Information Theory of Wireless Networks:A Deterministic Approach

David Tse

Wireless Foundations

U.C. Berkeley

CISS 2008

March 21, 2008

Joint work with Salman Avestimehr, Guy Bresler, Suhas Diggavi, Abhay Parekh.

The Holy Grail

• Shannon’s information theory provides the basis for all modern-day communication systems.

• His original theory was point-to-point.

• After 60 years we are still very far away from generalizing the theory to networks.

• We propose an approach to make progress in the context of wireless networks.

Modeling the Wireless Medium

• broadcast

• superposition

• high dynamic range in channel strengths between different nodes

• Basic model: additive Gaussian channel:

Gaussian Network Capacity: What We Know

Tx

Rx1

TxRx

Rx

Tx 1

Tx 2Rx 2

point-to-point (Shannon 48)

C = log2(1+ SNR)

multiple-access(Alshwede, Liao 70’s)

broadcast(Cover, Bergmans 70’s)

What We Don’t Know

Unfortunately we don’t know the capacity of most other Gaussian networks.

D

Tx 1

Relay

S

Tx 2 Rx 2

Rx 1

Interference

relay

(Best known achievable region: Han & Kobayashi 81)

(Best known achievable region: El Gamal & Cover 79)

30 Years Have Gone by…..

We are still stuck.

How to make progress?

It’s the model.

• Shannon focused on noise in point-to-point communication.

• But many wireless networks are interference rather than noise-limited.

• We propose a deterministic channel model emphasizing interaction between users’ signals rather than on background noise.

• Far more analytically tractable and can be used to determine approximate Gaussian capacity

Agenda

Warmup:

• point-to-point channel• multiple access channel• broadcast channel

The meat:

• relay networks (Avestimehr, Diggavi & T. 07)

• interference channels (Bresler &T. 08, Bresler,Parekh & T. 08)

Gaussian

Transmit a real number

If we have

Example 1: Point-to-Point Link

Deterministic

n / SNR on the dB scale

Cdet(n) = n

Least significant bits are truncated at noise level.

Cawgn(SNR) = 12 log(1+ SNR)

pSNR = 2n

Gaussian

Example 2: Multiple Access

Deterministic

SNR1 SNR2

SNR1 ¸ SNR2user 2

user 1

mod 2 addition

user 1 sends cloud centers, user 2 sends clouds.

Comparing Multiple Access Capacity Regions

Gaussian Deterministic

SNR1 SNR2

R1

R2

logSNR1

logSNR2

R1 + R2 =

¼logSNR1

SNR1 ¸ SNR2user 2

user 1

R1

R2

mod 2 addition

n2

n1

accurate to within 1 bit per user

log(1+ SNR1 + SNR2)(3;2)

Example 3: Broadcast

Gaussian Deterministic

SNR1 SNR2

user 2

user 1SNR1 ¸ SNR2

n1 = 5

n2 = 2

To within 1 bit

n1

log(1+SNR1)

n2

log(1+SNR2)

R1

R2

Agenda

Warmup:

• point-to-point channel• multiple access channel• broadcast channel

The meat:

• relay networks• interference channels

History

• The (single) relay channel was first proposed by Van der Meulen in 1971.

• Cover and El Gamal (1979) provided a whole array of achievable strategies.

• Recent generalization of these techniques to more than 1 relay.

• Do not know how far they are from optimal

• General upper bound: cutset bound ¹Ccutset

The Relay Channel

Gaussian Deterministic

S

R

D

hSR hRD

hSD

2

2

||

||

SD

RD

h

h

2

2

||

||

SD

SR

h

h

Decode-Forward is near optimal

Decode-Forward is optimal On average it is much less than 1-bit

xx

nSR nRD

nSD

gap = nSD + min¡(nSR ¡ nSD )+; (nR D ¡ nSD )+¢

¹Ccutset = min(max(nSD ;nSR );max(nSD ;nR D ))

Cutset bound is achievable.

Theorem (Avestimehr et al 07)Gap from cutset bound is at most 1 bit.

Generalization to Relay Networks

• Can the cutset bound be achievable in the deterministic model?

• Can one always achieve to within a contant gap of the cutset bound in the Gaussian case?

General Relay Networks

Main Theorem:

Cutset bound is achievable for deterministic networks.

S Sc

Crelay = ¹Ccutset = minS

rank(GS;S c )

(Avestimehr, Diggavi & T. 07)

Main Theorem

Theorem generalizes to arbitrary linear MIMO channels on finite fields.

In the case of wireline graph, rank is the number of links crossing the cut.

Our theorem is a generalization of Ford-Fulkerson max-flow min-cut theorem.

GS;S c

Crelay = minS

rank(GS;S c )

Connections to Network Coding

• Achievability: random linear coding at relays

• Proof style: similar to Ahlswede et al 2000 for wireline networks.

• Technical innovation: dealing with “inter-symbol interference” between signals arriving along paths of different lengths.

Back to Gaussian Relay Networks

Approximation Theorem:

There is a scheme that achieves within a constant gap to the cutset bound, independent of the SNR’s of the links.

(Avestimehr, Diggavi and T. 2008)

Agenda

Warmup:

• point-to-point channel• multiple access channel• broadcast channel

The meat:

• relay networks• interference channels

Interference

• So far we have looked at single source, single destination networks.

• All the signals received is useful.

• With multiple sources and multiple destinations, interference is the central phenomenon.

• Simplest interference network is the two-user interference channel.

Two-User Gaussian Interference Channel

• Capacity region unknown• Best known achievable region: Han & Kobayashi 81.

message m1

message m2

want m1

want m2

Gaussian to Deterministic Interference Channel

Gaussian Deterministic

Capacity can be computed usinga result by El Gamal and Costa 82.

In symmetric case, channeldescribed by two parameters:

SNR, INR

mn

n $ log2 SNR: m $ log2 INR

Symmetric Deterministic Capacity

®= 13 ®= 2

3

r = 23 r = 2

3

1

1/2® =

logINRlogSNR

=mn

r =

Back to Gaussian

• Theorem:

Constant gap between capacity regions of the two-user deterministic and Gaussian interference channels.

(Bresler & T. 08)

• A deeper view of earlier 1-bit gap result on two-user Gaussian interference channel (Etkin,T. & Wang 06).

• Bounds further sharpened to get exact results in the low-interference regime ( < 1/3)

(Shang et al 07,Annaprueddy&Veeravalli08,Motahari&Khandani07)

Extension:Many-to-One Interference Channel

Gaussian Deterministic

Deterministic capacity can be computed exactly .

Gaussian capacity to within constant gap, using structured codes and interference alignment.(Bresler, Parekh & T. 07)

Example

• Interference from users 1 and 2 is aligned at the MSB at user 0’s receiver in the deterministic channel.

• How can we mimic it for the Gaussian channel ?

Tx0

Tx1

Tx2

Rx0

Rx1

Rx2

• Suppose users 1 and 2 use a random Gaussian codebook:

Gaussian Han-Kobayashi Not Optimal

Tx0

Tx1

Tx2

Rx0

Rx1

Rx2Random Code

Sum of Two Random CodebooksLattice Code for Users 1 and 2

User 0 CodeInterference from users 1 and 2 fills the space: no room for user 0.

Lattice codes can achieve constant gap

Interference Channels: Recap

• In two-user case, we showed that an existing strategy can achieve within 1 bit to optimality.

• In many-to-one case, we showed that a new strategy can do much better.

• General K-user interference channel still open.

Evolution of Ideas

• deterministic network capacity in 1980’s:– broadcast channels (Marton 78, Pinsker 79)– 2-user interference channel (El Gamal & Costa 82)– single-relay channel (El Gamal & Aref 82)– relay networks with broadcast but no interference (Aref 79)

• inspired by network coding in early 2000’s:– finite-field model with erasures (Gupta et al 06)

but connection to Gaussian networks missing.

• 2-user Gaussian interference channel capacity to within 1 bit (Etkin, T & Wang 06)• Linear deterministic model (Avestimehr, Diggavi & T 07) and applied to relay networks.

Parting Words

• Main message:

If something can’t be computed exactly, approximate.

• Similar evolution has happened in other fields:

– fluid and heavy-traffic approximation in queueing networks

– approximation algorithms in CS theory

• Approximation should be good in engineering-relevant regimes.