uncoded transmission in mac channels achieves arbitrarily ......mainak chowdhury, andrea goldsmith,...

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


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



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



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



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



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



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



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 ?



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

)



Received space

Hx0

Hx1

Hx3

Hx2

Hx2NU−1

Transmitted point

Decision region

Incorrect point

View of the 2NU points in the projected space



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



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



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



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



Thank you for your attentionQuestions/Comments?


uncoded transmission in mac channels achieves arbitrarily ......mainak chowdhury, andrea goldsmith,...

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