channel aware sensor selection in distributed detection

8/12/2019 Channel Aware Sensor Selection in Distributed Detection

1/5

CHANNEL AWARE SENSOR SELECTION IN DISTRIBUTED DETECTION SYSTEMS

Hamid R. Ahmadi and Azadeh Vosoughi

ECE Department, University of Rochester, Rochester, NY 14627

ABSTRACT

We propose a novel censoring scheme for the distributed de-

tection problem in a wireless sensor network (WSNs) withNsensors, where the channels between the sensors and the fu-

sion center (FC) is subject to fading and noise. To achieve the

best tradeoff between energy efficiency and detection reliabil-

ity, the FC forms the maximum ratio combing (MRC) fusion

rule by integrating the partial knowledge of fading channel

state information (CSI) and the local sensor performance in-

dices, finds the best set ofK (K < N) sensors that maxi-mizes the total detection probability in the Neyman-Pearson

(NP) sense, and informs the selected sensors via one bit feed-

back. The FC learns the Rayleigh flat fading channels, utiliz-

ing training symbols sent by the sensors, via applying mini-

mum mean square error (MMSE) channel estimator. Assum-

ing the sensors employ BPSK to modulate their binary local

decisions, we derive the MRC fusion rule that depends on the

channel estimates and the sensors performance indices, and

incorporates the effect of channel estimation error. Simula-

tion results are provided to support the analytical derivations.

Index Terms Censoring, channel estimation error, dis-

tributed detection, partial channel state information.

1. INTRODUCTION

In deploying large-scale sensor networks, consisting of typ-

ically low-cost battery-powered devices, for the purposes of

monitoring or detection, we often need to address a funda-

mental tradeoff existing between energy efficiency and de-

tection reliability. Censoring networks have been introduced

in [1][2][3], as an alternative formulation of the original dis-

tributed detection (DD) problem proposed in [4]. In [1][2][3]

the authors raise the interesting question of deciding which

sensors should transmit their observations to the fusion center

(FC), assuming that the average number of transmitting sen-

sors is constrained. The sensors make local censoring deci-

sion and the uncensored sensors transmit a real valued func-

tion of their measurements (i.e., the likelihood ratio value),

with infinite precision, to the FC [1][2][3]. The references

[5][6][7] consider the asymptotic performance of these cen-

soring networks. Under Neyman-Pearson (NP) framework,

[8][9] propose an extreme censoring scheme with an on/off

local sensor signaling and investigate the optimal binary local

sensor quantizer designs (i.e., the optimal threshold for the

local LRT). Fusion of local decisions received from uncen-

sored sensors transmitted over wireless fading channels are

considered in [10], where the FC has noncoherent reception

and the uncensored sensors send one information bit, indi-

cating that their local LRT exceeds a threshold. There is a

similarity between the censoring sensors problems consid-

ered in the literature: the censoring decisions are made locally

at the sensors, based on local sensor performance indices anddoes not consider the quality of the wireless links between the

sensors and the FC. When censoring is implemented locally

at the sensors, any channel aware censoring requires feeding

back the CSI (available side information at the sensors [7])

from the FC to the local sensors with high precision.

In this paper we consider the problem of censoring sen-

sors from a different perspective. Consider a network ofNsensors where each sensor has a different performance index

(i.e., the kth sensor has detection probability Pdk and falsealarm probabilityPfk ). Furthermore, the fading channel co-efficient between the kth sensor and the FC (denoted as hk) is

different across sensors. The FC employs coherent receptionto fuse the binary local decisions received from the uncen-

sored sensors. We raise the question: given we wish to select

onlyKout of theNsensors, what is the best set ofK sen-sors, in terms of maximizing the detection probability? The

question does not have a trivial answer: a sensor with an infor-

mative observation may experience a poor quality link, while

a sensor with less informative measurement may have a bet-

ter quality link. The difficulty of the sensor selection problem

increases as we relax the perfect channel state information

(CSI) assumption at the FC. Estimation error due to imperfect

CSI affects the effective noise of the DD system, and hence

degrades the system detection performance. To address the

raised question we first derive a maximum ratio combining

(MRC) fusion rule, assuming that the sensors employ BPSK

modulation to send data. The transmission period consists of

two phases: in phase I all N sensors send a training symbolto facilitate channel estimation at the FC. Given the minimum

mean square error (MMSE) estimates of the complex fading

channels and the sensors performance indices, the FC selects

the set ofKsensors that would have maximized the detec-tion performance, and informs the selected sensors via one

bit feedback. In phase II the uncensored sensors send their

modulated binary local decisions. Our implicit assumption is

978-1-4244-3696-5/09/$25.00 2009 IEEE 71


2/5

that the fading channel is quasi-static, i.e., the fading coeffi-

cient is constant during the two phases of transmission.

The organization of the paper is as follows: in Section 2

we introduce our system model and the MMSE channel esti-

mation. In Section 3 we derive the MRC fusion rule and the

total detection probability at the FC, under the NP framework.

Section 4 includes the numerical examples for a specific DDsystem and our conclusive remarks.

2. SYSTEM MODEL AND MMSE CHANNEL

ESTIMATION

Suppose there are two hypothesesH1andH0under test. OurDD system consists ofNsensors, where the kth sensor has lo-cal performance indicesPfk (the false alarm probability) andPdk (detection probability). In general the pairs(Pfk , Pdk)need not be identical and are functions of the thresholds and

observation noise at the sensors. We assume that the observa-

tions at the sensors conditioned on each hypothesis are in-dependent. In phase I of transmission all N sensors sendone training symbol to enable the FC estimating all N fad-ing channels coefficients1. The received training symbol at

the FC from the kth(k= 1, 2,...,N)sensor is yk,t = skhk +nk,t. For simplicity of presentation we assume the trainingsymbol issk =

Eb. The fading channel coefficienthk =

kejk is modeled as zero mean complex Gaussian randomvariable with unit variancehk CN(0, 1),k is the channelamplitude with Rayleigh distribution and k is the channelphase with uniform distributionk [, +). We assumethat the FC forms the MMSE estimate ofhk using yk,t as[11]:

hk=E(hk|yk,t) = EbEb+ 2n

yk,t (1)

We model the channel estimation error as the difference be-

tween the actual and the estimate channel hk = hk hk.We assume that the estimation error is a zero mean complex

Gaussian variable and has a variance [11]:

2h

= (Eb2n

+ 1)1 (2)

We note that for the linear Gaussian model we have consid-

ered, the MMSE estimate ofhk givenyk,tis equivalent to theLMMSE estimate ofh

k[11].

In phase II of transmission the K uncensored sensorssend their local binary BPSK modulated decisions over Kchannels. We model each ofN channels as quasi-static flatRayleigh fading where the fading coefficients are fixed dur-

ing the two phases of transmission. The signals at the FC

received from thekth uncensored sensor is:

yk,d = ukhk+ nk,d k {1, 2,...,N} (3)1The accuracy of channel estimation can be enhanced by increasing the

number of training symbols, at the cost of consuming more energy for learn-

ing the channels.

The noisesnk,t, nk,d, and hks are all mutually independentandnk,t, nk,d CN(0, 2n), anduk {

Eb} is the dis-

crete time modulated binary decisions.

3. CHANNEL AWARE SENSOR SELECTION

To extend the lifetime of a DD system, consisting of battery-

powered sensors with limited communication capabilities,

that operates in a time varying wireless medium, one should

design adaptive censoring schemes that select the sensors

with greater impact on the total detection performance to

transmit their data, and turn off the sensors with insignificant

effect. The contribution of each sensor to the total detec-

tion performance depends on the local performance indices

(Pdk , Pfk) (which are related to the observation noise at thesensors), as well as the fading channel between the sensor and

the FC. Given the pairs(Pdk , Pfk)and the channel estimates

hk fork = 1, 2,...,N the FC selects the best Ksensors and

informs the uncensored sensors, via one bit feedback.In this section, we derive the optimal LRT fusion rule

LRTand the suboptimal MRC fusion rule , which is thelow SNR approximation ofLRT. The MRC rule dependson the pairs(Pdk , Pfk)and the channel estimates

hk for k =1, 2,...,N. Next, we find the total probability of detectionPD, by approximating the MRC rule as a Gaussian randomvariable. We derivePD as if allNsensors would have trans-mitted their modulated decisions during phase II. We use the

PD expression as a cost function whose maximization leadsus to the optimal set of sensors. For a given K, we select a set,among all the possible choices( NK ), that provides the largest

PD, under the constraint that the total false alarm probabilityPFat the FC is always smaller than or equal to a predeter-mined value. Only these selectedK sensors will actuallysend their modulated decisions to the FC during phase II.

3.1. LRT and MRC FUSION RULES

To capture the effect of channel estimation error we substitute

hk = hk+ hkin (3), wherehkis the MMSE channel estimategiven in (1). We can rewrite yk,das:

yk,d= ukhk+ wk,d k= 1, 2,...,N (4)

where

wk,d = ukhk+ nk,d

is the new noise term that is uncorrelated with the data symbol

uk and combines the effect of channel estimation error andthe AWGN. Recalling that the estimation error hk andnk,dare uncorrelated we find that the noise wk,d is a zero meancomplex Gaussian variable with variance:

2w =Eb2h

+ 2n = Eb

2n

Eb+ 2n+ 2n (5)

Givenhk and uk we find thatyk,d CN(ukhk, 2w).

72


3/5

Hence, we can write the LRT as the following:

LRT =Nk=1

f(yk,d|H1)f(yk,d|H0)

=

N

k=1

Pdkf(yk,d|uk =

Eb) + (1

Pdk)f(yk,d

|uk =

Eb)

Pfkf(yk,d|uk = Eb) + (1 Pfk)f(yk,d|uk = Eb)

=Nk=1

Pdke |yk,dhk

Eb|

2

2w + (1 Pdk)e |yk,d+hk

Eb|

2

2w

Pfke |yk,dhk

Eb|

2

2w + (1 Pfk)e |yk,d+hk

Eb|

2

2w

=Nk=1

Pdk + (1 Pdk)e4Eb

2wRe(yk,dh

k)

Pfk + (1 Pfk)e4Eb

2wRe(yk,dh

k)

(6)

In low SNR regime as 2n , we observe that 2w . By taking a logarithm from the LRT in (6) and using theapproximationse

x

1 x and log(1 + x) x for smallvalues ofx, we can simplify the above rule and reach to:

=Kk=1

PkRe(yk,dhk) (7)

wherePk = Pdk Pfk . Given hk we note that is a lin-ear combination of Gaussian random variables and thus has a

Gaussian distribution.

3.2. TOTAL PROBABILITY DETECTION

For a given threshold value the total probability of falsealarmPF at the FC can be written as:

PF =P r(> |H0) =Q( |H0

|H0)

whereQ(x) =x

12

et2

2 dt. AssumingPF = as the

NP constraint,can be derived in terms of and then detec-tion probabilityPD is written as:

PD =P r(> |H1)

=Q(Q1()|H0 + |H0 |H1

|H1) (8)

whereQ1(.)is the inverse ofQ(.)function. To findPD weneed to calculate |H0 , |H1 , |H0 , |H1 . Let zk =

Re(yk,dhk). Considering (7) and using the fact thatzks are

mutually independent, we find that:

|Hi =Kk=1

PkE(zk|Hi)

|Hi =Kk=1

P2k2zk|Hi i= 0, 1 (9)

Under hypothesisH0 mean ofzk would be equal to:

E(zk|H0) = |hk|2(2Pfk 1)

Eb (10)

To derive its variance under hypothesisH0, we can write:

E(z2k) =Eb|hk|4 + E

(Re(wk,dhk))

2

(11)

AsRe(wk,d

hk)is a zero mean Gaussian random variable with

variance 2w|hk|2

2 , (11) can be rewritten as:

E(z2k) =Eb|hk|4 +2w|hk|2

2

which leads to:

2zk|H0 = 4Eb|hk|4Pfk(1 Pfk) +2w|hk|2

2 (12)

To findE(zk|H1) and 2zk|H1 we should just substitute PfkwithPdk inE(zk|H0)and2zk|H0 . Therefore, thePDexpres-sion in (8) can be evaluated, given (Pdk , Pfk)and the channelestimateshk, as:

PD =QQ1()

Kk=1 P

2k

2zk|H0 2

EbK

k=1 P2k |hk|2K

k=1 P2k

2zk|H1

(13)

4. NUMERICAL RESULTS AND CONCLUSIONS

In this section we consider two algorithms to find the set of

Ksensors that provides the largest detection probability PDbetween all possible sets with

K sensors. There are

(

N

K )sets withKsensors that we can choose fromN sensors. Thefirst algorithm is brute force algorithm (so-called B.F. Alg.)

that examines all these sets to find the set that results in the

Table 1. Local performance indices of sensors and their corresponding channel coefficients. ( 2h= 1)

- S1 S2 S3 S4 S5 S6 S7 S8 S9 S10

Pdk .95 .9 .85 .8 .75 .7 .65 .6 .55 .5

Pfk .05 .05 .05 .05 .05 .05 .05 .05 .05 .05

hk 0.05ej2.95 0.6e

j1.25 0.78ej2.2 0.8ej

2.17 0.96ej2.47 1.1ej

0.99 1.14ej0.7 1.15e

j1.43 1.38ej2.8 1.4e

j1.2

73


4/5

1 2 3 4 5 6 7 8 9 100.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Selected Sensors (K)

ProbabilityofDetection

SNR =1dB, B.F. Alg.

SNR =1dB, So. Alg.

SNR =1dB, B.F. Alg.

SNR =1dB, So. Alg.

SNR =3dB, B.F. Alg.

SNR =3dB, So. Alg.

Fig. 1. Total probability of detection PD versus number of selectedsensors K, using brute force (B.F. Alg.) and sorting (So. Alg.) algo-

rithms. Perfect CSI is assumed at the FC, PF = 0.01and N = 10.

maximumPD. To find the maximumPDwe need to perform( NK )1 comparisons between the detection probabilities.The second algorithm is a sorting one (so-called So. Alg.)

that first evaluates the detection probability PD at the FC foreach individual sensor and finds the set containing the Ksen-sors with highest individualPDvalues. It is easy to show thatthis sorting algorithm needs to perform at most K(N K)comparisons to find the desired set, i.e., sorting method has

less complexity compared with that of brute force one. Sup-

pose we want to chooseK= 10sensors out ofN = 50sen-sors. The first algorithm requires1.03 1010 comparisonswhile the second one requires only400 comparisons. How-ever, the sorting algorithm finds the sensors for which their

individualPD values are good, that do not necessarily resultin the maximum value ofPD when working with other sen-sors. So the brute force algorithm is the optimal one with high

complexity, while sorting algorithm is a suboptimal one with

less complexity.

For the numerical examples we assume that there are N=10sensors with the local performance indices and channel co-efficients given in Table 1(the phases are in radian). We as-

sume that the complex channel has unit variance and SN R=Eb/2n. Also we assume that the total false alarm probabil-ity at the FC is fixed at = 0.01and sensors have the samefalse alarm probabilitiesPfk = 0.05for k = 1,...,N. Sen-sors in the field are indexed such that as the sensors indices

increase, the channel amplitudes|hk| increase while sensordetection probabilitiesPdk decrease, as it is shown in Table1. If the channel amplitudes |hk| and detection probabilitiesPdkwere both increasing the answer to selection problem wastrivial and both algorithms would choose the same set of sen-

sors: the set of sensors with larger |hk| orPdk values wouldhave been chosen. But in scenarios where|hk| is ascending

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Number of Selected Sensors (K)


SNR =1dB, B.F. Alg.

SNR =1dB, So. Alg.

SNR =1dB, B.F. Alg.

SNR =1dB, So. Alg.

SNR =3dB, B.F. Alg.

SNR =3dB, So. Alg.

Fig. 2. Total probability of detection PD versus number of selectedsensors K, using brute force (B.F. Alg.) and sorting (So. Alg.) al-

gorithms. Imperfect CSI is assumed at the FC, PF = 0.01 andN = 10.

andPdk is descending (e.g., Table 1), it is not clear which setwould be selected.

Figures 1 and 2 showPDin (13) versus the number of se-lected sensors Kfor both perfect and imperfect CSI in the FC.The sets of sensors selected by each algorithm are different,

and consequently their detection probabilities PD are differ-ent. As we expected the performance of the selected set by

sorting algorithm is never greater than that of the brute force

algorithm. Sets of chosen sensors forK = 3 and K = 6

atS N R =1dB, when we have perfect CSI at the FC areshown in Table 2. As it is shown different sets are selected

by different algorithms. Figures 3 and 4 illustrate detection

probabilityPD versus SNR forK= 3andK= 6values forboth perfect and imperfect CSI at the FC. We observe that at

low and high SNR values the algorithms are approaching to

each other and their selected sets are expected to be the same.

Table 2. selected sensors at SNR = 1dB.K selected sensors

3 brute force: S6, S5, S3

sorting: S6, S9, S76 brute force: S3, S4, S5, S6, S7, S8

sorting: S9, S10, S5, S6, S7, S8

In summary, we have proposed a novel censoring scheme

for the distributed detection problem in a WSNs with N sen-sors, where the channels between the sensors and the FC is

subject to fading and noise, and the sensors employ BPSK for

data modulation. Under the communication constraint that

only K out of the N sensors can transmit their local deci-sions to the FC, we have addressed the question of selecting

74


5/5

10 5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(dB)


K=3, B.F. Alg.

K=3, So. Alg.

K=6, B.F. Alg.

K=6, So. Alg.

Fig. 3. Total probability of detection PD versus SNR, using bruteforce (B.F. Alg.) and sorting (So. Alg.) algorithms. Perfect CSI is

assumed at the FC, PF = 0.01and N = 10.

which Ksensors should transmit, by integrating the local sen-sors performance indices as well as the MMSE estimates of

the fading channels between the sensors and the FC into the

MRC fusion rule. We have relied on numerical evaluations

to select the set from all ( NK ) possible choices. We haveused brute force and sorting algorithms to find the best set of

Ksensors, where sorting algorithm is a suboptimal methodwith less complexity, compared with brute force method. Fu-

ture work includes investigating heuristics, based on convex

optimization, for approximately solving this sensor selection

problem.

5. REFERENCES

[1] C. Rago, P. Willett, and Y. Bar-Shalom, Censoring sen-

sors: a low-communication-rate scheme for ditributed

detection, IEEE Trans. Aerospace Elec. Syst., vol. 17,

pp. 554567, April 1996.

[2] S. Appadwedula, V. V. Veeravalli, and D. L. Jones, Ro-

bust and locally-optimum decentralized detection with

censoring sensors, inProceedings of the Fifth Interna-

tional Conference on Information Fusion (ICIF), July

2002, vol. 1, pp. 5663.

[3] S. Appadwedula, V. V. Veeravalli, and D. L. Jones,

Energy-efficient detection in sensor networks, IEEE

Journal on Selected Areas in Communications, vol. 23,

no. 4, pp. 693702, April 2005.

[4] R.R. Tenney and N.R. Sandell Jr., Detection with dis-

tributed sensors, IEEE Trans. Aerospace Elec. Syst.,

vol. 17, pp. 501510, July 1981.

10 5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(dB)


K=3, B.F. Alg.

K=3, So. Alg.

K=6, B.F. Alg.

K=6, So. Alg.

Fig. 4. Total probability of detection PD versus SNR, using bruteforce (B.F. Alg.) and sorting (So. Alg.) algorithms. Imperfect CSI

is assumed at the FC, PF = 0.01and N = 10.

[5] J. F. Chamberland and V. V. Veeravalli, Asymptotic

results for decentralized detection in power constrained

wireless sensor networks, IEEE Journal on Selected

Areas in Communications, vol. 22, no. 6, pp. 1007

1015, August 2004.

[6] J. F. Chamberland and V. V. Veeravalli, Decentralized

detection in sensor networks, IEEE Trans. on Signal

Processing, vol. 51, no. 2, pp. 407416, February 2003.

[7] W. P. Tay, J. N. Tsitsiklis, and M. Z. Win, Asymp-totic performance of a censoring sensor network, IEEE

Trans. on Information Theory, vol. 53, no. 11, 2007.

[8] F. Gini, F. Lombardini, and L. Verrazzani, Decen-

tralised detection strategies under communication con-

straints, IEE Proceedings, Radar, Sonar and Naviga-

tion, vol. 145, no. 4, pp. 199 208, August 1998.

[9] Y. Lin, B. Chen, P. Willett, and B. Suter, Distributed

binary quantizers for communication constrained large-

scale sensor networks, inProceedings of the Ninth In-

ternational Conference on Information Fusion (ICIF),

2006.

[10] R. Jiang and B. Chen, Fusion of censored decisions

in wireless sensor networks, IEEE Trans. on Wireless

Communications, vol. 4, no. 6, pp. 2668 2673, Novem-

ber 2005.

[11] S. M. Kay, Fundamentals of Statistical Signal Pro-

cessing I: Estimation Theory, Prentice-Hall, Englewood

Cliffs, NJ, 1993.

75

channel aware sensor selection in distributed detection

Documents