COOPERATIVE COMMUNICATION STRATEGIES FOR SENSOR NETWORKS

Chulhan Lee and Sriram VishwanathDepartment of Electrical and Computer EngineeringThe University of Texas at Austin, Austin, TX 78712

{chlee, sriram}@ece.utexas.edu

ABSTRACTThis paper focuses on a relay-based sensor network

where there is no direct link between the sensing elementsand the destination, with a cluster-head acting as therelaying element. The destination in this scenario may beanother cluster-head or the ultimate receiving-station. Ourgoal is to enable cooperative communication in such ascenario, allowing the nodes in the cluster-head to workcooperatively amongst themselves. We find this enablementproblem to be a minmax optimization problem whosesolution is at the intersection of two capacity regions.

I. INTRODUCTIONThe concept of a sensor network brings to mind a

multitude of low power objects jointly gathering infor-mation from their surrounding environment. Such devicescould have been deployed with varying objectives, somemore stringent than the others. An increasingly importantconfiguration that is seen as highly efficient is one wheresensors are organized into clusters, each cluster having acluster-head as the leader. The communication within acluster must travel through the cluster-head, which then isforwarded to a neighboring cluster-head until it reaches itsultimate intended receiver. This gives rise to the relayingproblem depicted in Figure 1. One or more sensors collectsinformation and forwards it to the cluster-head. The headprocesses this information, forwarding it to the next station.Our goal is to enable cooperation in an information-theoretic sense in such a scenario, thereby enhancing theoverall system throughput.

Enabling cooperation with distributed information hasbeen studied extensively in information theory [1], [6]-[10], [12], [13], [15]. An important result in this settingis that of [5], where it was found that unless the transmitdata is perfect correlated, the gains of cooperation is verylimited. To perfectly correlate data in our setting, one mustadopt a decode-and-forward strategy at the cluster-head.Thus, we utilize a decode-and-forward strategy for ouranalysis in this paper as a pivotal cooperation-enablingtechnique.

Another factor that influences our analysis is the correla-tion between data gathered by the sensing elements, sincethe exact placement of multiple sensors is often difficult,

sensors almost always gather correlated data - i.e. data thatis either exactly the same, or is probabilistically related. Weincorporate this fact in our analysis as explained in the nextsection.The remainder of this paper is organized as follows.

Section II details the system model and formulates ourcapacity problem. Section III delves into the analysis for acase when there are two sensors in the cluster-head. SectionIV handles the multiple sensors in the cluster-head case,and section V concludes the paper.

II. SYSTEM MODEL AND PROBLEMFORMULATION

Sensors in the cluster-head

Figure 1. Sensors in the cluster-head obtain information from sensingsensors and send it to the destination

Sensor 1

Sensor i Destination

2Sensor 2S0

Figure 2. Two sensors in a cluster-head and each sensor has only oneantenna

The system model in Figure 1 details how informationfrom a group of sensors is forwarded to a cluster-head, withthen reaches back to the destination. A decode-and-forwardstrategy is used at each one of the cluster-head nodes. Thisstrategy decouples the entire system into two parts. Thetransmission from the sensors to the cluster-head comes avirtual MIMO broadcast channel (BC), while that from the

cluster-head onwards becomes a multiple access channelwith common information. The virtual MIMO broadcastchannel results from the fact that the sensor nodes shareinformation they are gathering from the correlated field,and can act as a virtual multiple antenna transmitter. Thesevirtual multiple antenna transmitters communicate withthe cluster-head in a manner that ensures that common-information is shared by the nodes of the cluster-head.The cluster-head nodes, in turn, utilize this common-information to cooperatively reach-back to the ultimatereceiver, thus resulting in a multiple-access channel (MAC)with common-information.To simplify matters in our first paper on this topic, we

restrict ourselves to the case where the BC portion of thischannel is a single-antenna system. A power constraintof Po is associated with the sensing-source, and a powerof P with each of the sensors in the cluster-head. Eachreceiver in our model is assumed to be inflicted by additiveGaussian noise which is normalized to be of unit variance.The channels in the BC are depicted using hi for Receiveri and in the MAC using gj for Transmitter j.To gain an intuitive understanding of the dynamics of

such a system, a simple example is highly instructive.Let us begin with a system that has one sensing nodeand a cluster-head with two sensors. This simple modelis illustrated in Figure 2.

sensor 1hl 0 WI,W2:f3Po

0 h2sensor 20 W2:(1-f)PO

Figure 3. From the source to sensors can be seen as a broadcastchannel, h, > h2

sensor 1wI,W20P\

sensor 2 gW2 0

p

sensor 1 91

uP

sensor 1W20 gl

(1-)P 0

sensor 2W2 0

p

Figure 4. Separated multiple access channel: A point-to-point systemfor W1 and a MISO system for W2

that can be achieved using cooperation is identical to thatof determining the optimal resource-allocation policies forthe BC and MAC jointly. We desire to maximize the sumof the rates with which W1 and W2 can be communicatedto the destination. Let the rates of W1 and W2 be R1 andR2, respectively.

Capacity region of BC

1.5

10

0.5-

0.5 1

RI1.5 2

Figure 5. Capacity region of BC is under 45° line and using only thethe better channel gives the maximum sum rate.

Since we are working with additive Gaussian noise, thechannel on the left hand side of the cluster-head in Figure2 becomes a stochastically degraded broadcast channel(BC) [4] as depicted inFigure 3. Assuming, without loss ofgenerality that hl > h2 in Figure 3, it is well known that thecapacity can be achieved by sending the common message

W2 to both sensors in the cluster-head and the additionalmessage W1 to the "better" sensor [2], [4].The channel on the right-hand side of the cluster-head

is an unusual multiple-access channel (MAC). Specifically,it is a MAC with a common message W2 that has to besent cooperatively by both transmitters to the destination.One can understand the MAC with common informationas composed of two channels (as in Figure 4): a MISOsystem for communicating W2 and a point-to-point systemfor W1.The problem of determining the maximum throughput

,From [2], [4], it is well known that the capacity regionboundary of the BC is expressed in terms of the followinginequalities:

R1 < log (I +fPPh)2

R2 < 21log(1 ) h)2 (1)

where /3 parametrizes each point on the boundary (0 < /3 <1). As before, Po is the transmitting power and noise poweris normalized to 1.

It is also known that the sum-rate maximizing point forsuch a BC lies at one extreme point, where the entire rateis allocated to the receiver with the "better" channel [4](See Figure 5). Although this is a maximizing solutionfor the BC alone, it is clearly not always the sum-ratemaximizing solution for the overall system. For the sake ofcompleteness, we present a proof of this statement below.

O _0

Proposition 1: The BC capacity region of Figure 3expressed as (1) is under the 450 line, R1 + R2 =I log (1 + Poh2) . Therefore, the maximum sum rate occurson the point where (R1,R2) (logg(1+Pohl2)0) i.e.,where we turn on only the better channel sensor (/3 1).

Proof: Consider the boundary described by:

1 2RI = - log (1 + PPohl)21 (I- )Ph

J?2 1o ~I'+ )Ph 22 o 1+ PPOh2J

It is enough to show that

dR2_df3

df3

- l R<R <OforO<p < 1.dRJ -,

-Poh2(1+PPoh2)

Pohl21 + PPoh2

Therefore

dR2 dR2 da _ h2(1+PPoh2)dR1 dp dR1 (1 +PPoh2)h12

Because hl > h2 > 0,

<h2(I+PPoh2)<(I +PPoh2)h2

Therefore

-1<,R <0,dR,

for 0 < /3 < 1..

Next, we focus on characterizing the capacity regionof the cooperative MAC (Figure 4). It is easy to showthe optimality of Gaussian inputs in this scenario. It iswell known that successive decoding achieves the capacityregion boundary for the MAC [4]. In addition, we canshow that the sum-capacity of this channel is achievedwhen we concentrate all our power in transmitting W2 alone(Proposition 3). Thus, the entire capacity region for thecooperative MAC can be expressed as [11]

Proposition 2: The input covariance matrix E in (2)which characterizes points on the capacity region boundaryis

miax - [L( P P -

In the mathematical expression,

Emax - argmax log )1 92

argmax ([g1 g21] )

_ (I -a)P"I-aP[(1 caP p1

Proof:

sensor 1:X1W20 91(1-u)P

sensor 2:X2 g 0W2 0

p

Figure 6. MISO channel

The input covariance matrix E is the following equationfrom Figure 6

LEff 2] E[X1X2]J[XXiLE[XIX2] E[X2]Iwhere X1 and X2 are modulated signals from sensor 1 and2, respectively.

E[X12] < (1 c)P, E[X2] < P from the power constraint.By Schwartz inequality [14], we have the following

relation

E[XlX2] < E[X]E[X22] 1 P.

In addition,element of E is

Therefore

Emax

[gl g2] [gl] is maximized when everymaximized.

E maxE[Xi] maxE[XIX2] 1maxE[X1X2] maxE[X22]

[ (1 - a)P 1 aP]L1~aP P

R1 < Ilog(1+ aPg2)2

R2 < max -2l+ogP )' (2)

where E is the input covariance matrix of MISO channeland 0 < a < 1.

Let us now determine covariances E corresponding tothe boundary of the MAC's capacity region.

UNext, we show that the capacity region of the cooperative

MAC has its sum capacity point when only the cooperativesignal is being transmitted.

Proposition 3: The MAC capacity region of (2) is underthe 450 line given by the equation:

R1 +R2 Ilog (1 +P[g g2] [I gl2 \\[11] Lg2J

12.

Capacity region of MAC

1.8

1.6

1.4

1.2

0.8

0.6

0.4

0.2

0 0.5 1R

I

1.5

Figure 7. Capacity region of MAC is under 45° line and sendingcommon message W2 through MISO gives us maximum sum rate.

Therefore, the maximum sum rate happens where(R1,R2) = (0, 10g(1 +P(g1 +g2)2)), i.e., when we useonly MISO channel (a = 0).

Proof:In Figure 7, let us consider the following two points:

III. ANALYSIS OF COMBINED CHANNEL

The problem now reduces to determining a and /3 thatachieve the maximum possible rate for W1 and W2, i.e.,extremize R1 +R2. The solution to this problem lies inthe intersection of the two capacity regions - the BCand the cooperative MAC. Determining the sum capacitymaximizing point of this intersection determines the bestoperating point for the overall system.To further analyze this, we break the system into multi-

ple cases based on channel conditions.

A. Case I: Full Cooperation

Capacity region of BC and MAC

MAC=BC2.5

N 1.5

(R1 ,A, R2,A)

( log (I+ p log [g121]max.. [g))(R1,B,R2,B)

= 2log (I + apgl2),I

log + p(g, + g2)22g 1 2g 11+aPg2

where (R1,A,R2,A), (R1,B,R2,B) are the points on the BCcapacity region boundary and 450 line respectively.To establish our result, it suffices to show that R2,B >

R2,A for 0 < a < 1.

R2,B -R2,AI

1 I( +p(g, +g2)2) 1loI1+= log (--log

1~~~~~~~~~~

= 2log(l + P(g + g2)2)

- 2 log(l +Pgl +2g1g2P +Pga )

[9 21+ a. [g2]9+cg2J

Here, since log is an increasing function, it is enough tocheck

1 +P(gl +g2)2 (1 +pg +2gg21 +Pg2) > 0.

The above inequality is clear since 2glg2P-2glg2P a >O for 0 < a < 1.

0.5

0 _0 0.5 1.5

RI2 2.5

Figure 8. Case I: The MAC capacity region is a subset of the BCcapacity region

For full cooperation, one requires that (a,/3) = (0,0)and hence the entire transmit-resources are devoted to W2.

If hl = h2, the best performance of the BC is obtainedby sending same data to all sensors. In this case, MAChas only MISO channel. Therefore the best power policyis (a,3) = (0,0).

In general, when the cooperative MAC's capacity regionis a subset of the BC's capacity region, their intersection re-sults in the entire cooperative MAC region. Thus, the sum-capacity maximizing solution for the MAC now becomesthe cooperative solution for the overall system. Since thesum capacity of the MAC lies at one extreme where exactlyone message (W2) is sent to the receiver, we have a fullycooperative solution. Note that in this solution, the sensortransmits W2 alone which is decoded by both the receive-sensors in the cluster-head. In other words, receivers 1 and2 in the cluster-head share exactly same information fromthe source. We refer to this case as the full-cooperation.

This case happens when R2-intercept value of BC bound-ary is greater than or equal to R2-intercept value of MACboundary, i.e., log(1 +Poh2) > I10g(1 +P(g1 +g2)2)from (1),(2). If the sensor power is equal to the power of

sensors in the cluster-head, i.e, Po = P, then the conditionfor the full-cooperation is

h2 >_g1 +9g2 (3)B. Case II: No Cooperation

Capacity region of BC and MAC

MAC*BC

2.50.5

0 0.5

111

,

1, t

\ \~~~~~~~1

1.5 2 2.5

N 1.5

Figure 10. Case III: BC capacity region and MAC capacity region hasan intersection point.

0.5

1.5R

I

2.5To find exact intersection point, we need to get a and

/3 to satisfy the following equations from (1) and (2),

Figure 9. Case II: MAC capacity region contains BC capacity region.

If the BC capacity region is a subset of the cooperativeMAC capacity region as in Figure 9, their intersectionand hence the possible set of rates is restricted to the BCcapacity region. Sending only W1 without W2, i.e., (a, 3) =(1, 1) is the best policy for sum rate from Proposition 1.In other words, we turn off sensor 2 in the cluster-head.

This case occurs if RI -intercept value of MAC boundaryis greater than or equal to RI -intercept value of BC bound-ary, i.e., log(1 +Pg2) > 1log(1 +Poh2) from (1),(2).Assuming the source-sensor's power Po is same as P, thenthe condition for non-cooperation is

gi > hi. (4)

C. Case III: Limited Cooperation

The most interesting and probable case is when the BCand cooperative MAC capacity region boundaries intersect.This leads to an intersection of the two regions that is a

strict subset of each region, making the problem of findingthe optimum point relatively non-trivial.

Combining Proposition 1 and 3, it is clear that theintersection point of the BC and MAC capacity boundariesis the sum rate maximization point. This can be verifiedby realizing that the 450 line meets the region at theintersection point of the two boundaries Figure 10.

Similar to the analysis in III-A and III-B, if we assume

Po = P, the condition for existence of the intersection pointis

h2 < gl +g2

h, > gl. (5)

/Pohl

(1 -P)Poh21 + PPoh2

apg2 (6)1

[1 g21 [(1 a)P 1i up] [g9li ucP P L2J

I + apg2

since log is a monotonic function. The solution of the aboveequations can be determined in closed form, although itresults in extremely complicated expressions and for thatreason are not included in this paper.

Here's a simple example that helps us build intuitionabout the solution of (5). Assume g = h2 = gl g2 = h, /2and Po = P. Then (6) and (7) simplify to

1 0s

4

1 + °g2p

1a+ 1)21 + cxPg2

(8)

30 -

25 -

20 -

o

"- 15-.

10 -

5-.

°120.4

g1 -h2-g2 h,12CX

Figure 11. The relation among power, channel gain and a

2.5Capacity region of BC and MAC

MAC=BC

1.5

0 0.5

Figure 11 shows the relation among power, channel gaing and a = 4/3 according to (8).

g=i Sensor

Cluster-head

Figure 14. m sensors in a cluster-head

10 15P P0

20 25 30

Figure 12. The relation between power and a

Q Pipohi O1POh

,,0~ 2PO

Source h Of33PO0p0 \ 1

hm n3m-1PO

n3mPO

Figure 15. Broadcast channel of m sensorsP=5

Wl,W2, ,Wm

se

W2,. *,Wm ( Destination

Wm i,Wm

5 10 15 20 25 30 35

g

Figure 13. The relation between channel gain and a

Figure 12 illustrates that a reduces with increasing P.This means that the optimum solution demands a greaterdegree of cooperation as the power constraints on thesensors increases.

Similarly, Figure 13 implies that a decreases as g

increases, which also indicates an increase in the need forcooperation as the channel gains increase.

IV. MULTIPLE (> 2) SENSORS IN ACLUSTER-HEAD

Figure 14 shows the m sensor model. The system iscomposed of a BC and a MAC.The BC model is illustrated in Figure 15. Assume hl >

h2 > ... > hm without loss of generality. The power alloca-tion policy /3 (/3', . , /3m) satisfies /3i > 0 and 7m I pi = 1.The MAC in Figure 16 is divided into a point-to-point

and MISO channels as Figure 17.

Figure 16. Multiple access channel of m sensors

WmUp0

0

0

sensor 1 Destination

Figure 17. Separated multiple access channel according to the sharedinformation

sedsor 1

0.95

0.9

0.85

0.8

t 0.75

0.7

0.65

0.6

0.55

0.50

Destination

0.9

0.8 -

0.7

0.6

t 0.5

0.4

0.3

0.2

0.1

0 _0

The power allocation policy iik= (ak,1, ***k,m+ -k) ofsensor k in Figure 17 satisfies aXk,i > 0 and Em l-k aCk,i 1.

A. Case I: Full-Cooperation

As in the two-receiver case, this happens if the cooper-ative MAC capacity region is a subset of the BC capacityregion. Here, the cooperative MAC acts as a bottlneck,and so its sum-capacity achieving point is the maximumthroughput of the overall system. Thus, from Proposition 3,the entire system has only a single message(Wm) which isdecoded by the entire cluster-head.

B. Case II: No-Cooperation

Again, if the BC capacity region is a subset of theMAC capacity region, the BC capacity region is the criticalcapacity region. From Proposition 1, we again have asingle-message policy, i.e. sending only W1 to Receive-sensor 1 is the policy that maximizes the sum rate. Allsensors other than Sensor 1 can be turned off in the cluster-head.

C. Case III: Limited Cooperation

In this case, multiple messages must be directed from thetransmitting sensor to the cluster-head, which in turn mustbe cooperative relayed by the cluster-head to the receiver.

This case is even more computationally complex thanthe two-sensor case analyzed before. This is because theintersection of the two capacity regions (cooperative MACand BC) is no longer a single point, but is in fact itselfa region. however, not every point in this intersection isa valid solution to our problem, and further steps must betaken to identify the appropriate set of points that maximizethe throughput of the overall system. In other words, thenumber of variables that descrobe the boundaries of the twocapacity regions (Ck,i and f3i) is greater than the numberof equations available by equating the rates. Figure 16illustrates this fact. The total number of akUi and f3i is[(-1)+(m -2)+...+1]+(m -1)= (m-l)(m+2)Othother hand, from equating the rates Ri,MAC = Ri,BC fori1=l...,m, we only have m equations. Therefore, wemust resort to convex optimization techniques to determinethe solution. This optimization problem is of the form[3]: max 1mIRi, subject to the constraint of the formR1,R2,...,Rm) E Intersection region between cooperativeMAC and BC. We do not delve into the solution of thisoptimization problem in this paper due to space limitations.Algorithmic solutions to this problem as well as to thegeneral virtual-MIMO transmission problem can be found

V. CONCLUSION

Under the cluster-based communication techniques forsensor networks, cooperation within the cluster at the phys-ical layer can provide large MIMO-like gains in throughputfor the overall system. The gains from cooperation lieat the intersection of the capacity regions of the BCand cooperative MAC that form the components of thisnetwork. The region defined by the intersection of theboundaries of the two capacity regions is the region ofinterest, and the points of intersection with the hyperplanewhich has all its 2-dimension projections leading to 450lines are the optimal operating points for our problem.These operating points are explicitly characterized for thecase when there are two sensors in the cluster-head, andimplicitly characterized for the general m-sensor case.

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