wireless network information theory

7/31/2019 Wireless Network Information Theory

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June 30, 2009 , P. R. Kumar

Wireless Network information Theory

P. R. Kumar

Dept. of Electrical and Computer Engineering, and

Coordinated Science LabUniversity of Illinois, Urbana-Champaign

Email: [email protected]: http://decision.csl.illinois.edu/~prkumar

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License.Based on a work at decision.csl.illinois.edu

See last page and http://creativecommons.org/licenses/by-nc-nd/3.0/


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What is really the best way to

operate wireless networks?

And what are the ultimate limits to

information transfer over wirelessnetworks?


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Outline

Reappraising multi-hop transport 4

What is information theory? 11

Network information theory 22

Model for wireless network information theory 33

Results when absorption or relatively large path loss 45

Order optimality of multi-hop transport 65

The effect of fading 80

Low path loss 82

A quick survey of more recent results 94

Remarks 99

References 100


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Reappraising multi-hop transport


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Reappraising multi-hop transport

Nodes fully decode packets at each stage

Treating interference as noise

But why should nodes Decode and Forward?

Why not just Amplify and Forward?

Interference+

Noise

Interference+

Noise

Interference+

Noise

S DR1 R2 R3

R

S

D

Why should intermediate nodes be able to decode the packets?

Why go digital?


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Why treat interference as noise?

Interference is not interference

Subtract

loudsignal

Interference is information

Packets do not destructively collide

Why not use multi-user decoding?

How much benefit can multi-user decoding give for wireless

networks?


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Should we try to do active interferencecancellation?

Why not reduce the denominator in the SINR rather than increase the

denominator?

A

BC X

Reduce by cancellation

Signal

Interference + Noise


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Why even take small hops?

Why not use long range communicationwith multi-user decoding?


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In fact is the notion of spatial reuseappropriate for wireless networks?

Spatial reuse of frequency

If spatial reuse of frequency isthe goal, then is a sharper pathloss better for wirelessnetworks?

0Distance

Attenuation

1

r8

1

r4

1

r8

better for wireless networks than1

r4

?

Is

Or worse?

Arejungles better for wireless networking than deserts?


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Wireless networks are not wirednetworks

There are more things in heaven and earth, Horatio,Than are dreamt of in your philosophy.

Hamlet

Wireless networks are formed by nodes with radios

There is no a priorinotion of links

Nodes simply radiate energy

Nodes can cooperate in many complex ways

So how should information be transported in wireless networks?

What should be the architecture of wireless networks?

What are the limits to information transfer?

Maxwell rather than Kirchoff

Need an information theory to provide strategic guidance for wireless networks


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What is Information Theory?


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Model of communication

InformationSource

InformationTransmitter

Channel Receiver InformationSink

Noise

Message

Receivedsignal

TransmittedSignalMessage


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Shannons Information Theory

Question that Shannon posed and answered

Given a noisycommunication channel

Channel Modeled byp(y|x)

Called a Discrete Memoryless Channel

Question: How many bits per transmission can be reliablysent?

Call this the capacity of the channel

How can we achievethis capacity over the channel?

Channelp(y|x)x y


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Shannons formulation

There are a set of 2nR messages

1 2

4

6

7

3

2nR-1 2nR

5

One message Win {1, 2, , 2nR}is picked by the source out ofthese 2nR messages

This is encoded as a codeword{X1,X2, ,Xn}

5

Channelp(y|x)

Xk Yk

Xk is transmitted on the k-thtransmission

Yk is received on the k-th

transmission

So in n uses of the channel{X1,X2, ,Xn}is sent, and

{Y1, Y2, , Yn}is received


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Shannons formulation

Channelp(y|x)

{X1, ,Xn}

{Y1, , Yn}

1 2

4

6

7

3

2nR-1 2nR

55


The receiver decodes {Y1, Y2, , Yn}as W


One message Win {1, 2, , 2nR}is picked by the source out ofthese 2nR messages

This is encoded as a codeword{X1,X2, ,Xn}

Xk is transmitted on the k-thtransmission

Yk is received on the k-th

transmission

So in n uses of the channel{X1,X2, ,Xn}is sent, and

{Y1, Y2, , Yn}is received


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Definition of Achievable RateR

Let Perror = Prob(WW)

Suppose we can make Perror smaller than any we desire bychoosing n large

Then we say that the channel can support a Rate ofR bitsper transmission

Overall scheme

Choose encoder E: {1, 2, , 2nR} Xn

Choose decoder D: Xn {1, 2, , 2nR}

Want Perror smaller than a desired Then we can reliably transmitR bits per transmission

DE

2nR

messages

W W{X1,X2, ,Xn} {Y1, Y2, , Yn}Channelp(y|x)


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Shannons Answers

Capacity Theorem

Given Channel Modelp(y|x)

Capacity = Max I(X;Y) bits/transmission

Where is called the mutual information

This is the supremum of the achievable rates

Shannons architecture for digital communication

Channel

p(y|x)x y

I(X;Y) = p(x,y)x, y

logp(X,Y)

p(X)p(Y)

p(x)

Channel Source decode(Decompression)Decode

Encode

for thechannel

Source code(Compression)

2nR messages 2nR messages


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Capacity of Gaussian Channel

Gaussian Channel

Yi=Xi+Zi

ZiN(0, 2)

Independent, identically distributed noise

Power constraint P on transmissions:

Capacity =

Channel

p(y|x)x y

X Y

+

Z ~ N(0,2) = Noise

1

nX

i

2

i=1

n

P

1

2log 1+

P

2

bits per transmission


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Capacity of Continuous AWGNBandlimited Channel

AWGN NoiseZ(t)with Power Spectral Density

Band Limited Channel [-W,+W]

Power constraint P on signal transmitted:

Capacity =

1

T

X2(t)

0

T

P

W log 1+P

WN

bits per second

X(t) Y(t)+

Z(t)WhiteGaussianNoise with PSDN

-W +W

1

N

2


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Limitations of Shannons result

Does not address the issue of latency

Delay incurred by block coding

What is the joint tradeoff between

Throughput and Delay (and Error Rate)


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The classic references

C. E. Shannon, "A mathematical theory of communication", BellSyst. Tech. J.", Vol 27, pp. 379--423", 1948.

C. E. Shannon, "Communication in the presence of noise",

Proceedings of the IRE, vol. 37, pp. 10--21, 1949.

C. E. Shannon and W. Weaver The Mathematical Theory ofInformation, University of Illinois Press, Urbana, 1949.

R. G. Gallager, Information Theory and Reliable

Communication, John Wiley and Sons, New York, 1968.

T. Cover and J. Thomas, Elements of Information Theory, Wileyand Sons, New York, 19103.


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Network Information Theory


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23

The Multiple Access Channel

Model

Node 1 sends

Node 2 sends

The receiver receives generated as

Senders and their Rates

Message 1:

Sends

Message 2:

Sends

Decoder: and

What rate vectors are feasible?


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24

Solution

Capacity region:

All rate vectors satisfying

for some distribution are feasible


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25

Interpretation and coding strategy

At point A

A

Node 2 acts as a purefacilitator


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26

Interpretation and coding strategy

At point B

Receiver first decodes

Possible since

Then decodes

Possible since

B

Successive subtraction anddecoding strategy (CDMA)


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27

The Scalar Gaussian BroadcastChannel

Goal

To send to Receiver 1

To send to Receiver 2

Simultaneously

Through one broadcast

Power constraint

Receiver 1 receives

Decodes

Receiver 2 receives

Decodes

What rate vectors are feasible?


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28

Solution

Assume Receiver 1 is better than Receiver 2

So Receiver 1 can decode anything that Receiver 2 can

So Receiver 1 can decode

Capacity region: All vectors satisfying

for some

Sender uses power for Receiver 1, and power for Receiver 2

Receiver 2 has signal strength and noise

Receiver 1 first decodes and then subtracts it. So signal in noise


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General broadcast channel

General Broadcast channel capacity unknown

Vector Gaussian channel capacity recently established


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30

Max Flow - Min Cut Theorem

Theorem (El Gamal Ph. D. Thesis)

Suppose is feasible vector of rates.

Then

Example: Relay Channel

S Sc

X

X1,Y

1

Y


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31

The Slepian-Wolfe Problem:Distributed Source Coding

To reconstruct (X,Y) at thedestination, it is sufficientto have

So X and Y can code separately and still achieve the same

result as though they were cooperating


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Network information theory

Gaussian broadcast channel

Unknowns

The simplest interference channel

Networks being built (ad hoc networks, sensor nets) are much more complicated

Multiple access channel

Triumphs

The simplest relay channel


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Model for Wireless Network

Information Theory


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Model of system: A planar network

Introduce distance Node locations

Distances between nodes,

Attenuation as a function of distance

n nodes in a plane

ij= distance between nodes i andj

Signal attenuation with distance is

> 0 is the path loss exponent

G0 is the absorption constant

Generally > 0 since the medium is absorptive unless over a vacuum

Corresponds to a loss of 20log10e db per meter

ijmin

i

j

e


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)

Transmitted and received signals

N(0,2)

= fi ,t(yit1

,Wi )

{1,2,3,,2TRik}

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i

(R1,R2,...,Rl ) is feasible rate vector if there is a sequence of codes withMax

W1,W2 ,...,Wl

Pr(WiW

ifor some i W1,W2 ,...,Wl ) 0 as T

Wi = symbol from to be sent by node i in Ttransmissions

xi(t) = signal transmitted by node i time t

yj(t) = signal received by nodejat time t

Destinationjuses the decoder

Error if

(

Individual power constraint Pi Pind for all nodesI. or Total power constraint

Transport Capacity bit-meters/second or bit-meters/slot


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i


W1,W2 ,...,Wl

Pr(WiW






Error if

(



{1,2,3,,2TRik}

xi yj


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i


W1,W2 ,...,Wl

Pr(WiW






Error if

(



{1,2,3,,2TRik}

xi yj


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i


W1,W2 ,...,Wl

Pr(WiW






Error if

(



{1,2,3,,2TRik}

xi yj


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i


W1,W2 ,...,Wl

Pr(WiW






Error if

(



{1,2,3,,2TRik}

xi yj


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i


W1,W2 ,...,Wl

Pr(WiW






Error if

(



{1,2,3,,2TRik}

xi yj


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i


W1,W2 ,...,Wl

Pr(WiW






Error if

(



{1,2,3,,2TRik}

xi yj


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i

(R

1,R

2,...,R

l ) is feasible rate vector if there is a sequence of codes withMax

W1,W2 ,...,Wl

Pr(WiW






Error if

(

Individual power constraint Pi Pind for all nodesI. Or Total power constraint


{1,2,3,,2TRik}

xi yj


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CT = sup

(R1,R2 ,

,Rn(n1))

Ri

i=1

n(n1)

i

Wi = gj(yjT,Wj)


xi yj

N(0,2)

= fi ,t(yit1

,Wi )

=eij

ij

i=1

i j

n

xi (t)+ zj(t)

Pi

i=1

n

Ptotal

WiW

i

(R

1,R

2,...,R

l ) is feasible rate vector if there is a sequence of codes withMax

W1,W2 ,...,Wl

Pr(WiW






Error if

(

Individual power constraint Pi Pind for all nodesI. Or Total power constraint


{1,2,3,,2TRik}


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Results when there is absorption or arelatively large path loss


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Total transmitted power bounds thetransport capacity

Theorem: Bit-meters per Joule bound (Xie & K 02)

Suppose > 0, there is some absorption,

Or > 3, if there is no absorption at all

Then for all Planar Networks

where

CTc1(,,

min)

2P

total

c1(,, min) =22+7

2min2+1

e

min2 (2 e

min2 )

(1 emin

2 )

if > 0

=2

2+5(3 8)

( 2)2( 3)min21 if = 0 and > 3


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where

CTc1(,,

min)

2P

total

c1(,, min) =22+7

2min2+1

e

min2 (2 e

min2 )

(1 emin

2 )

if > 0

=2

2+5(3 8)

( 2)2( 3)min21 if = 0 and > 3


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where

CTc1(,,

min)

2P

total

c1(,, min) =22+7

2min2+1

e

min2 (2 e

min2 )

(1 emin

2 )

if > 0

=2

2+5(3 8)

( 2)2( 3)min21 if = 0 and > 3


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where

CTc1(,,

min)

2P

total

c1(,, min) =22+7

2min2+1

e

min2 (2 e

min2 )

(1 emin

2 )

if > 0

=2

2+5(3 8)

( 2)2( 3)min21 if = 0 and > 3

Energy cost of communicating one bit-meter in a sensor network


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where

CTc1(,,

min)

2P

total

c1(,, min) =22+7

2min2+1

e

min2 (2 e

min2 )

(1 emin

2 )

if > 0

=2

2+5(3 8)

(

2)2

(

3)min21 if = 0 and > 3

Energy cost of communicating one bit-meter in a wireless network


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O(n) upper bound on TransportCapacity

Theorem: Transport capacity is O(n) (Xie & K 02)




Same as square root law based on treating interference as noise

since areaA grows like (n)

So multi-hop with decode and forward with interference treated as noise is

order optimal architecture whenever (n) can be achieved

CTc1(,,

min)P

ind

2n

An( ) = n( )


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CTc1(,,

min)P

ind

2n

An( ) = n( )


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CTc1(,,

min)P

ind

2n

An( ) = n( )

J P R K


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Same as square root law base on treating interference as noise




CTc1(,,

min)P

ind

2n

An( ) = n( )


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June 30 2009 P R Kumar


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CTc1(,,

min)P

ind

2n

An( ) = n( )

Ptotal = Pind n

June 30 2009 P R Kumar


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CTc1(,,

min)P

ind

2n

An( ) = n( )

Ptotal = Pind n


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62

Idea behind proof

A Max-flow Min-cut Lemma

N= subset of nodes

Then

Rl{l:d

lNbut s

lN}

1

22 lim inf

TPNrec

(T)

PNrec

(T) = Power received by nodes inNfrom outside N

=

1

TE

xi (t)

ij

iN

jN

t=1

T

2

Prec(T)N

R1R2

R3N



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, ,

63

To obtain power bound on transportcapacity

Idea of proof

Consider a number of cutsone meter apart

Every source-destination

pair (sl,dl) with source ata distance l is cut by aboutl cuts

Thus

l

Rlll

c Rl{l is cut by Nk}

Nk

c

22liminf

TPNk

rec(T)

cPtotal

2

Nk



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64


Theorem




where

CT c1(,,min )Pind

2n

c1(,, min ) =22+7

2min2+1

e

min2 (2 e

min2 )

(1emin

2 )

if > 0

=2

2+5(3 8)

( 2)

2( 3)

min21 if = 0 and > 3


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67

Random traffic

Multihop can provide bits/second

for every source

with probability 1

as the number of nodes n

Nearly optimal since transport

capacity achieved is

Order optimality of multihop transportin a randomly chosen scenario

1

n logn

n

logn



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68

Random traffic


for every source

with probability 1





1

n logn

n

logn



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69

Random traffic


for every source

with probability 1





1

n logn

n

logn



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70

Random traffic


for every source

with probability 1





1

n logn

n

logn



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71

Random traffic


for every source

with probability 1





1

n logn

n

logn



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72

Random traffic


for every source

with probability 1





1

n logn

n

logn



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73

Random traffic


for every source

with probability 1





1

n logn

n

logn



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74

Random traffic


for every source

with probability 1




So Random case Best Case


1

n logn

n

logn



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75

What can multihop transportachieve?

Theorem

A set of rates (R1,R2, ,Rl) can besupported by multi-hop transport if

Traffic can be routed, possibly overmany paths, such that

No node has to relay more than

where is the longest distance of a hop

and

S

e2

Pind 2c3(,,min )Pind+

2

c3(,,min) =23+2

emin

min1+2 if > 0

=2

2+2

min2

(

1)

if = 0 and > 1



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76

Multihop transport can achieve (n)

Theorem



Then in a regular planar network

where

CT Se2

Pind

c2 (,)Pind +2

n

c2(,) =4(1+4)e

24e

4

2(1 e2 )if > 0

=16

2+ (216)

(1)(21)if = 0 and >1

n sources each sending

over a distance n



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77

Optimality of multi-hop transport

Corollary

So if > 0 or > 3

And multi-hop achieves (n)

Then it is optimal with respect to the transport capacity- up to order

Example



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78

Multi-hop is almost optimal in arandom network

Theorem

Consider a regular planar network

Suppose each node randomly chooses a destination

Choose a node nearest to a random point in the square

Suppose > 0 or> 1

Then multihop can provide bits/time-unit for every

source with probability 1 as the number of nodes n

Corollary

Nearly optimal since transport achieved is

1

n logn

n

logn



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79

Idea of proof for random source -destination pairs

Simpler than Gupta-Kumar sincecells are square and containone node each

A cell has to relay traffic if a randomstraight line passes through it

How many random straight lines

pass through cell?

Use Vapnik-Chervonenkis theoryto guarantee that no cell is overloaded


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82

What happens when the attenuationis very low?


A f ibl f h G i


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83

A feasible rate for the Gaussianmultiple-relay channel

Theorem

Suppose ij = attenuation from i toj

Choose power Pik = power usedby i intended directly for node k

where

Then

is feasible

Proof based on coding

ij

i

j

Piki k

R < min1 jn

S 1

2

ij Piki=0

k1

2

k=1

j

Pikk=i

M

Pi



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84

A group relaying version

Theorem

A feasible rate for group relaying

R


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85

A dichotomy: Optimal architecturedepends on attenuation by medium

When =0 and small (XK 04)

Transport capacity can grow superlinearly like (n) for > 1

Coherent multi-stage relaying with interference cancellation can beoptimal

Unbounded transport capacity for fixed total power



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86

Coherent multi-stage relaying with interference subtraction(CRIS)

All upstream nodes coherently cooperate to send a packet to

the next node

A node cancels all the interference caused by all transmissions

to its downstream nodes

Another strategy

k-1 k-2 k-3k

k k-1 k-2k+1



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87


All upstream nodes coherently cooperate to send a packet to

the next node



Another strategy

k

kk+1



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88


All upstream nodes coherently cooperate to send a packet tothe next node



Another strategy

k k-1 k-2k+1


U b d d t t it


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89

Unbounded transport capacity canbe obtained for fixed total power

Theorem

Suppose = 0, there is no absorption at all,

And < 3/2

Then CTcan be unbounded in regular planar networkseven for fixed Ptotal

Theorem

If = 0 and < 1 in regular planar networks Then no matter how many many nodes there are

No matter how far apart the source and destination are chosen

A fixed rate Rmincan be provided for the single-source destinationpair



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90

Idea of proof of unboundedness

Linear case: Source at 0, destination at n

Choose

Planar case

Pik =P

(k i)k

0

1

i

k

n

Pik

Source Destination

Source

0 iq rq

Destination

(i+1)q

iq-1


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92

Idea of proof

Consider a linear network

Choose

A positive rate is feasible from source to destination for all n

By using coherent multi-stage relaying with interference cancellation

To show upper bound

Sum of power received by all other nodes from any nodej is bounded

Source destination distance is at most n

0

1

i

k n

Pik

Source Destination

Pik =P

(k i)

where 1


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Low path loss

Theorem (Unbounded path loss)

Suppose = 0 and < 3/2

Then CT

can be unbounded in regular planar networks even for fixed Ptotal

Theorem (Superlinear scaling)

Suppose = 0. Then for every 1/2


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Recent work


Low path loss Scaling behavior for


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Low path loss: Scaling behavior forpath loss exponent < 3

For what path loss exponents smallerthan 3 is CT = (n)?

Jovicic, Viswanath and Kulkarni 04:

Xie and K 06:

So the question remains for 1 1

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What is the scaling behavior in the


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What is the scaling behavior in therange

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1< < 2

Ozgur, Leveque and Tse 07: Lower bound

Based on cooperation- Long range MIMO between blocks of nodes

- Intra-cluster cooperation

- Transmit and receive cooperation

- Xie 08: Exact study of pre-constant and shows it is o(1)

Niessen, Gupta and Shah 08: Arbitrarily spaced nodes

n(n) cn2

for 1 3

2

n(n) c ' n for3

2 2

Aeron and Saligrama 07: How to achieve a total

throughput of in a dense network n2 /3( )


Is channel the right model for


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Is channel the right model formassive cooperation?

Franceschetti, Migliore, Minero 08

Number of information channels is only

Scaling law per node

Limitation in spatial degrees of freedom

Not based on empirical path-loss models and stochastic fading models

Depends only on geometry

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O n( )

O log

2n

n


Paper by Lloyd Giovannetti and


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Paper by Lloyd, Giovannetti andMaccone

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Remarks

Studied networks with arbitrary numbers of nodes Explicitly incorporated distance in model

Distances between nodes

Attenuation as a function of distance

Distance is also used to measure transport capacity

Make progress by asking for less Instead of studying capacity region, study the transport capacity

Instead of asking for exact results, study the scaling laws The exponent is more important

The preconstant is also important but is secondary - so bound it

Draw some broad conclusions Optimality of multi-hop when absorption or large path loss

Optimality of coherent multi-stage relaying with interference cancellation when noabsorption and very low path loss

Open problems abound What happens for intermediate path loss when there is no absorption

The channel model is simplistic, ...

..



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References-1

C. E. Shannon, "A mathematical theory of communication", Bell Syst. Tech.

J.", Vol 27, pp. 379--423", 1948.

C. E. Shannon, "Communication in the presence of noise", Proceedings ofthe IRE, vol. 37, pp. 10--21, 1949.

C. E. Shannon and W. Weaver The Mathematical Theory of Information,

University of Illinois Press, Urbana, 1949. R. G. Gallager, Information Theory and Reliable Communication, John Wiley

and Sons, New York, 1968.

T. Cover and J. Thomas, Elements of Information Theory, Wiley and Sons,New York, 19103.

R ~Ahlswede, ``Multi-way communication channels, in Proceedings of the

2nd Int. Symp. Inform. Theory (Tsahkadsor, Armenian S.S.R.), (Prague), pp.23-52, Publishing House of the Hungarian Academy of Sciences, 1971.

H. Liao, Multiple access channels. PhD thesis, University of Hawaii,Honolulu, HA, 1972. Department of Electrical Engineering.

T. Cover, Broadcast channels, IEEE Trans. Inform. Theory, vol. 18, pp.2-14, 1972.



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P. Bergmans, ``Random coding theorem for broadcast channels with

degraded components,' IEEE Trans. Inform. Theory, vol. 19, pp. 197207,1973.

P. Bergmans, ``A simple converse for broadcast channels with additive whiteGaussian noise,' IEEE Trans. Inform. Theory, vol.~20, pp. 279-280, 1974.

E. C. Van der Meulen, Three-terminal communication channels, Adv. Appl.Prob., vol. 3, pp. 120-154, 1971.

T. Cover and A.~E. Gamal, ``Capacity theorems for the relay channel,' IEEETrans. Inform. Theory, vol.~25, pp.~572--584, 1979

M. Franceschetti, J. Bruck, and L. J. Schulman, A random walk model ofwave propagation, IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1304

1317, May 2004. Liang-Liang Xie and P. R. Kumar, New Results in Network Information

Theory: Scaling Laws for Wireless Communication and Optimal Strategies for

Information Transport, Proceedings of 2002 IEEE Information Theory

Workshop, Bangalore, India, pp. 2425, October 20-25, 2002.

References-2



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References-3

Liang-Liang Xie and P. R. Kumar, A Network Information Theory for

Wireless Communication: Scaling Laws and Optimal Operation, IEEETransactions on Information Theory, vol. 50, no. 5, pp. 748767, May 2004.

Piyush Gupta and P. R. Kumar, Towards an Information Theory of Large

Networks: An Achievable Rate Region, IEEE Transactions on InformationTheory, vol. 49, no. 8, pp. 18771894, August 2003.

Liang-Liang Xie and P. R. Kumar, An Achievable Rate for the Multiple-

Level Relay Channel, IEEE Transactions on Information Theory, vol. 51,no. 4, pp. 13481358, April 2005.

Feng Xue and P. R. Kumar, Scaling Laws for Ad Hoc Wireless Networks:An Information Theoretic Approach. NOW Publishers, Delft, The

Netherlands, 2006.

Liang-Liang Xie and P. R. Kumar, On the Path-Loss Attenuation Regimefor Positive Cost and Linear Scaling of Transport Capacity in Wireless

Networks, Joint Special Issue of IEEE Transactions on Information Theoryand IEEE/ACM Transactions on Networking on Networking and Information

Theory, pp. 23132328, vol. 52, no. 6, June 2006.



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References-4

Liang-Liang Xie and P. R. Kumar, Multisource, multidestination, multirelay

wireless networks, IEEE Transactions on Information Theory, Special issueon Models, Theory and Codes for Relaying and Cooperation in

Communication Networks, vol. 53, no. 10, pp. 35863595, October 2007.

Feng Xue, Liang-Liang Xie, and P. R. Kumar, The Transport Capacity ofWireless Networks over Fading Channels, IEEE Transactions on

Information Theory, vol. 51, no. 3, pp. 834847, March 2005.

O. Lvque and I. E. Telatar, Information-theoretic upper bounds on thecapacity of large, extended ad hoc wireless networks, IEEE Trans. Inf.

Theory, vol. 51, no. 3, pp. 858865, Mar. 2005.

A. Jovicic, P. Viswanath and S. R. Kulkarni,. Upper Bounds to TransportCapacity of Wireless Networks,. IEEE Transactions on Information Theory,50(11):2555--2565, 2004.



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References-5

S. Aeron, V. Saligrama, Wireless Ad-hoc networks: Strategies and scaling

laws in Fixed SNR regime, IEEE Trans. on Info Theory (to appear)

Ayfer Ozgr, Olivier Lvque, and David N. C. Tse, Hierarchical

Cooperation Achieves Optimal Capacity Scaling in Ad Hoc Networks, in

IEEE Transactions on Information Theory, vol. 53, no. 10, Oct 2007,

Liang-Liang Xie, On Information-Theoretic Scaling Laws for WirelessNetworks, arXiv:0809.1205v2 [cs.IT], 2008

Urs Niesen, Piyush Gupta, and Devavrat Shah, On Capacity Scaling inArbitrary Wireless Networks, to appear in IEEE Transactions on InformationTheory arXiv:0711.2745v2 [cs.IT]

Massimo Franceschetti, Marco D. Migliore, Paolo Minero, The Capacity of

Wireless Networks: Information-theoretic and Physical Limits, Forty-FifthAnnual Allerton Conference Allerton House, UIUC, Illinois, September

26-28, 2007. IEEE Trans. on Information Theory, in press.


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wireless network information theory

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