OPTIMAL ENERGY ALLOCATION FOR DETECTION IN WIRLESS

SENSOR NETWORKS

Krishna Gunturu and Mort Naraghi-Pour

Dept. of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA 70803, USA

Keywords:

Hypothesis Testing, Distance measure, Lagrange multiplier.

Abstract:

The problem of binary hypothesis testing in a wireless sensor network is studies in the presence of noisy

channels and for non-identical sensors. We have devised an energy allocation scheme for individual sensors

in order to optimize a cost function with a constraint on the total network energy. Two cost functions were

considered; the probability of error and the J-divergence distance measure. We have also designed a mathe-

matically tractable fusion rule for which optimal energy allocation can be achieved. Results of optimal energy

allocation and the resulting probability of error are presented for different sensor network conﬁgurations.

1 INTRODUCTION

In recent years wireless sensor networks (WSN) have

attracted a great deal of attention from the research

community. Typical applications of WSNs include

environmental monitoring, surveillance, intruder de-

tection and denial of access, target tracking, and struc-

ture monitoring, among others. Wireless sensor net-

works can also serve as the ﬁrst line of detection for

various types of hazards, such as toxic gas or radia-

tion.

The nodes in wireless sensor networks are pow-

ered by batteries for which replacement, if at all pos-

sible, is very difﬁcult and expensive. Thus in many

scenarios, wireless sensor nodes are expected to oper-

ate without battery replacement for many years. Con-

sequently , constraining the energy consumption in

the nodes is a very important design consideration.

In (Luo and Giannakis, 2004), the authors consider

quantization of sensor data and energy allocation for

the purpose of estimation under a total energy con-

straint. Optimal modulation with minimum energy

requirements to transmit a given number of bits with

a prescribed bit error rate (BER) is considered in (Cui

et al., 2005).

In this paper we consider the problem of binary

hypothesis testing using wireless sensor networks un-

der energy constraint. Traditionally, the decentral-

ized detection problem has been investigated assum-

ing identical sensor nodes. For example the work re-

ported in (Zhang et al., 2002; Tsitsiklis, 1988; Varsh-

ney, 1997), considers identically distributed observa-

tions for all the sensor nodes and error-free transmis-

sions from the nodes to the fusion center. In this

paper we do not assume identically distributed ob-

servations. In particular the observation noise expe-

rienced by each sensor may be different. Further-

more, the wireless channels between the sensor nodes

and the fusion center is assumed to be a noisy chan-

nel. Speciﬁcally, it is assumed that the nodes’ deci-

sion is transmitted using a modulation scheme over

an AWGN channel. Our goal is to design a fusion

rule and an energy allocation for the nodes so as to

minimize a cost function subject to a limit on the total

energy of all the nodes. We consider two types of cost

functions. The probability of error at the fusion center

as well as the divergence distance measure.

The remainder of this paper is organized as fol-

lows. In Section 2 we present the system model. The

problem of energy allocation for the probability of er-

ror and the J-divergence cost functions is studied in

Sections 3 and 4, respectively. The results are pre-

sented in Section 5 and the conclusions are drawn in

Section 6.

17

Gunturu K. and Naraghi-Pour M. (2007).

OPTIMAL ENERGY ALLOCATION FOR DETECTION IN WIRLESS SENSOR NETWORKS.

In Proceedings of the Second International Conference on Wireless Information Networks and Systems, pages 17-21

DOI: 10.5220/0002150200170021

Copyright

c

SciTePress

2 SYSTEM MODEL

Let H be a binary random variable with prior prob-

ability distribution given by P(H = H

0

) = q

0

and

P(H = H

1

) = q

1

. We consider a network of n wire-

less sensors with sensor k acquiring a measurement

X

k

about the state of H. Gaussian observations are as-

sumed although the results can be extended to other

cases. With this assumption we have

p

X

k

(x|H

0

) ∼

N (0,σ

2

k

)

p

X

k

(x|H

1

) ∼

N (d,σ

2

k

) (1)

Sensor k computes a local binary decision u

k

ac-

cording to

u

k

=

1, if ln(

p

X

k

(x|H

1

)

p

X

k

(x|H

0

)

) ≥ λ

k

0, if ln(

p

X

k

(x|H

1

)

p

X

k

(x|H

0

)

) < λ

k

For the given distribution in (1) the optimal value

of λ

k

is given by

λ

k

=

d

2

+ σ

k

2

(q

0

−q

1

)

2d

(2)

The channel between sensor k and the fusion cen-

ter is modeled by a binary symmetric channel with

cross over probability ε

k

. The value of u

k

is trans-

mitted to the fusion center over this channel and z

k

denotes the received bit. Let E

T

denote the total en-

ergy available to all the sensors. The fraction of en-

ergy allocated to sensor k is given by x

k

E

T

where

0 ≤x

k

≤ 1 and

∑

n

i=1

x

i

= 1. For the sake of concrete-

ness we assume that the sensors use a BPSK modu-

lation scheme. The value of ε

k

is then given by ε

k

=

Q(

p

2E

T

x

k

/N

0

) where Q(x) =

1

√

2π

∞

x

e

−x

2

/2

dx.

3 ERROR PROBABILITY

CRITERION

The fusion center receives the sequence z =

(z

1

,z

2

,··· , z

n

) and must decide on the state of H.

Finding the optimum decision rule for H based on z is

mathematically intractable. We therefore choose the

following fusion rule.

ψ(z) =

H

1

,

∑

n

i=1

α

i

z

i

≥ τ

H

0

,

∑

n

i=1

α

i

z

i

< τ

(3)

Our motivation here is that for α

i

=

1

σ

2

i

, i = 1, 2,··· ,n,

this is the optimal rule if the fusion center had ac-

cess to the observations {X

k

}. Our goal is to choose

the values of α

i

,x

i

,i = 1,2,··· ,n, and τ such that the

probability or error is minimized subject to the con-

straint that the total energy of the sensor network for

a single measurement and transmission does not ex-

ceed E

T

.

Evaluation of the performance of this rule requires

the distribution of Z =

∑

α

i

z

i

. We invoke the central

limit theorem and assume that, given either hypothe-

sis, Z is a Gaussian random variable (Eremin, 1999).

The conditional moments of Z are then evaluated as

follows.

E(Z|H

ℓ

) =

∑

α

i

E(z

i

|H

ℓ

), ℓ = 0,1 (4)

and

var(Z|H

ℓ

) =

∑

α

i

var(z

i

|H

ℓ

), ℓ = 0,1 (5)

Now

E(z

i

|H

0

) = P(z

i

= 1|H

0

)

= P(z

i

= 1|u

i

= 0, H

0

)P(u

i

= 0|H

0

)

+P(z

i

= 1|u

i

= 1, H

0

)P(u

i

= 1|H

0

)

= ε

i

1−Q

λ

i

σ

i

d

+

d

2σ

i

+(1−ε

i

)Q

λ

i

σ

i

d

+

d

2σ

i

(6)

where ε

i

= Q(

p

2E

T

x

i

/No) is the crossover probabil-

ity for the ith channel. Let E(z

i

|H

0

) = ω

i0

. Then

var(z

i

|H

0

) = ω

i0

(1−ω

i0

) (7)

Similarly for hypothesis H

1

, we have

E(z

i

|H

1

) = P(z

i

= 1|H

1

)

= ε

i

1−Q

λ

i

σ

i

d

−

d

2σ

i

+(1−ε

i

)Q

λ

i

σ

i

d

−

d

2σ

i

(8)

Let E(z

i

|H

1

) = ω

i1

. Then

var(z

i

|H

1

) = ω

i1

(1−ω

i1

) (9)

Let var(z

i

|H

ℓ

) = γ

iℓ

2

. The probability of false

alarm is now given by

P

f

= P(Z ≥ τ|H

0

) = Q

τ−

∑

n

i=1

α

i

ω

i0

p

∑

n

i=1

α

i

2

γ

i0

2

!

(10)

and the probability of detection is given by

P

d

= P(Z ≥ τ|H

1

) = Q

τ−

∑

n

i=1

α

i

ω

i1

p

∑

n

i=1

α

i

2

γ

i1

2

!

(11)

Finally the probability of error is given by

P

e

= q

0

P

f

+ q

1

(1−P

d

) (12)

WINSYS 2007 - International Conference on Wireless Information Networks and Systems

18

Let x = (x

1

,x

2

,··· , x

n

) and y = (α

1

,α

2

,··· , α

n

).

Under the probability of error criteria, we can formu-

late the following optimization problem.

Minimize P

e

(τ,x, y)

Subject to

∑

x

i

= 1

x

i

≥ 0

(13)

Now the Lagrangian is given by

L(τ,x, y,{κ

i

},µ) =

P

e

(τ,x, y) +

n

∑

i=1

κ

i

x

i

+ µ(

n

∑

i=1

x

i

−1) (14)

The Karush-kuhn-Tucker(KKT) (Boyd and Van-

denberghe, 2004) conditions dictate that there must

exist {κ

i

}

n

i=1

and µ such that

x

i

≥ 0, γ

i

≥ 0, κ

i

x

i

= 0, i = 1,2, ..,n. (15)

n

∑

i=1

x

i

= 1 (16)

∇P

e

(τ,x, y) + ∇

n

∑

i=1

κ

i

x

i

+ ∇µ(

n

∑

i=1

x

i

−1) = 0 (17)

where ∇ denotes gradient. By solving this problem

we can obtain the optimal energy allocation x, and

the decision rule (τ,y).

4 DISTANCE MEASURE

CRITERION

In general we would like to perform the energy alloca-

tion using the probability of error as the cost function.

However, as noted in the previous section obtaining

the optimal detection rule may be intractable. In ad-

dition, while currently we are only considering a one

bit quantization of the sensor observations, we would

like to extend our results allowing the sensors to use

more generalized quantizers. In this case obtaining an

expression for the error probability that is suitable for

energy allocation is difﬁcult. Therefore, in this sec-

tion we opt for an alternative cost function, namely

the J-divergence distance measure which belongs to

the class of Ali-Silvey distance measures between

probability measures. Theorems relating the maxi-

mum distance to the minimum probability of error

justify the application of distance measures in our set-

ting (Poor and Thomas, 1977). A lower bound for the

error probability in terms of the J-divergence distance

measure is given in (Kailath, 1967). For more discus-

sion on the Ali-Silvey class of distance measures and

their application for the design of generalized quan-

tizer we refer the reader to (Poor and Thomas, 1977).

The J-divergence distance measure is given by

J(x) = E

H

1

[T(z)] −E

H

0

[T(z)] (18)

where T(z) is the log-likelihood ratio function given

by ln

p(z|H

1

)

p(z|H

0

)

and E

H

ℓ

is expectation operation under

the hypothesis H

ℓ

. We can write

T(z) = ln

p(z|H

1

)

p(z|H

0

)

=

n

∑

i=1

ln

p(z

i

|H

1

)

p(z

i

|H

0

)

(19)

Thus J(x) =

∑

n

i=1

j(x

i

), where

j(x

i

) = (ω

i1

−ω

i0

)ln

ω

i1

ω

i0

+ (ν

i1

−ν

i0

)ln

ν

i1

ν

i0

and where ν

iℓ

= P(z

i

= 0|H

ℓ

) = 1−ω

iℓ

, for ℓ = 0,1.

The optimization problem is now formulated as

follows.

Maximize

n

∑

i=1

j(x

i

) (20)

Subject to

n

∑

i=1

x

i

= 1 (21)

x

i

≥ 0 (22)

The Lagrangian is given by

L(x,{κ

i

},µ) = −

n

∑

i=1

J(x

i

) +

n

∑

i=1

κ

i

x

i

+ µ(

∑

x

i

−1)

(23)

The KKT conditions dictate that there must exist

{κ}

n

i=1

and µ such that:

x

1

≥ 0, κ

i

≥ 0, κx

i

= 0, i = 1,2, ....,n

n

∑

i=1

x

i

= 0

−∇(

n

∑

i=1

J(x

i

)) + ∇(

n

∑

i=1

κ

i

x

i

) + ∇(µ(

n

∑

i=1

x

i

−1)) = 0

(24)

By solving this problem we can obtain the opti-

mal energy allocation x. In this case the fusion rule is

given by

ψ(z) =

H

1

, T(z) > τ

P(H

1

) = a, T(z) = τ

H

0

, T(z) < τ

(25)

where H

1

is chosen with probability a when T(z) = τ.

5 NUMERICAL RESULTS

5.1 Error-Free Channels

To show the efﬁcacy of the prediction rule in (3), we

consider the case of error free channels. The WSN is

OPTIMAL ENERGY ALLOCATION FOR DETECTION IN WIRELESS SENSOR NETWORKS

19

assumed to have ﬁve sensors with the noise variances

given in Table 1. We plot the error probability for the

optimal values of {α

i

}(given in Table 2) as a function

of τ in Figure 1. For comparison we have also plotted

the error probability for α

i

= 1/σ

2

i

. It can be seen that

both cases result in similarly small error probabilities

albeit for different values of τ. This indicates that if

optimization over τ is performed then α

i

= 1/σ

2

i

re-

sults in good performance.

Table 1: σ values for different sensor index.

Sensor node index(i) 1 2 3 4 5

σ 1 2 3 4 5

Table 2: Optimumα values for different sensor index.

Sensor 1 2 3 4 5

α 2.023 0.52 0.33 0.26 0.23

−4 −3 −2 −1 0 1 2 3 4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

global threshhold(tau)

probability of error

optimized alpha values

alpha=1/sigma

2

Figure 1: Probability of error vs global threshold.

5.2 Energy Allocation for Noisy

Channels

In this case ,we are interested in optimal energy al-

location for non-identical sensors. For sensor i the

channel is a binary symmetric channel with crossover

probability ε

i

= Q(

p

2E

T

x

i

/N

0

). The WSN conﬁgu-

rations are given in Tables 1 and 3. The optimal value

of energy fractions obtained through analytical for-

mulation are depicted in Figures 2 and 3. In these

ﬁgures we also show the resulting error probabilities.

As expected the sensors with smaller noise variance

are allocated a higher fraction of the energy.

We obtained similar results for the case N=8, with

following σ

i

values.

Fig.3 shows the results for the case N=8.

1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

energy fraction allocated

sensor index

τ=0.4327,Probability of error=0.0099

Figure 2: Optimal energy fractions allocated for N=5.

Table 3: σ values for different sensor index.

Sensor 1 2 3 4 5 6 7 8

σ 1 2 3 4 5 4 3 2

5.3 Energy Allocation Using the

Distance Measure

For the WSN conﬁgurations in Tables 1 and 3 we have

obtained the optimal energy allocation using the J-

divergence distance measure. The energy allocations

are shown in Figures 4 and 5, respectively. It is in-

teresting to note that in these cases the nodes with a

large noise variance are not allocated any energy and

thus are prevented from transmitting their decision to

the fusion center. These nodes are censored. In these

ﬁgures we also show a lower bound on the error prob-

ability obtained from the J-divergence distance mea-

sure.

6 CONCLUSION

We have studied the problem of binary hypothesis

testing in a wireless sensor network in the presence

of noisy channels and for non-identical sensors. We

have designed a mathematically tractable fusion rule

for which optimal energy allocation for individual

sensors can be achieved. The objective is to opti-

mize a cost function with a constraint on the total

network energy. Two cost functions were considered;

the probability of error and the J-divergence distance

measure. Results of optimal energy allocation and the

resulting probability of error are presented for differ-

ent sensor network conﬁgurations.

WINSYS 2007 - International Conference on Wireless Information Networks and Systems

20

1 2 3 4 5 6 7 8

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

energy fraction allocated

sensor index

τ=0.5256,Probability of error=0.006

Figure 3: Optimal energy fractions allocated for N=8.

1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

probability of error=0.008

sensor index

energy fraction allocated

Figure 4: Optimal energy fractions allocated for N=5 using

Distance Measure.

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OPTIMAL ENERGY ALLOCATION FOR DETECTION IN WIRELESS SENSOR NETWORKS

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