uncoded transmission in mac channels achieves arbitrarily ...mainakch/talks/ericsson_talk.pdfmainak...

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

Introduction

Table of contents

1 Introduction

2 Our work

3 Proofs

4 Extensions

5 Conclusions

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

Introduction

System model

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||

Introduction

System model

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

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

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

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

Introduction

Main question

Intuition

Introduction

Main question

Intuition

Results

Table of contents

1 Introduction

2 Our work

3 Proofs

4 Extensions

5 Conclusions

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

Error probability Pe → 0 with NU →∞

This is due to a combination of receiver diversity and spatial multiplexing

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

Error probability Pe → 0 with NU →∞

This is due to a combination of receiver diversity and spatial multiplexing

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

NU≥ 2

logNU + 2c

Can we achieve similar ratios for reliable uncoded systems also ?

Results

Equal rate capacity of the system is.= 1

2NR logNU + cNR

NR ≥2NU

2c+ logNU

Smallest NRNU

NU≥ 2

logNU + 2c

Results

2NR logNU + cNR

NR ≥2NU

2c+ logNU

Smallest NRNU

NU≥ 2

logNU + 2c

Results

2NR logNU + cNR

NR ≥2NU

2c+ logNU

Smallest NRNU

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

Proofs

Table of contents

1 Introduction

2 Our work

3 Proofs

4 Extensions

5 Conclusions

Proofs

Received space

Hx2NU−1

Transmitted point

Decision region

Incorrect point

View of the 2NU points in the projected space

Proofs

Proof outline: Union bound

Let’s say x0 is transmitted.

Pe ≤∑x 6=x0

P (x̂ = x)

Group wrong codewords by the number of positions in which they differfrom x0

Pe ≤NU∑i=1

∑x:d(x,x0)=i

P (x̂ = x),

where d(·, ·) computes the hamming distance between its two arguments

Proofs

Proof outline: Union bound

Let’s say x0 is transmitted.

Pe ≤∑x 6=x0

P (x̂ = x)

Group wrong codewords by the number of positions in which they differfrom x0

Pe ≤NU∑i=1

∑x:d(x,x0)=i

P (x̂ = x),

where d(·, ·) computes the hamming distance between its two arguments

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

(||∑i

j=1 2hb(j)||2σ

)b(j) is the jth position where xi and x0 differ,

hb is the bth column of H

(−||∑i

j=1 2hb(j)||28σ2

Proofs

Proof outline: Average pairwise error probability

Average over channel realizations

EH(P (x̂ = xi)) ≤ EH

(−||∑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

)−NR2

Depends only on i, i.e. number of columns, not on particular columnsconsidered

Proofs

Proof outline: Average pairwise error probability

Average over channel realizations

EH(P (x̂ = xi)) ≤ EH

(−||∑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

)−NR2

Depends only on i, i.e. number of columns, not on particular columnsconsidered

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

≤NU∑i=1

)−NR2

≤ NU maxi∈{1,...,NU}

)−NR2

≤ NU

i∈{1,...,NU}2NUH2

)−NR2

Proofs

Proof outline: Asymptotic limits

Define

α =NR

Decoding error probability

Pe ≤ 2NUg(NU )

g(n) , max1≤i≤n

)− α

log n− log 2

, max1≤i≤n

g(i, n)

Proofs

Proof outline: Behaviour of g(n), constant α

0 10 20 30 40 50i

−0.4

−0.2

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

Proofs

0 20 40 60 80 100i

−0.4

−0.2

α = 0.3, σ2 = 0.5, n = 100, g(n) = max1≤i≤n g(i, n) = 0.08

Message

g(n) ≤ c(α) < 0

Proofs

0 50 100 150 200i

−0.4

−0.2

α = 0.3, σ2 = 0.5, n = 200, g(n) = max1≤i≤n g(i, n) = −0.1

Message

g(n) ≤ c(α) < 0

Proofs

0 100 200 300 400 500 600 700i

−0.4

−0.2

α = 0.3, σ2 = 0.5, n = 700, g(n) = max1≤i≤n g(i, n) = −0.37

Message

g(n) ≤ c(α) < 0

Proofs

Message

g(n) ≤ c(α) < 0

Proofs

Proof outline: Behaviour of g(n), α = 2+εlog n

0 20 40 60 80 100i

−0.4

−0.2

ǫ = 0.1, σ2 = 0.5, n = 100, g(n) = max1≤i≤n g(i, n) = 0.03

Pe ≤ 2−cNUlogNU

Proofs

0 50 100 150 200 250 300i

−0.4

−0.2

ǫ = 0.1, σ2 = 0.5, n = 300, g(n) = max1≤i≤n g(i, n) = −0.01

Pe ≤ 2−cNUlogNU

Proofs

Message

g(n) ≤ c(ε) < 0,

c(ε) goes down as Θ(− 1logn)

Pe ≤ 2−cNUlogNU

Extensions

Table of contents

1 Introduction

2 Our work

3 Proofs

4 Extensions

5 Conclusions

Extensions

So far discussion focused on

BPSK constellations: Each user transmits from {−1,+1}Gaussian channel statistics (Rayleigh fading)

Extensions

Arbitrary finite constellation C

Pairwise error probability can be bounded by

P (x̂ = xi) ≤1

(−||∑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

Extensions

Arbitrary fading statistics

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√

Can be shown thatPe,i ≤ Ci−

for all i ≥ i0Thus

Asymptotic behaviour of Pe,i in i same as that for Gaussian fading

Extensions

Arbitrary fading statistics

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√

Can be shown thatPe,i ≤ Ci−

for all i ≥ i0But Pe =

∑i0i=1 Pe,i +

∑NUi=i0+1 Pe,i ?

Terms with less than i0 mismatches → 0 with large NR

Summary

Table of contents

1 Introduction

2 Our work

3 Proofs

4 Extensions

5 Conclusions

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

Summary

Thank you for your attentionQuestions/Comments?

uncoded transmission in mac channels achieves arbitrarily ...mainakch/talks/ericsson_talk.pdfmainak...

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