound on total error probability
LetEH(P (x̂ = xi)) = Pe,i,
since it is independent of xi. Then the total error probability is
Pe ≤NU∑i=1
∑x:d(x,x0)=i
P (x̂ = x) =
NU∑i=1
1
2
(NU
i
)Pe,i
≤NU∑i=1
1
2
(NU
i
)(1 +
i
σ2
)−NR2
≤ NU maxi∈{1,...,NU}
1
2
(NU
i
)(1 +
i
σ2
)−NR2
≤ NU
2max
i∈{1,...,NU}2NUH2
(i
NU
)(1 +
i
σ2
)−NR2
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 18 / 28
Proofs
Proof outline: Asymptotic limits
Define
α =NR
NU
Decoding error probability
Pe ≤ 2NUg(NU )
where
g(n) , max1≤i≤n
H2
(i
n
)− α
2log
(1 +
i
σ2
)+
log n− log 2
n
, max1≤i≤n
g(i, n)
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 19 / 28
Proofs
Proof outline: Behaviour of g(n), constant α
0 10 20 30 40 50i
−0.4
−0.2
0.0
0.2
0.4
g(i,n
)
α = 0.3, σ2 = 0.5, n = 50, g(n) = max1≤i≤n g(i, n) = 0.26
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 20 / 28
Proofs
Proof outline: Behaviour of g(n), constant α
0 20 40 60 80 100i
−0.4
−0.2
0.0
0.2
0.4
g(i,n
)
α = 0.3, σ2 = 0.5, n = 100, g(n) = max1≤i≤n g(i, n) = 0.08
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 20 / 28
Proofs
Proof outline: Behaviour of g(n), constant α
0 50 100 150 200i
−0.4
−0.2
0.0
0.2
0.4
g(i,n
)
α = 0.3, σ2 = 0.5, n = 200, g(n) = max1≤i≤n g(i, n) = −0.1
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 20 / 28
Proofs
Proof outline: Behaviour of g(n), constant α
0 100 200 300 400 500 600 700i
−0.4
−0.2
0.0
0.2
0.4
g(i,n
)
α = 0.3, σ2 = 0.5, n = 700, g(n) = max1≤i≤n g(i, n) = −0.37
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 20 / 28
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 20 / 28
Proofs
Proof outline: Behaviour of g(n), α = 2+εlog n
0 20 40 60 80 100i
−0.4
−0.2
0.0
0.2
0.4
g(i,n
)
ǫ = 0.1, σ2 = 0.5, n = 100, g(n) = max1≤i≤n g(i, n) = 0.03
Thus
Pe ≤ 2−cNUlogNU
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 21 / 28
Proofs
Proof outline: Behaviour of g(n), α = 2+εlog n
0 50 100 150 200 250 300i
−0.4
−0.2
0.0
0.2
0.4
g(i,n
)
ǫ = 0.1, σ2 = 0.5, n = 300, g(n) = max1≤i≤n g(i, n) = −0.01
Thus
Pe ≤ 2−cNUlogNU
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 21 / 28
Proofs
Proof outline: Behaviour of g(n), α = 2+εlog n
Message
For fixed α > 0 and large enough n,
g(n) ≤ c(ε) < 0,
c(ε) goes down as Θ(− 1logn)
Thus
Pe ≤ 2−cNUlogNU
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 21 / 28
Extensions
Table of contents
1 Introduction
2 Our work
3 Proofs
4 Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 22 / 28
Extensions
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 23 / 28
Extensions
Arbitrary finite constellation C
Pairwise error probability can be bounded by
P (x̂ = xi) ≤1
2exp
(−||∑i
j=1 dhb(j)||28σ2
),
where d is the minimum distance of the constellation i.e.
d = minx∈C,y∈C,x 6=y
||x− y||
In general loose
Same scaling Θ(
1logNU
)for the number of receiver antennas per user
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 24 / 28
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√
n) convergence, i.e.
supx|Fn(x)− Fg(x)| ≤ C̃√
n
Can be shown thatPe,i ≤ Ci−
NR2 ,
for all i ≥ i0Thus
Asymptotic behaviour of Pe,i in i same as that for Gaussian fading
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 25 / 28
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√
n) convergence, i.e.
supx|Fn(x)− Fg(x)| ≤ C̃√
n
Can be shown thatPe,i ≤ Ci−
NR2 ,
for all i ≥ i0But Pe =
∑i0i=1 Pe,i +
∑NUi=i0+1 Pe,i ?
Terms with less than i0 mismatches → 0 with large NR
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 25 / 28
Summary
Table of contents
1 Introduction
2 Our work
3 Proofs
4 Extensions
5 Conclusions
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 26 / 28
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 27 / 28
Summary
Thank you for your attentionQuestions/Comments?
M. Chowdhury, A. Goldsmith, T. Weissman Uncoded transmission in MAC channels 28 / 28