DEFENSE 1
Efficient Data Communication Protocols for Wireless
NetworksDISSERTATION DEFENSE
Engin Zeydan
December 08, 2010Advisor: Prof. Cristina Comaniciu(Stevens Institute of Technology)
Co-Advisor: Dr.Didem Kivanc(West Virginia University Institute of Technology)
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MAIN CONTRIBUTIONS:
I- Energy-efficient Routing for Correlated Data in Wireless Sensor Networks (WSNs). (PROPOSAL)II- Throughput maximizing Routing for Correlated Data in WSNs.III- Beamforming for multi-user MIMO Ad Hoc Networks.
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OUTLINE Introduction for routing in WSNs
System Model & Concepts for WSNs
Game Theory Background
PART II- Throughput Maximizing for Correlated Data in
WSNs
a)Facility Cost Selection for the Congestion Game
b)Simulation Results
PART III- Iterative Beamforming and Power Control
a)Cooperative Algorithm
b)Noncooperative Regret-Matching Algorithm
Conclusions & Future Work
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Sink Node
Sensor Node
Efficient distributed routing algorithms are of utmost importance for connectivity and resource allocation of wireless sensor networks (WSNs) . Many routing protocols emphasizing various metrics depending on the application and design specifications.
I. Introduction
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Sink Node
A simple game-theoretic model with utility function that account for data correlation for energy minimization and throughput maximization problems.
I. Introduction
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Each source node Yi generates a certain amount of data Ψ(Yi), where Ψ(Yi) is the data rate (encoding rate) of source Yi. The units is bits/symbol.
II. System Model
Sink Node
YφN-2
Sensor Node
Y1Yi
YφN
Y2
YφN-1
Relay Node#
# #
source
relay source
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The nodes can either send their own raw data directly into the sink, or, they can aggregate.
II. System Model
Sink Node
YφN-2
Sensor Node
Y1Yi
YφN
Y2
YφN-1
Relay Node
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Synchronous direct-sequence CDMA (DS-CDMA) where all nodes use a variable spreading sequences of length L.
II. System Model
Sink Node
YφN-2
Sensor Node
Y1Yi
YφN
Y2
YφN-1
Relay Node
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The energy per bit Ebi,j for packet transmissions
between nodes Yi and Yj can be defined as:
,
, ( )i j ib
i j c
MPE
mR P
M the packet length, m the information bits in a packet, Pi the constant transmit power for all i, Pc(γ) is the probability of a correct reception, which depends on the achieved SIR, γ. Rij : bit throughput of the link between Yi and Yj
Joule
bits
II. System Model
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Rij : bit throughput --> Rij= W/Lij
W : the system bandwidth, Lij : The minimum spreading gain to reach a target
SIR γ*.
*
1, ,, * 2
N
kj kk k i j
i jij i
h P
Lh P
where the link gain hij = 1/dij2,
• dij is the distance of between the nodes Yi and Yj, • σ2 is the thermal noise.
sec
bits
II. System Model
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The joint data rate Ψ(Yi, Yj) of two sources Yi and
Yj after data aggregation, is,( , ) max( ( ), ( )) (1 ) min( ( ), ( ))i j i j i j i jY Y Y Y Y Y
II. System Model
Ψ(Yj)
where ρi,j correlation coefficient, and if Gaussian random field data correlation model is used ρ= exp(-d2
Yi,Yj /c)
Ψ(Yi,Yj)
Source Yi
Source YjΨ(Yi)
Data Aggregation at Yj
Correlation constant
0 1
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Ψfinal(Ys)
Source Y1 Source Y2
Ψ(Y2)
Ψ(Y1)Data Aggregationat source Ys
. . .
Source Ym
Ψ(Ym)
Ψtemp_1 (Ys)=Ψ (Ys)
ρ=exp(-d2ij/c)
(Ys)
(Ys,Y1)
(Ys,Y2)
Sources Entropy
(Ys,Ym)
Ψtemp_2(Ys)= max (Ψtemp_1 (Ys), Ψ(Y1)) +(1-ρ) min (Ψtemp_1 (Ys), Ψ(Y1))Ψtemp_3(Ys)=
max(Ψtemp_2(Ys), Ψ(Y2))+(1- ρ) min (Ψtemp_2(Ys), Ψ(Y2))
.
.
. Ψfinal (Ys)=max(Ψ(Ym),Ψ(Ytem
p_m (Ys))) + (1- ρ) min (Ψ(Ym),,Ψ(Ytemp_m (Ys)))
Ψtemp_1(Ys)Ψtemp_2(Ys)Ψtemp_3(Ys)
Source Ys
II. System Model
Final Result
Case I: Multiple source Data aggregation Algorithm
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II. System Model
Forgetting factor
Ψ(Yi,Yj)YiYj
Data Aggregation
0 1 where
Case II: Multi-hop data aggregation
,
( , ) max( ( ), ( ))
(1 ) min( ( ), ( ))
agg i j agg i agg j
i j agg i agg j
Y Y Y Y
Y Y
Ψ(Yi)=Ψ(Yi)+Ψagg(Yi)
Ψ(Yj)=Ψ(Yj)+Ψagg(Yj)
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Sink Node
Yi
, , ( )k l k ls b k k
Joule Joule bitsE E Y
symbol bits symbol
Y2
Y3 Y4Y6
Y7
Y5
SiEb
i,2
Energy per symbol
Y4
Y6
Y5
Ψ(Y4)
Ψ(Y6)
Ψ(Y5)
Energy per Symbol : Eb
2,3Eb
3,4
Eb4,5
Eb6,5
Eb5,7
Eb7,sink
Esi,2
Es2,3
Es3,4
Es4,5
Es6,5
Es5,7
Es7,sink
II. System Model
15
Sink Node
Yi
Λ1,2
Λ2,3
Λ3,4 Λ4,5 R6,
5
Λ5,7 Λ7,sin
k
,
,,
secsec ( )
( )
k l
k lk l k k
k k
bitsR
symbol W
L YbitsY
symbol
Y2
Y3 Y4Y6
Y7
Y5Si
Ri,2
R2,3
R3,4R4,5
Λ6,5
R5,7
R7,sink
Symbol Throughpu
t
Y4
Y6
Y5
Ψ(Y4)
Ψ(Y6)
Ψ(Y5)
Symbol Throughput:
II. System Model
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Sink Node
Y1
Sensor Node Y2
Y3 Y4
Y6
Y7
Y5
Bottleneck link
Throughput of Y1 Λ7,sink
Λ1,2
Λ2,3
Λ3,4 Λ4,5
Λ7,sink
Λ6,5
Λ5,7
II. System Model
Λ7,sink
Λ7,sink
Λ7,sink
Λ7,sinkΛ7,sin
kΛ7,sink
S1
Bottleneck Throughput:
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Game Theory Background
Finite Strategic Form Games:
1, ,{ }Nm mP A U
the set of players1 2{ , ,..., }NP P P P
1 2{ , ,..., }NA A A A represents the set of actions
1{ } :Nm mU A is the utility function
m ma A is the action of player Pm
1 2( , ,..., ) ( , )N m ma a a a a a action of all players
18
Forms of Equilibrium
Nash Equilibrium
An action profile * * *( , ) max ( , )
m mm m m m m m
a AU a a U a a
*a A is called a pure NE if
Mixed-Strategy Nash EquilibriumA strategy profile
* ( )A is called a mixed-strategy NE
where
( )A is the set of probability space
1 2( , ,..., )N is the joint probability distribution of all players
, m P
* * *
( )( , ) max ( , )
m mm m m m m m
AU U
, m P if
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Forms of Equilibrium
Correlated Equilibrium: The probability distribution
( , ) ( , ) ( ' , ) ( , )m m m m
m m m m m m m m m ma A a A
U a a a a U a a a a
( )a is a correlated equilibrium if
Coarse –Correlated Equilibrium
where '
( ) ( ' , )m m
m m m ma A
a a a
( , ) ( , ) ( ' , ) ( )m m
m m m m m m m m m ma A a A
U a a a a U a a a
The probability distribution
( )a is a coarse- correlatedequilibrium if
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Classes of Games
Identical Interest Games:The players utilities are same, i.e. for some function
( , ) ( , )m m m m mU a a a a
Potential Games
: A R
An exact potential function Pot(.) is defined as
: , , 'm m mPot A P P a a A
( , ) ( ' , ) ( , ) ( ' , )m m m m m m m m m mu a a u a a Pot a a Pot a a
mP P a A
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Classes of Games
Congestion Games:
1 1, ,{ } ,{ } fmNi i f fP F A w
: the set of players1 2{ , ,..., }NP P P P
1 2{ , ,..., }NA A A A represents the set of actions
1{ } :fm
f fw A is the cost of using facility f
{1,2,..., }fF m : the set of facilities
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Classes of Games
Congestion Games:
Define
using facility f
( ) { | }f i ma P f a
Then, the utility function is
as the subset of players
( , ) ( ( ))m
m m m f ff a
u a a w a
:mu a
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Iterative Updating Schemes
Let T: AA be any mapping from a subset A a=(a1,a2,…,aN) is the set of actions.
Define the updating scheme of actions:a(n+1)=T(a(n)).
The most common updating strategies are: 1-) Jacobi scheme: All components of a=(a1,a2,
…,aN) are updated simultaneously. 2-) Gauss-Seidel scheme: All components of
a=(a1,a2,…,aN) are updated sequentially. 3-) Totally asynchronous scheme: All components
of a=(a1,a2,…,aN) are updated totally asynchronous way.
NR
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PART II:
Throughput maximized Routing for Correlated Data in Wireless Sensor
Networks
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Motivation:
•Time Critical applications: Deadline (early disaster warning applications or timely detection of events)
•Maintain a certain throughput in order to satisfy the quality-of-service (QoS) requirements and stability requirements under latency constraints in a practical system.
• Trade-off between energy and throughput
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Sink Node Senso
r Node
Branch B1
Branch B2
Y1
Yi
YN
Y2
YN-1
YN-2
ΛB1ΛB2
Bottleneck Link Bottleneck
LinkΛB1
ΛB1
ΛB1 ΛB1
ΛB1
ΛB1
Throughput B1:ΛB1nB1
ΛB2
ΛB2
ΛB2ΛB2
ΛB2Throughput B2: ΛB2 nB2
Bottleneck Throughput
Number of sources
Efficient Routing for Throughput Maximization
Branches of a tree:
Si
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An NP complete optimization problem!!! A game theoretic formulation Convergence to a local optimal solution with relatively low complexity and in a distributed fashion.
Efficient Routing for Bottleneck Throughput Maximization
Maximize
subject to *,k lSIR , kP C
where ,,
mini
i k lk l B
1
N
ii
i iS X
and iX other possible routes for Yi
DEFENSE 28
Sink Node
Sensor Node Yi
N = {Y1,…….,Yφ}
The congestion game Γ is a tuple (N, F, (Si)iЄN ,(wf) fЄF)
(Si)iЄφF = {1,….,m} (wf) fЄF
(w1)
(w2)(w3)
(w4)
(w5)
Si
Efficient Routing for Throughput Maximization
Relay Node
DEFENSE 29
The utility function for source Yi in our congestion game is:
( , ) ( , )i
i i i f i if S
u S S w S S
where S-i = (S1, S2,………, Si-1, Si+1 ,….,Sφ).
The game performance is influenced by the selection of cost functions wf (Si, S-i) for facilities.
Efficient Routing for Energy Minimization
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In setting up the costs for facilities, we can consider several parameters:
Obtained energy or throughput for relaying bits on outgoing route from the facility,
Impact of interference awareness, Opportunity for aggregation by exploiting data
correlation
a) Facility Cost Selection for the Congestion Game
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A. Minimum Energy Routing (MER)
a) Facility Cost Selection for the Congestion Game
Sink Node
Sensor Node
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For MER, the following utility function is used
where Ebf is the cost of facility f through the strategy
(or route) Si and S-i.
Source Yi
Facility f1
Ebf1
Facility f2
Ebf2
Strategy Si
Ebf0
( , )i
fi i i b
f S
u S S E
a) Facility Cost Selection for the Congestion Game
Sink Node
a) Facility Cost Selection for the Congestion Game
Dr
Interference Aware Routing (IAR) (*)
nf1
nf2
nf3
nf4
nf5
nf6
nf7
nf8
nf9
nf10
nf11
nf12
nf13
Sink Node
Sensor Node
Relay Node
(*) H. Mahmood, C. Comaniciu, Interference aware cooperative routing for wireless ad hoc networks Ad Hoc Networks, vol. 7, no. 1, pp. 248–263, 2009
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a) Facility Cost Selection for the Congestion Game
For IAR, the following utility function is used
Source Yi
Facility f1
Ebf1
Facility f2
Ebf2
Strategy Si
i i -iu (S , S ) = -i
ff b
f S
E
nf2nf1
Dr
Ebf0
nf0Sink Node
DEFENSE 35
Maximum Utility
U1U2
U3
U4
U5
Throughput Maximizing Correlation Aware Routing (T-CAR)
Sink Node
Sensor Node
a) Facility Cost Selection for the Congestion Game
DIRECT
DEFENSE 36
Throughput Maximizing Correlation Aware Routing (T-CAR)
Sink Node
Sensor Node
a) Facility Cost Selection for the Congestion Game
T-CAR
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Utility Function of source Yi forT-CAR:
D,Sink Node
Sensor Node
Y1
Yi
YN
ΛB1
New Bottleneck Link
Bottleneck Throughput
Prune sub-tree
of Yi 1 1 1 1( , ) ( ) '
ii i i B B B Y Bu S S n n n
Λ'B1
1 11 B Bu n
nYi
Number of sources at subtree Yi
Branch B1
a) Facility Cost Selection for the Congestion Game
1 2( , )i i iu S S u u
1 12 ( ) 'iB Y Bu n n ( )NY1( )NY
2( )Y
1( )i Y
1( )Y
( )iY
Y2
YN-1
1
1, 1( )BD
W
L Y
1
1, 1
'( )B i
D
W
L Y Si
DEFENSE 38
A Potential Game Formulation for T-CAR
a) Facility Cost Selection for the Congestion Game
It can be shown that T-CAR is an exact potential game with the potential function,
1
( , )N
i i ii
P S S
1 1 1 1( , ) ( ) '
ii i i B B B Y Bu S S n n n
Therefore, T-CAR converges!
DEFENSE 39
Difference from T-CAR is starting tree:
T-ICAR starting tree IART-CAR starting tree MER or DIRECT
Throughput maximizing interference and Correlation Aware Routing (T-ICAR)
The utility of source Yi:
a) Facility Cost Selection for the Congestion Game
1 1 1 1( , ) ( ) '
ii i i B B B Y Bu S S n n n
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Proposition 1: Let Sdirect be the selected strategy for each source Yi, ∀ i ∈ N where all nodes communicate directly with sink node. Then, the set of state Sdirect = {S1,S2, ..., Sφ} is NE solution
Condition for the Nash equilibrium (NE) strategy
where Li,D is the minimum spreading gain from node Yi to sink D,Li,j is the minimum spreading gain between nodes Yi and Yj.
,i j
i j
Y Y N
Y Y
,, ,
, ,
2
( )j D
i j i ji D j D
L
L L
, , (2 )i j j D ijL L
elseif
if
, , (2 )i j j D ijL L ,
, ,, ,
2
( )j D
i j i ji D j D
L
L L
and
and
DEFENSE 41
Proof of Proposition 1:
Yi
D
Yj
Yi
D
Yj
DIRECT
,
( , )i i ii D
Wu S S
L , ,
,
2( ' , )
( (1 ) )i i ij D i j
j D
Wu S S
L
W
L
S’iSi
Ψ(Yi) = Ψ(Yj) = μ (bits/symbol)
Bottleneck link
1-)
CASE 1:
DEFENSE 42
Proof of Proposition 1:
Yi
D
Yj
Yi
D
Yj
DIRECT
S’iSi
, , ,(2 )i j j D i j
W W
L L
Bottleneck link
, , ,(2 )i j j D i jL L 2-)
CASE 1:
DEFENSE 43
Proof of Proposition 1:
Yi
D
Yj
Yi
D
Yj
DIRECT
SiS’i
,i j i jY Y N Y Y ,
, ,, ,
2
( )j D
i j i ji D j D
L
L L
Therefore, if , , ,(2 )i j j D i jL L
and
Bottleneck link
CASE 1:
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Proof of Proposition 1:
Yi
D
Yj
Yi
D
Yj
DIRECT
,
( , )i i ii D
Wu S S
L , ,
2( ' , )i i i
i j j D
W Wu S S
L L
S’iSi
Ψ(Yi) = Ψ(Yj) = μ (bits/symbol)
Bottleneck link
1-)
CASE 2:
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Proof of Proposition 1:
Yi
D
Yj
Yi
D
Yj
DIRECT
S’iSi
, , ,(2 )j D i j i j
W W
L L
, , ,(2 )i j j D i jL L 2-)
Bottleneck link
CASE 2:
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Proof of Proposition 1:
Yi
D
Yj
Yi
D
Yj
DIRECT
SiS’i
,i j i jY Y N Y Y
,, ,
, ,
2
( )j D
i j i ji D j D
L
L L
Therefore, if , , ,(2 )i j j D i jL L
and
Bottleneck link
CASE 2:
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Corollary: For a special case when
Condition for the Nash equilibrium strategy
Then, Sdirect is a NE point.
2,
, exp( )i ji j
d
c
2,
,
1log( )
i j
i j
dc
,i j
i j
Y Y N
Y Y
, , (2 )i j j D ijL L
elseif
if
, , (2 )i j j D ijL L
2,
,
1log( )
i j
i j
dc
DEFENSE 48
Condition for the Nash equilibrium strategy
Corollary 2: For a special case when c=0 (no correlation), and
state Sdirect is a NE point
, ,2i j j DL L , ,i j i jY Y N Y Y
DEFENSE 49
The number of nodes in the network is selected to be N = 4 to 40, uniformly distributed over a square area of dimension 100m X 100m.
The target SINR is selected to be γ*= 5 (7 dB) Constant transmit Power Pi =10-2 Watts (10 dBm),
σ2= 10-13 Watts, W=1 Mhz, forgetting factor=0.8. Each symbol is represented with 1 bits, i.e.
Ψ(Yi)=1bits/symbol for all Yi. The spatial correlation of data is chosen to be
c=0 (no aggregation), c = 100 (low correlated) and c=1000 (highly correlated). ρij=exp(-d2
ij/c).
b) Simulation Results
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Start TCAR with MER when φ=0.5 (half relay nodes, half source nodes) and with DIRECT when φ=1.0 (all source nodes)
T-ICAR always starts with IAR. Use Dr=16m, for the best performance of IAR
(Trade-off between energy and throughput). MER and IAR perform data aggregation
opportunistically based on the routes set-up, i.e. whenever the routes meet
Average over 1000 different topologies
b) Simulation Results
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b) Simulation Results
(a) MER (b) IAR
N=24, φ=0.5 (half source, half relay), and c=100, Dr =16m
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Relay
SourceSink
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Relay
Source
Sink
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b) Simulation Results
(c) T-CAR (d) T-ICAR
N=24, φ=0.5 (half source, half relay), and c=100, Dr =16m
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Relay
Source
Sink
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Relay
Source
Sink
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b) Simulation Results
10 15 20 25 30 35 40150
200
250
300
350
400
Number of nodes (N)
Tota
l th
roughput
(kbps)
T-CAR
DIRECT
MER
T-CAR throughput improvements MER DIRECT
• N=10 (72 %) (16 %) • N=40 (70 %) (54 %)
70 %
φ=1 (all sources), and c=1000.
DEFENSE 54
b) Simulation Results
10 15 20 25 30 35 4010
-3
10-2
10-1
100
101
Number of nodes (N)
Tota
l energ
y (
Joule
/bits)
DIRECT
T-CAR
MER
TCAR Energy loss MER• N=10 (81 %) • N=40 (55 %)
φ=1 (all sources), and c=1000.
55 %
81 %
DEFENSE 55
b) Simulation ResultsN=24, φ=0.5 (half source, half relay), and c=100, Dr =16m
IAR Throguhtput Improvements MER
8.23 %
8.23 %
1 2 3 4 5 6 7 890
100
110
120
130
140
150
160
170
180
Number of iteration
Tot
al t
hrou
ghpu
t (k
bps)
T-ICAR
T-CARIAR
MER
DEFENSE 56
b) Simulation Results
Throguhtput improvements T-ICAR T-CAR IAR MER
4.36% 70% 84%
84 %
N=24, φ=0.5 (half source, half relay), and c=100, Dr =16m
1 2 3 4 5 6 7 890
100
110
120
130
140
150
160
170
180
Number of iteration
Tota
l th
roughput
(kbp
s)
T-ICAR
T-CARIAR
MER
DEFENSE 57
b) Simulation Results
Energy Loss of IAR MER 9.09 %
9 %
N=24, φ=0.5 (half source, half relay), and c=100, Dr =16m
1 2 3 4 5 6 7 810
-3
10-2
10-1
100
Number of iteration
Tot
al e
nerg
y (J
oule
/bits
)
T-ICAR
T-CAR
IAR
MER
DEFENSE 58
b) Simulation Results
Energy Loss of T-ICAR & TCAR IAR MER 97.06%
97.32%
97 %
N=24, φ=0.5 (half source, half relay), and c=100, Dr =16m
1 2 3 4 5 6 7 810
-3
10-2
10-1
100
Number of iteration
Tot
al e
nerg
y (J
oule
/bits
)
T-ICAR
T-CAR
IAR
MER
DEFENSE 59
b) Simulation Results
0 100 200 300 400 500 600 700 800 900 100050
100
150
200
250
300
350
Correlation constant (c)
Tot
al t
hrou
ghpu
t (k
bps)
TCAR (N=20)
TCAR (N=10)MER (N=10)
MER (N=20)
TCAR improvements over MER c=200 c=800
• N=10 (51.30 %) (43.24 %)
• N=20 (56.38 %) (46.52 %)
φ=1All sources
56%
46%
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b) Simulation Results
1 2 3 4 5 6 7 8 9 10 11 12 13 14 151
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Number of iterations
Nor
mal
ized
thr
ough
put
N=40
N=30N=20
N=10
TCAR improvements over DIRECT (TCAR/DIRECT)
61
b) Simulation Results
Comparisons with Optimal Throughput Maximizing Routing (search over 125 different trees) for N+1=5, φ=1, and c=1000
OPT TCAR MER DIRECT
Throughput (kbps)
311.6 303.26 236.47 280.03
Energy (nJ/bits)
3.0 3.4 0.6 5.3
97 %
76 %
80 %82 %Loss
DEFENSE
DEFENSE 62
The problem of efficient transmission structure in WSNs to minimize the total energy and to maximize throughput.
The impact of correlation structure in establishing routing paths towards the sink.
Distributed iterative protocols based on a game theoretic framework which are shown to converge within a couple of iterations.
Significant energy reductions and throughput gains over classic approaches can be achieved.
c) Conclusions
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c) Future Work1) Similar design approach can be used for: Rate Distortion using multi-hop routing., Network lifetime maximization., end-to-end delay minimization
through mutual information accumulation, data accuracy,
latency, data security, capacity, etc…2) Aggregation Cost can also be incorporated into utility function easily,3) Pareto optimal tree configuration, 4) Applying adaptive learning algorithms (Regret Matching, Simulated Annealing, Genetic Algorithms,
etc.)5) Comparison with Cluster-based approaches
DEFENSE 64
PART III:
Joint Iterative Beamforming and Power Adaptation for MIMO Ad-hoc
Networks
DEFENSE 65
Introduction
Interference management by performing a joint iterative transmit beamforming and power adaptaion in a multi-user MIMO ad hoc network.
Objective: Minimize the total transmit power in the network considering the interference from other nodes in the network.
“Transmit beamformers” are selected from a “predefined codebook” known for both Tx’s and Rx’s for multi-user MIMO networks.
The nodes optimize their performance by modifying their beamform patterns and powers
Grassmanian subspace packing codebook generation,[*]
1
2
1
( )
( )
.
.
( )
k
k
k
k T Tx
t
t
t
t
Codeword
where Tkt
21kt
Goal: Maximize SNR and therefore minimize BER!
T: number of antennas
1 2{ , ,..., }k k kt t t
Codebook
size γ
(*) Love, D.J., Heath Jr., R.W. , Strohmer, T. “Grassmannian beamforming for multiple-input multiple-output wireless systems “ IEEE Transactions on Information Theory, 2003
DEFENSE 67
Grassmanian subspace packing codebook generation (*)
T: 2, γ=4
-0.1612 - 0.7348i -0.0787 - 0.3192i -0.2399 + 0.5985i -0.9541, , ,
-0.5135 - 0.4128i -0.2506 + 0.9106i -0.7641 - 0.0212i 0.2996
0.7939 + 0.0590i 0.2189 + 0.0654i 0.3087 - 0.4341i 0.5915 - 0.1175i
-0.4126 - 0.0807i , 0.1844 - 0.3191i , -0.2454 - 0.6507i , 0
-0.0853 - 0.4269i -0.8804 - 0.1921i 0.4817 + 0.0258i
.3113 + 0.6635i
0.2941 + 0.1128i
T: 3, γ=4
(*) http://cobweb.ecn.purdue.edu/~djlove/grass.html
DEFENSE 68
Codeword selection in Single User Scenario
, ,arg max( )k
H Hk k k k k k k
tt t H H t
0
, ,k H H
k k k k k k
Pt H H t
0 : Target SNR
DEFENSE 69
Choose Tx beamformer with minimum Power
Rx 1Tx 1 Rx 2Tx 2
Rx kTx kRx NTx N
User 1
User k User N
User 2
1 1( , )P t 2 2( , )P t
( , )k kP t ( , )N NP t
0 : Target SINR
Interference
DEFENSE 70
Multi-User Case
Node pair m
Tx
Rx
, ,m m m m m m i m i i i mi m
r P H t b PH t b n
Signal Interference
+Noise
1 2{ , ,..., },m m mm mt t t t Transmit
Beamformer
1,
1,
,|| ||
m m m mm
m m m m
R H tw
R H t
Normalized Receive Beamformer
2|| || 1mw
Multi-user Interferenc
e
2, ,( , ) H H
m m m i m i i i m ii m
R P PH t t H I
Interference+Noise Covariance Matrix
1 2 1 2[ , ,..., ], [ , ,..., ]N Nt t t P P P P Define
( , )m mP t
2|| || 1mt
DEFENSE 71
Multi-User Case
2, 1
, ,2 2,
| |
| |
Hm m m m m H H
m m m m m m m m mHi m m i i
i m
P w H tP t H R H t
P w H t
Received signal-to-interference plus noise ratio (SINR)
01
, ,m H H
m m m m m m m
Pt H R H t
Power can also be adjusted based on received normalized SINR
arg maxm m
optm m
tt
Receiver Selects the best index of transmit beamformer:
Limited-Rate Feedback: (Extension to Multi-user case)
Convergence is not guaranteed!
DEFENSE 72
Optimization Problem
Minimize 1
N
mm
P
|| || || || 1m mw t
subject to 0m
min max ,mP P P {1,2,..., }m N
,m mt
,P
DEFENSE 73
Game Theory Interpretation
: the set of players (node pairs)
The game
1, ,{ }Nm mN C U
{1,2,..., }N N
1 2{ , ,..., }NC C C C : represents the set of actions (transmit beamformer and powers)
1{ } :Nm mU C : the utility function
min max[ , ]mP P PActions are:1 2{ , ,..., }m m mm mt t t t
mc C
DEFENSE 74
Centralized SolutionMinimize
* *
, 1
( , ) arg min ( , )N
m mP m
P P P
* * * * * * * *
1 2 1 2[ , ,..., ], [ , ,..., ]N Nt t t P P P P
where0
1, ,
( , )m m H Hm m m m m m m
P Pt H R H t
Note that * * * *1 2[ , ,..., ]Nt t t * * * *
1 2[ , ,..., ]NP P P Pgives
DEFENSE 75
Maximize
1
( , ) ( , )N
network m mm
U P P P
Identical Interest Game: All users utilities are same.
Cooperative Power Minimization Algorithm (COPMA)
1
( , ) ( , )N
m network m mm
U U P P P m N
DEFENSE 76
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
• Initialize with index one for transmit beamformers and maximum powers for all node pairs 1
m m mt t
maxmP P
1n n nt t
maxnP P
1k k kt t
maxkP P1
l l lt t
maxlP P
DEFENSE 77
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
1currentm mt t
maxcurrentmP P
updatedm mt updated
mP
Node pair n
Node pair l
Node pair k
[ , , ]current updatedm mm P P
[ , , ]current updatedn nn P P
[ , , ]current updatedk kk P P
[ , , ]current updatedl ll P P
Randomly choose
DEFENSE 78
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
Calculate1
Nupdated
updated mm
P P
Calculate1
Ncurrent
current mm
P P
DEFENSE 79
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
1Pr( )
1 exp(( ) / )updated currentP P
Keep with probabilityupdatedmt Smoothing
factor ~ 1/n2
DEFENSE 80
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
current updatedm mP P m N
If is keptupdatedmt
else no change ! (don’t update)
DEFENSE 81
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
1currentn nt t current
nPupdatedm mt updated
mPNode pair n
Node pair l
Node pair k
[ , , ]current updatedm mm P P
[ , , ]current updatedn nn P P
[ , , ]current updatedk kk P P
[ , , ]current updatedl ll P P
Randomly choose
DEFENSE 82
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
Calculate1
Nupdated
updated ii
P P
Calculate1
Ncurrent
current ii
P P
DEFENSE 83
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
1Pr( )
1 exp(( ) / )updated currentP P
Keep with probabilityupdatednt
DEFENSE 84
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
current updatedn nP P n N
If is keptupdatednt
else no change ! (don’t update)
DEFENSE 85
Cooperative Power Minimization Algorithm (COPMA)
Node pair m
Node pair n
Node pair l
Node pair k
DEFENSE 86
Cooperative Power Minimization Algorithm (COPMA)
Inititalization: The initial index of transmit beamformer is one and initial transmit power is maximum for all node pairsRepeat: Sequentially choose a node pair m. Denote tm(n) as the current transmit beamformer at iteration n for the m’th node pair.
Until: Predefined number of iterations
1-) Set tm(n)=tm(n-1).
1
Ncurrent
current ii
P P
2-) Randomly choose a transmit beamformer and calculate
1
Nupdated
updated ii
P P
updatedm mt
3-) Keep with probabilityupdatedmt
1
1 exp(( ) / )updated currentP P
and
DEFENSE 87
Theorem: COPMA converges to optimal profile
with arbitrarily high probability. In other words,
Cooperative Power Minimization Algorithm (COPMA)
* * * *1 2[ , ,..., ]Nt t t
Proof: Based on Markov chain analysis
* *
0lim lim ( ( ) ) 1
kP k
is the transmit beamformer profile at iteration k
1 2( ) [ ( ), ( ),..., ( )]Nk t k t k t k where
DEFENSE 88
11 12 1 j 1
21 22 2 j 2
1i 2i ij i
1 2 j
Two Players Markov Chain
1 111 1 2[ , ]t t 1 1
11 1 2( ) [ , ]P P P
Optimal
DEFENSE 89
11 12 1 j 1
21 22 2 j 2
1i 2i ij i
1 2 j
Two Players Markov Chain
12 11Pr ( | ) 1 11Pr ( | )
DEFENSE 90
* * * *11 1 11 1 11 1
2 2
Pr ( ) Pr ( | ) Pr ( ) Pr ( | )p p pp p
Cooperative Power Minimization Algorithm (COPMA)
The stationary distribution is calculated from
~2
*~
( )
exp( ( ( )) / )Pr ( ( ))
exp( ( ( )) / )k
P kk
P k
Then
1 111 11
1Pr ( | )
2 (1 exp(( ( ) ( )) / ))ppP P
* *
0lim lim Pr ( ( ) ) 1
kk
and
*Pr (.)
DEFENSE 91
Non-Cooperative Power Control : For non-cooperative selection, convergence of transmit beamformer is problematic under constant SINR requirements!• Noncooperative beamforming for multi-user MIMO ad-hoc networks lacks the quality of “strategic complementarities” that are found in power control-only games.• If a node pair adjusts its transmit beamformer to increase its own received SINR, this change may either increase or decrease the received SINR of every other node pair
Noncooperative beamforming and Power Control
Regret Matching based Selection Game (RMSG)
1 1( 1) ( ) ( ( , ( )) ( ( ), ( ))m mt t
m m m m m m m m
kR k R k U t t i U t i t i
k k
1
1
1( ) ( ( , ( )) ( ( ), ( ))
1m
ktm m m m m m m
i
R k U t t i U t i t ik
• We study a non-cooperative learning algorithm called the regret matching adaptive algorithm from (*), in which the players choose their actions based on the regret for not choosing particular actions in the past. 1 2[ , ,..., ]m m m mt t t tDenote
as the vector of all strategies
Define the average regret vector for an action vectormt
Update the average regret vector based on formula:
(*) S. Hart and A. Mas-Colell, “A simple adaptive procedure leading to correlated equilibrium,” Econometrica, vol. 68, no. 5, pp. 1127–1150, 2000
DEFENSE 93
Regret Matching based Selection Game (RMSG)
[ ( )]( ) Pr ( ( ) )
[ ( )]
m
m
m
m m
tt mm m m t
mt
R kk ob t k t
R k
1, ,( , ( )) log( ( , ) )H H
m m m m m m m m m m m mU t t k t H R P H t
• Choose an action according to probability distribution:
• The utility function for noncooperative users is:
• Local information
DEFENSE 94
Inititalization: The initial index of transmit beamformer are chosen randomly and initial transmit power is maximum for all node pairs
For k=1,….,ITER
Regret Matching based Selection Game (RMSG)
For m=1,….,N1) Calculate
1 1( 1) ( ) ( ( , ( )) ( ( ), ( ))m mt t
m m m m m m m m
kR k R k U t t i U t i t i
k k
2) Update the probability distribution[ ( )]
( ) Pr ( ( ) )[ ( )]
m
m
m
m m
tt mm m m t
mt
R kk ob t k t
R k
Next mNext k
3) Updatemt0
1, ,( , )m H H
m m m m m m m m m
Pt H R P H t
Regret Matching based Selection Game (RMSG)
Facts: 1-) Every finite strategy game has a mixed strategy Nash equilibrium.2-) For all finite games, using a proper learning algorithm, the game can be shown to converge to the fixed points of probability. 3-)The advantage of regret matching- based selection is that it is distributed and requires limited information exchange4-) The time-averaged behavior of regret-matching game converges almost surely (with probabilityone) to the set of coarse-correlated equilibrium (*)
(*) H. P. Young, “Strategic learning and its limits,” in Oxford University Press, 2005.
DEFENSE 96
The number of node pairs in the network is selected to be N = 4 (small network) , uniformly distributed over a square area of dimension 30m X 30m and N=10 (large network), uniformly distributed over a square area of dimension 100m X 100m.
The target SINR is selected to be γ*= 10 dB. Transmit Power range Pi = [1mW, 5mW, 20mW,
30mW, 50 mW, 100mW ], σ2= -95 dBm. Each entry in the channel matrix is i.i.d. Gaussian
random variables and don’t vary throughout iterations.
For each node pair, codebook size is ϒ=16 and number of antennas is T=3.
b) Simulation Results
DEFENSE 97
10 20 30 40 50 60 70 80 90 100 110 12010
-2
10-1
100
Iteration
To
tal t
ran
smit
Po
we
r (W
)
RMSG
COPMACentralized
Simulation Results
68 %
N=4 (Small Network), τ=0.1/k2 ,
52 % Close to optimal
DEFENSE 98
Simulation Results
0 20 40 60 80 100 1200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Iteration
Tra
nsm
it P
ow
ers
(W
)
User 1
User 2User 3
User 4
COPMA Transmit Power
DEFENSE 99
Simulation Results
0 20 40 60 80 100 1200
2
4
6
8
10
12
14
16
Iteration
Tra
nsm
it B
ea
mfo
rme
r In
de
x
User 1
User 2
User 3
User 4
COPMA Transmit Beamformer Index
DEFENSE 100
Simulation ResultsRegret-Matching probability mass function (p.m.f)
0 5 10 15 200
0.02
0.04
0.06
0.08Iteration 1
Index Value
P.m
.f
0 5 10 15 200
0.1
0.2
0.3
0.4Iteration 12
P.m
.f
Index Value
0 5 10 15 200
0.2
0.4
0.6
0.8Iteration 50
P.m
.f
Index Value0 5 10 15 20
0
0.5
1Iteration 100
P.m
.f.
Index Value
DEFENSE 101
Simulation Results
0 10 20 30 40 50 60 70 80 90 1000
10
20
30
40
50
60
70
80
90
100
Receiver
Transmitter
COPMA
N=10(Large Network), TRANSMIT BEAMPATTERNS
DEFENSE 102
Simulation Results
76 %
200 400 600 800 1000 1200 14000
0.1
0.2
0.3
0.4
0.5
Iteration
To
tal t
ran
smit
Po
we
r (W
)
RMSG
COPMA
N=10(Large Network), τ=200/k2 ,
Close
DEFENSE 103
Simulation ResultsN=10, Probability mass function, RMSG
0 5 10 15 200
0.02
0.04
0.06
0.08Iteration 1
Index Value
P.m
.f
0 5 10 15 200
0.1
0.2
0.3
0.4Iteration 500
P.m
.f
Index Value
0 5 10 15 200
0.2
0.4
0.6
0.8Iteration 1000
P.m
.f
Index Value0 5 10 15 20
0
0.5
1Iteration 1500
P.m
.f.
Index Value
DEFENSE 104
Conclusion Joint Beamforming (Codebook selection) and
Power Control in multi-user Wireless Ad-hoc Networks.
Cooperative Power Control : Choose Tx beamform randomly and keep the current one with high probability if it gives lower total network power.
Non-Cooperative Power Control : For non-cooperative selection, convergence of transmit beamformer is problematic!
Learning : Regret-matching based learning is used to for probabilistic convergence.
The selection is performed based on ‘’regret’’.
DEFENSE 105
FUTURE WORK Better Codebook: Codebook adaptation using
evolutionary learning (Genetic Algorithms, etc.) Modifying regret-matching for convergence to
NE or Pareto Optimal point for regret matching techniques
Apply same learning strategies to other joint transmission adaptation parameters (joint CDMA waverform design and power adaptation, channel selection and continous beamforming update, rate and modulation adaptations, etc. )
DEFENSE 106
PUBLICATIONS (*)
(*) http://www.ece.stevens-tech.edu/~ezeydan/
•E. Zeydan D. Kivanc and U. Tureli, “Iterative Beamforming and Power Control for MIMO Ad Hoc Networks” in Proc of IEEE GLOBECOM'10, Miami, FL, Dec. 2010.
•E. Zeydan D. Kivanc, C. Comaniciu and U. Tureli, “Bottleneck Throughput Maximization for Correlated Data Routing: A Game Theoretic Approach” in Proc. of CISS'10, Princeton, NJ, March 2010.
•E. Zeydan D. Kivanc and C. Comaniciu, “Efficient Routing for Correlated Data in Wireless Sensor Networks” in Proc of IEEE MILCOM’’08, San Diego, CA, Nov. 2008.
•E. Zeydan, D. Kivanc and U. Tureli, “Cross Layer Interference Mitigation using a convergent Two-Stage Game for Ad Hoc Networks” in Proc. of CISS'08, Princeton, NJ, March 2008.
•E. Zeydan D. Kivanc and U. Tureli, “Unitary and Nonunitary Differential Space Frequency coded OFDM” in Proc of IEEE Wireless Communications and Networking Conference (WCNC'08), Las Vegas, NV, April, 2008.
•E. Zeydan, D. Kivanc and U. Tureli, “Joint Iterative Channel Allocation and Beamforming Algorithm for Interference Avoidance in Multiple-Antenna Ad Hoc Networks” in Proc. of 2007 IEEE MILCOM, Orlando, FL, Oct. 2007.
•E. Zeydan, U. Tureli "Differential Space-Frequency Group Codes for MIMO-OFDM " in Proc. of 41st Annual Conference on Information Sciences and Systems (CISS'07), Baltimore, MD, March 2007.
DEFENSE 107
Thank You !
Questions, Comments !
DEFENSE 108
DEFENSE 109
PART I:
Energy Efficient Routing for Correlated Data in Wireless Sensor
Networks
DEFENSE 110
Minimize
subject to *,k lSIR , kP C
An NP complete optimization problem!!! A game theoretic formulation Convergence to a local optimal solution with relatively low complexity and in a distributed fashion.
Efficient Routing for Energy Minimization
1
( )N
kb k k
k
E Y
i iS X
where iX other possible routes for node Yi
DEFENSE 111
In setting up the costs for facilities, we can consider the following parameters:
Energy spent for relaying bits on outgoing links from the facility,
Opportunity for aggregation by exploiting the data correlation
a) Facility Cost Selection for the Congestion Game
DEFENSE 112
Maximum Utility
MER
U1U2
U3
U4
U5
B. Correlation Aware Routing (CAR)
Sink Node
Sensor Node
a) Facility Cost Selection for the Congestion Game
DEFENSE 113
CAR
Sink Node
Sensor Node
a) Facility Cost Selection for the Congestion Game
DEFENSE 114
Utility Function of source Yi for CAR:
Sink Node
Sensor Node
Y1
Yi
YN
Prune sub-tree
of Yi ( , ) - ( ) ( )i
f ii i i b f f f f
f S
u S S E Y Y
1 - ( )i
fb f f
f S
u E Y
a) Facility Cost Selection for the Congestion Game
1 2( , )i i iu S S u u
2 - ( )i
f ib f f
f S
u E Y
Si
( )NY
YN-1
1( )NY
Y2
2( )Y
( )iY
1( )Y 1( )i Y
DEFENSE 115
A Potential Game Formulation for CAR
An exact potential function P(.) is defined as: , , 'i iP S R i N S S S
( , ) ( ' , ) ( , ) ( ' , )i i i i i i i i i iu S S u S S P S S P S S
A best response strategy in a potential game converges to a Nash equilibrium.
a) Facility Cost Selection for the Congestion Game
It can be shown that CAR is an exact potential game with the potential function,
1
( , )N
ii i s
i
P S S E
Therefore, CAR converges!
( , ) - ( ) ( )i
f ii i i b f f f f
f S
u S S E Y Y
DEFENSE 116
Minimum Energy Data Gathering Algorithm (MEGA) [Rickenbach et. al .DIALM-POMC’04]
a) Facility Cost Selection for the Congestion Game
Yj
Yi
YN ,,( , ) - ( ) (1 ) ( )
j
i j fi i i b i i i j i i b
f M
u S S E Y Y E
MiMinimum Energy
route
Ci
Coding route
( )i iY
,(1 ) ( )i m m mY
Ym
Coding routeCi
MER route of Yj
Sink Node
Coding routeMER route
DEFENSE 117
The number of nodes in the network is selected to be N = 10 to 40, uniformly distributed over a square area of dimension 40m X 40m.
The target SINR is selected to be γ*= 5 (7 dB)
Constant transmit Power Pi =10-2 Watts (10 dBm), σ2= 10-13 Watts, W=1 Mhz, forgetting factor=0.8.
Each symbol is represented with 1 bits, i.e. Ψ(Yi)=1bits/symbol for all Yi.
MER constructs route and aggregates data opportunistically.
b) Simulation Results
DEFENSE 118
The spatial correlation of data is chosen to be c=0 (no aggregation), c = 100 (low correlated) and c=1000 (highly correlated). ρij=exp(-d2
ij/c). To compare CAR, MER and MEGA fairly, total
energy required for a total symbol throughput of 100 kbps is compared. (“effective energy”)
φ=1 (all sources, no relays!) Average over 100 different topologies.
b) Simulation Results
DEFENSE 119
b) Simulation Results
(a) MER (b) CAR
N=30, c=1000
0 5 10 15 20 25 30 350
5
10
15
20
25
30
35
40
Source
Sink
(c) MEGA
0 5 10 15 20 25 30 350
5
10
15
20
25
30
35
40
Source
Sink
0 5 10 15 20 25 30 350
5
10
15
20
25
30
35
40
Source
Sink
DEFENSE 120
c=100 improvement of CAR compared to MER MEGA
N=10 (4.31 %) (14.30 %)
N=40 (25.35 %) (52.18 %)
b) Simulation Results
10 15 20 25 30 35 4010
-3
10-2
10-1
100
Number of Nodes (N)
Tot
al E
ffec
tive
Ene
rgy
(Jou
le/s
ymbo
l)
MER, CAR, MEGA (c=0)MEGA (c=100)
MER (c=100)
CAR (c=100)
MEGA (c=1000)
MER (c=1000)CAR (c=1000)
52 %25 %
c=100
c=1000
c=0
DEFENSE 121
c=0 (no aggregation) improvements of CAR
@ c=100 (93.15 %)
@ c=1000 (96.29 %)
b) Simulation Results
10 15 20 25 30 35 4010
-3
10-2
10-1
100
Number of Nodes (N)
Tot
al E
ffec
tive
Ene
rgy
(Jou
le/s
ymbo
l)
MER, CAR, MEGA (c=0)MEGA (c=100)
MER (c=100)
CAR (c=100)
MEGA (c=1000)
MER (c=1000)CAR (c=1000)
96 %
c=0 (no aggregation)
DEFENSE 122
CAR improvements (N=20) compared to MER MEGA • c=200 (16.73 %)
(31.46%)• c=800 (16.36 %) (16.61
%)
b) Simulation Results
0 100 200 300 400 500 600 700 800 900 1000
10-3
10-2
Correlation Constant (c)
Tota
l E
ffective E
nerg
y (
Joule
/sym
bol)
MEGA (N=20)
MER (N=20)
CAR (N=20)MEGA (N=10)
MER (N=10)
CAR (N=10)
16 %
31 %
DEFENSE 123
b) Simulation Results
1 2 3 4 5 60.75
0.8
0.85
0.9
0.95
1
Number of Iterations
Nor
mal
ized
Eff
ectiv
e E
nerg
y
N=10
N=20
N=30
N=40
CAR improvements over MER (CAR/MER)