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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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
What is Information Theory?
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Model of communication
InformationSource
InformationTransmitter
Channel Receiver InformationSink
Noise
Message
Receivedsignal
TransmittedSignalMessage
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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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
There are a set of 2nR messages
The receiver decodes {Y1, Y2, , Yn}as W
There are a set of 2nR messages
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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 )
=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
{1,2,3,,2TRik}
xi yj
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June 30, 2009 , P. R. Kumar
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 )
=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
{1,2,3,,2TRik}
xi yj
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June 30, 2009 , P. R. Kumar
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 )
=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
{1,2,3,,2TRik}
xi yj
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June 30, 2009 , P. R. Kumar
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 )
=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
{1,2,3,,2TRik}
xi yj
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June 30, 2009 , P. R. Kumar
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 )
=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
{1,2,3,,2TRik}
xi yj
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June 30, 2009 , P. R. Kumar
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 )
=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
{1,2,3,,2TRik}
xi yj
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June 30, 2009 , P. R. Kumar
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 )
=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
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
{1,2,3,,2TRik}
xi yj
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June 30, 2009 , P. R. Kumar
CT = sup
(R1,R2 ,
,Rn(n1))
Ri
i=1
n(n1)
i
Wi = gj(yjT,Wj)
Transmitted and received signals
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
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
{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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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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June 30, 2009 , P. R. Kumar
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
Energy cost of communicating one bit-meter in a sensor network
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June 30, 2009 , P. R. Kumar
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
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)
Suppose > 0, there is some absorption,
Or > 3, if there is no absorption at all
Then for all Planar Networks
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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June 30, 2009 , P. R. Kumar
O(n) upper bound on TransportCapacity
Theorem: Transport capacity is O(n) (Xie & K 02)
Suppose > 0, there is some absorption,
Or > 3, if there is no absorption at all
Then for all Planar Networks
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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June 30, 2009 , P. R. Kumar
O(n) upper bound on TransportCapacity
Theorem: Transport capacity is O(n) (Xie & K 02)
Suppose > 0, there is some absorption,
Or > 3, if there is no absorption at all
Then for all Planar Networks
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( )
J P R K
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June 30, 2009 , P. R. Kumar
O(n) upper bound on TransportCapacity
Theorem: Transport capacity is O(n) (Xie & K 02)
Suppose > 0, there is some absorption,
Or > 3, if there is no absorption at all
Then for all Planar Networks
Same as square root law base 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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June 30, 2009 , P. R. Kumar
O(n) upper bound on TransportCapacity
Theorem: Transport capacity is O(n) (Xie & K 02)
Suppose > 0, there is some absorption,
Or > 3, if there is no absorption at all
Then for all Planar Networks
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( )
Ptotal = Pind n
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June 30, 2009 , P. R. Kumar
O(n) upper bound on TransportCapacity
Theorem: Transport capacity is O(n) (Xie & K 02)
Suppose > 0, there is some absorption,
Or > 3, if there is no absorption at all
Then for all Planar Networks
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( )
Ptotal = Pind n
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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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O(n) upper bound on TransportCapacity
Theorem
Suppose > 0, there is some absorption,
Or > 3, if there is no absorption at all
Then for all Planar Networks
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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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
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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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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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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71
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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72
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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73
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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74
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
So Random case Best Case
Order optimality of multihop transportin a randomly chosen scenario
1
n logn
n
logn
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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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Multihop transport can achieve (n)
Theorem
Suppose > 0, there is some absorption,
Or > 1, if there is no absorption at all
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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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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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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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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What happens when the attenuationis very low?
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A f ibl f h G i
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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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A group relaying version
Theorem
A feasible rate for group relaying
R
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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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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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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
kk+1
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Coherent multi-stage relaying with interference subtraction(CRIS)
All upstream nodes coherently cooperate to send a packet tothe next node
A node cancels all the interference caused by all transmissions
to its downstream nodes
Another strategy
k k-1 k-2k+1
June 30, 2009 , P. R. Kumar
U b d d t t it
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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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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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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
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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( )
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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
June 30, 2009 , P. R. Kumar
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
June 30, 2009 , P. R. Kumar
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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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