OPTIMAL POWER ALLOCATION OVER A FADING MAC WITH VARYINGOBSERVATION SNRS IN RESOURCE CONSTRAINED WIRELESS SENSOR NETWORK

R. Muralishankar, H. N. Shankar, Aniketh Venkat and Manisha Sinha

MIEEE, Department of Electronics and Communications Engineering,CMR Inst. of Technology, Bangalore, INDIA.

MIEEE, Dean Academics and Research, CMR Inst. of Technology, Bangalore, INDIA.Department of Electronics and Communication, PES Inst. of Technology, Bangalore, INDIA.

Department of Computer Science and Engineering, PES Inst. of Technology, Bangalore, INDIA.muralishankar@cmrit.ac.in hnshankar@cmrit.ac.in anikethvenkat@gmail.com manisha.sinha8@gmail.com

ABSTRACT

In a distributed multiple sensor multiple antenna setup, transmis-sion over a fading multiple access channel results in incoherent fu-sion of sensor observation. With an overall sensor power constraint,uniform power allocation (PA) over the sensors is optimal when thechannel between the sensors and the fusion center (FC) is AdditiveWhite Gaussian Noise (AWGN). However, with varying observa-tion signal-to-noise ratios (SNR) at the sensors, uniform PA is notoptimal even with AWGN channel. In this paper, we consider im-perfect communication between the sensors and the FC and corre-lated noise at the sensors. We propose a joint power optimizationscheme that maximizes the probability of detection (PD) at the FC,subject to an overall power constraint. Also, we derive a closedform expression for the optimization problem when the channel isAWGN. We verify the performance of our algorithm by conductingnumerical experiments. We show that the maximum power is allo-cated to a sensor which is operating with the best SNR and channelcondition. We further show that with optimal PA, we never undulydrain our on-board sensor energy and hence extend the battery lifewithout compromising on the PD. Finally, we show that with op-timal PA, the total system power requirement is brought down tonearly 10% of that with uniform PA, even when achieving the samePD.

Keywords: Wireless Sensor Networks, Fading Multiple Ac-cess Channel, Correlated Sensor Noise, Sensor Power Constraint.

1. INTRODUCTION

In distributed detection over a wireless sensor network (WSN), thesensing devices are usually inexpensive and are built using lowpower on-board batteries. Performance and life span of such net-works depend critically on the energy conservation scheme used.Challenges in designing practically implementable sensor networksinvolve the integration of (i) communication and decision fusionfunctions, (ii) constraints on the total power to attain a low prob-ability of intercept, (iii) accommodating imperfect channel modelswith correlated sensor noise, (iv) optimal PA and sensor positioningschemes, and, more recently, (v) multiple antennas at the FC toachieve better detection rates. However, by often relaxing one ormore of the above stated constraints, a reasonably tractable prob-lem formulation is achieved. The immediate benefits of optimallyallocating power to each sensor are prolonged network life and con-current enhanced detection rate. Optimally allocating power over

Sensor1

Sensor2

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+

+

yi

y2

y1

yN

1

2

i

N

1

2

j

L

h(i, j)Sensorj

SensorL

Fig. 1. Schematic of the Setup

random fading channel envelope with power and FA rate constraintsis investigated with iid sensor noise [1], using the system model asillustrated in Fig 1. The noisy observations from the sensors aretransmitted over fading channels to a FC. Neyman-Pearson algo-rithm is adopted to detect the observed event / parameter. Gain interms of power saved at each sensor and increase in detection rateat the FC is demonstrated. Further, the study enables to chooseoptimal system parameters to achieve a certain detection rate fora specified FA rate. If the source noise is non-iid, this may lead towastage of system resources. Our goal here is to develop an optimalPA scheme when the sensor noise is correlated.

Section 2 describes the system model and the power constraint.The detection algorithm and its basic formulation comprise Sec-tion 3. Overall optimal PA scheme is discussed in Section 4. Simu-lation results are presented in Section 5 and concluding remarks inSection 6.

2. SYSTEM DESCRIPTION AND MODEL

Xiao et al. [2] propose an optimal power scheduling strategy for adecentralized estimation problem. They optimize the quantizationlevels and transmission power at each node by jointly imposing con-straints on the channel gains and observation noise levels. The ap-proach is shown to achieve good energy saving over uniform quan-tization strategy at each node. Aravinthan et al. [3] propose an opti-mal PA scheme in a distributed estimation problem. There, the sen-sors with poor SNR do not transmit their observations whereas the

978-1-61284-233-2/11/$26.00 2011 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

1 2 3 4 5 6 7 8 9 100.82

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Total Power (PT)

Prob

ablit

y of

Det

ectio

n (P

D)

Optimal PA vs Uniform PA across PT, H = AWGN, L = 5, N = 6, p1 = 0.6, PFA = 0.01.

PD with Optimal Power AllocationPD with Uniform Power Allocation

Fig. 2. PD vs PT with L = 5, N = 6 and PFA = 0.01 overAWGN channel

rest transmit quantized observations. Vincent Poor et al. [4] achieveoptimal PA for transmission over imperfect multiple-input multiple-output channels with uncorrelated sensor noise. They quantify thetradeoff between the quality of the local sensor decisions and thatof the communication channels between the sensors and the FC.Since the detectors at the sensors and the FC target the same FAprobability that is fixed a priori, the method is not optimal in termsof detector operating points. Besides, the system performance de-pends on the observation SNRs. An approximate J-divergence isused as the performance measure. Krithika et al. [5] go further andpropose elemental J-divergence and elemental L2 distance as per-formance measures; these are simple relative to PA considering J-divergence [4]. However, the noise statistic there is considered tobe independent and the covariance matrix to be diagonal. The chan-nel matrix is assumed to be identity! Simulation results comparevarious distance measures for optimal PA, but do not discuss theadvantage of optimally allocating power in a WSN.

When the sensed data is transmitted in the digital mode, obser-vations are quantized into one or more bits of information and aretransmitted to the FC [6,7]. Since quantization is an integral part ofthe detection process here, it is critical to the detection performance.In the analog transmission mode, an amplify-and-forward strategyis used to relay the sensed observations to the FC [8]. In this paper,we consider transmission of sensor observations in the analog modeto the FC. The central theme of this paper is optimizing the detec-tion performance in a fading environment with varying observationSNRs. We propose a study to develop insights into the design of op-timal sensor gains under constraints on the overall on-board power.

2.1. The Sensor

The sensor network consists of L sensors transmitting over NLchannels to N antennas as in Fig. 1 [9]. The sensors observe {0, }.

=

{H0 : 0 w.p. p0H1 : w.p. p1.

The additive noise, , at the th sensor is correlated; i.e.,

1 2 3 4 5 6 7 8 9 100.93

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1.01

Total Power (PT)

Prob

ablit

y of

Det

ectio

n (P

D)

Optimal PA vs Uniform PA across PT, H = AWGN, L = 5, N = 6, p1 = 0.6, PFA = 0.04.

PD with Optimal Power AllocationPD with Uniform Power Allocation

Fig. 3. PD vs PT with L = 5, N = 6 and PFA = 0.04 overAWGN channel

CN (0, 2). The noisy signal is amplified by the ith sensor by C, thus accounting for phase-shift. The th sensor output is thus

( + ), = 1, 2, , L.We impose a constraint, PT , on the overall sensor power as

PT = E

[L

=1

| ( + n) |2]

=L

=1

2(p12 + 2 ),

where E[] is the expectation operator. Let = {i2}Li=1, the Her-mitian of x be xH , the identity matrix of size M be IM and D(b)be the diagonal matrix with the vector b as its diagonal. Then,

PT = H [p12IL +D()].

2.2. Multiple Access Channels (MAC)Each sensor feeds into antennas through multiple channels. Chan-nels may be flat fading or orthogonal [10]. To the nth antennafrom the th sensor, we consider a fading channel with random gain,h(n, ).

2.3. Antennas and the Fusion Center

There are N antennas at the receiving end. The additive noise,vn CN (0, v2), at the nth antenna is assumed to be iid, n =1, 2, , N . For simplicity, let the antenna noise have unit vari-ance. Signals from the N antennas are fused at the FC. The vectorsignal, y, received at the FC is given by

y = H +HD() + v, (1)where H = {h(n, )} CNL; , CL1; v CN1 is thevector of {vn}Nn=1 and D() CLL. The output, {0, },of the FC is the parameter received. The problem is to detect the

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10.82

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Probablity of False Alarm (PFA)

Prob

ablit

y of

Det

ectio

n (P

D)

ROC for Optimal PA vs Uniform PA , H = AWGN, L = 5, N = 6, p1 = 0.6, PT = 1.

PD with Optimal Power AllocationPD with Uniform Power Allocation

Fig. 4. Receiver Operating Characteristics (ROC) for AWGN chan-nel with L = 5 and N = 6

parameter emitted by the source and to analyze the system perfor-mance.

3. DETECTION ALGORITHM

The detection algorithm adopted in this paper is similar to that in [1].For readability, we highlight the important points here. We employNeyman-Pearson formulation at the FC by freezing the FA proba-bility, PFA, and minimizing the miss detection probability, PD . Atthe FC, y is gaussian distributed under the two hypotheses:

H0 : y CN (0N,R)H1 : y CN (H,R). (2)

Here, 0N is the N 1 zero vector and R is the N N covariancematrix of the received signal given by

R = HD()D()D()HHH + IN. (3)Given y, the FC selects the appropriate hypothesis thus:

yHR1HH1H0

1

22HHHR1H + , (4)

where is the decision threshold. Define errors as follows.Type-I Error: Probability of false-alarm, PFA, is given by

PFA = Q

(

2+

), (5)

where

= HHHR1H and Q(x) = 12

x

et2/2dt.

Type-II Error: Probability of miss detection, PMD , is givenby

PMD = 1 , (6)

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Probablity of False Alarm (PFA)

Prob

ablit

y of

Det

ectio

n (P

D)

ROC Optimal PA vs Uniform PA, H = RCE, L = 5, N = 6, p1 = 0.6, PT = 1.

PD with Optimal Power AllocationPD with Uniform Power Allocation

Fig. 5. Receiver Operating Characteristics (ROC) for RCE withL = 5 and N = 6

where the probability of detection, PD , is given by

PD = = Q(Q1(PFA)

). (7)

Fixing PFA, we may then have

=

Q1(PFA) 2

2.

4. OPTIMAL POWER ALLOCATION ALGORITHM

With fixed observation SNRs and fading channels, the optimal PAscheme of [1] leads to wastage of resources under uniform powerdistribution with non-ideal channels. Moreover, with varying andnon-iid sensor SNRs, the scheme of [1] may not be optimal. Allo-cating power to a sensor based exclusively on a measure of good-ness of the channel ahead is not desirable since if the observationSNR is poor, the resultant noise amplification at the FC contributesto detection error. For mission critical applications where the chan-nel and the observation SNR at each sensor vary dynamically, ajoint optimization scheme is required. Here, we allocate power toonly those sensors that have good observation SNR and are trans-mitting over a good channel, subject to the total power constraint.The refined optimization problem is cast as:

OPT = argmax

[HHHR1H

],

subject to PT =L

=1

[2 (p12 + 2)] = L=1

2,(8)

where =

(p12 + 2 ). Since p1, and are known a priori, can be fixed . Now, with = ,

PT =

L=1

2. (9)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

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Prob

ablit

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ectio

n (P

D)

Optimal PA vs Uniform PA across PT, H = RCE, L = 5, N = 6, p1 = 0.6, PFA = 0.04.

PD with Optimal Power AllocationPD with Uniform Power Allocation

Fig. 6. PD vs PT with L = 5, N = 6 and PFA = 0.04 over RCE

We find , = 1, 2, , L, by sampling a sphere of radius

PTand centered at the origin in an L dimensional complex space, andthence, get = /. We numerically solve by search for OPT .Since varies with , if the observation SNR at any sensor changes,we recompute to get the new OPT . By maximizing the argumentHHHR1H with respect to for a fixed PFA, we minimizePMD , and therefore maximize PD .

4.1. Performance over AWGN channels

With a fixed observation SNR, uniform PA over AWGN channel isoptimal [1]. However, with varying SNR, even though the channelgain is unity, uniform PA is sub-optimal. We derive an analyti-cal expression for the optimization problem with the channel beingAWGN. We then find OPT for the AWGN channel. Further, set-ting the channel gain h(n, ) = 1, n, , we getH = 1NL. Then,

(3) yields R =[

L=1

22]1N + IN. Using the Matrix Inver-

sion Lemma (ShermanMorrisonWoodbury formula), we get

R1 = IN

L

=1

22

1 + NL

=1

22

1N111N . (10)

From (10), for the AWGN case,

=N

1 + NL

=1

22H1LL,

=

N

(

L=1

)2

1 + N

L=1

22. (11)

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Prob

ablit

y of

Det

ectio

n (P

D)

Optimal PA vs Uniform PA across PT, H = RCE, L = 5, N = 6, p1 = 0.6, PFA = 0.1.

PD with Optimal Power AllocationPD with Uniform Power Allocation

Fig. 7. PD vs PT with L = 5, N = 6 and PFA = 0.1 over RCE

Table 1. Optimal Sensor Gains with L = 5 and N = 6 over AWGN

SNR 2 Optimal PT(dB) alphas 1 2 5 7 10

5 0.91 12 0.10 0 0 0 010 0.70 22 0.14 0.31 0.79 1.11 00 1.60 32 0 0 0 0 015 0.63 42 0.63 1.40 3.51 4.92 3.1620 0.61 52 0.65 1.45 3.64 5.10 13.11

Therefore, for the AWGN channel, the optimization problem is

OPT = argmax

N

(

L=1

)2

1 + NL

=1

22

.

Finally, with OPT = | = OPT , the corresponding probabilityof detection becomes

PDAWGN = Q[Q1(PFA)

OPT

]. (12)

5. RESULTS AND DISCUSSIONS

We simulated with 5 sensors, 6 antennas, probability of the nullhypothesis, p0 = 0.4, and the parameter being sensed, = 1.We arbitrarily set the observation SNR to 5dB, 10dB, 0dB, 15dBand 20dB at the 1st, 2nd, 3rd, 4th and 5th sensor respectively. Wesimulated the random channel as in [1]. We searched in the 2L di-mensional real space for subject to (9) and correspondingly found = /. We ran the optimization algorithm to find the opti-mal through brute-force grid search. Using the detection algo-rithm at the FC, we computed the probability of detection for var-ious FA rates and total power. We evaluated the performance with(i) AWGN channel and (ii) Random Channel Envelope (RCE) bycomparing the detection rates obtained using (a) optimal PA and (b)uniform PA over the same channel and same observation SNR. Theresults were averaged over several channel realizations to make thedesign independent of the channel model.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

In the remaining part of this section we discuss the results andsuggest as to how the design parameters can be chosen to achieve adesired performance. From Fig. 2, Fig. 3 and Fig. 4, we see thatuniform PA is non-optimal for an AWGN channel. Table 1 givesthe optimal sensor gains for the AWGN channel; it also shows thatwe allocate zero power to sensors 1 and 3. However, maximumpower is allocated to sensor 5 operating at 20dB SNR. This confirmsthat the optimization algorithm prefers those sensors that have betterobservation SNR to transmit. From Table 1 and Table 2, it can be

verified thatL

=1

2 =L

=1

2 = PT . We distribute the

total power PT amongst the L sensors as {2i }Li=1 subject to (9)and we compute the optimal sensor gains {i}Li=1 as weighted s.Further, with total power PT = 1, 5 and 10, we gain in terms ofthe detection rate using optimal PA over uniform PA in the orderof 15%, 6% and 5% respectively for a FA of 1%. However, with4% FA (Fig. 3) and PT = 1, 5 and 10, we gain in the performancein the order of 5.8%, 1.5% and 1.2% respectively. Therefore, asinferred in [1], the gain with optimal PA over uniform PA reduceswhen total on-board power increases for a fixed FA. The strategiessuggested in [1] for trading off between FA and total power can beused here as well.

From Fig. 4 and Fig. 5, we see that with a FA rate of 1% and10%, the gain using optimal PA over uniform PA is in the order of15% and 2% when the channel is AWGN and 56% and 22% whenthe channel is random. This demonstrates that optimal PA is moreeffective with the fading channel and when the system resourcesare tightly constrained. In Table 2, we present the optimal sensorgains for RCE, showing that the algorithm allocates, on an average,maximum power to sensor 5 (20dB SNR). Moreover, with PT = 2,we see that sensor 4 has been allocated more power than sensor5, because the net effect of the channel and the SNR at sensor 4is better. This justifies the joint optimization scheme by allocatingmore power to a sensor only if it is operating at a good SNR and istransmitting over a good channel. Fig. 6 and Fig. 7 demonstrates theutility of optimal PA for a changing channel and a tightly resourceconstrained system. Finally, from Fig. 2 Fig. 3 and Fig. 6- 7, wesee that to achieve a certain detection rate, with optimal allocation,we require nearly 10% power as required with uniform PA.

6. CONCLUDING REMARKS

On a fading MAC, we proposed and studied an optimal sensor gaindesign, based on a joint optimization over channel and observationSNR at each sensor, subject to a total power constraint. We fixedthe global probability of FA and computed the probability of detec-tion for transmissions by sensors over AWGN and random channelenvelope with non-iid sensor noise. We proposed an optimal PAscheme with a goal of maximizing the detection performance at theFC. We derived an analytical expression for the optimization prob-lem over the AWGN channel, and showed that uniform PA there isnon-optimal. Through simulations, we verified that our algorithmallocates less power to sensors that operate at low SNRs and max-imum power to the sensor operating at the highest SNR. Since theobservation SNR is assumed to vary, we never unduly drain our on-board energy. However, by intelligently allocating power to sensorsbased on the SNR and the channel fading, we extend the battery lifewithout compromising on the detection rate. We discussed the ad-vantage of optimal PA when the system had a total power constraint,FA rate, or both. We saw that as FA increased, the performanceof the system enhanced. However, the advantage with optimal PA

Table 2. Optimal Sensor Gains with L = 5 and N = 6 over RCE

SNR 2 Optimal PT(dB) alphas 1 2 5 7 10

5 0.91 12 0.06 0.12 0.36 0.50 010 0.70 22 0.17 0.63 0.95 1.33 2.380 1.6 32 0.03 0 0 0 015 0.63 42 0.39 1.87 2.11 2.95 2.6320 0.61 52 0.81 1.45 4.37 6.12 10.92

over uniform was significant only when the system was tightly con-strained. We also saw that to achieve a desired detection rate, withoptimal sensor gains, the system power requirement was about 10%of that with uniform PA. Further, this joint optimization algorithmcan be implemented with any channel model and SNR scenario. Fi-nally, findings reported in this paper enable the choice of systemdesign parameters such as total power and FA rate to achieve a de-sired detection rate when the channel and the sensor SNRs vary.

7. REFERENCES

[1] Aniketh Venkat, H. N. Shankar, and Muralishankar R., Opti-mal Sensor Gain Design over Fading MAC using DistributedSensing on a Wireless Sensor Network with False Alarm RateConstraint, Proc. 16th Asia-Pacific Conference on Commu-nications (APCC 2010), pp. 311315, Oct. 31-Nov. 4, 2010.

[2] Jin-Jun Xiao and Shuguang Cui, Joint Estimation in SensorNetworks under Energy Constraints, Proc. IEEE first confer-ence on Sensor and Ad Hoc Communications and Networks,pp. 264271, 2004.

[3] V. Aravinthan, S. K. Jayaweera, and K. Al-Tarazi, Dis-tributed estimation in a power constrained sensor network,VTC Spring, pp. 10481052, 2006.

[4] H.V. Poor X. Zhang and M. Chiang, Optimal Power Alloca-tion for Distributed Detection over MIMO Channels in Wire-less Sensor Networks, IEEE Trans. Signal Process., vol. 56,no. 9, 2008.

[5] K. Rajan and B. Natrajan, A Distance Based Comparison ofOptimal Power Allocation to Distributed Sensors in a WirelessSensor Network, Proc. IEEE ICC08, pp. 4391 4395, 2008.

[6] A. Ribeiro and G. B. Giannakis, Bandwidth-ConstrainedDistributed Estimation for Wireless Sensor Networks Part I:Gaussian Case, IEEE Transactions on Signal Processing,vol. 54, no. 3, pp. 11311143, March 2006.

[7] Z.-Q. Luo, Universal Decentralized Estimation in a Band-width Constrained Sensor Network, IEEE Transactions onInformation Theory, vol. 51, no. 6, pp. 22102219, June 2005.

[8] M. Gastpar and M. Vetterli, Source-channel communica-tion in sensor networks, proceedings of the 2nd Interna-tional Workshop on Information Processing in Sensor Net-works (IPSN03), pp. 162177, April 2003.

[9] Mahesh K. Banavar, A. D. Smith, C. Tepedelenlioglu, andA. Spanias, Distrubuted Detection over Fading MACs withMultiple Antennas at the Fusion Center, Proc. ICASSP10,pp. 2894 2897, March 2010.

[10] S. Cui, J. Xiao, A. Goldsmith, Z.-Q. Luo, and H. V. Poor, Es-timation Diversity and Energy Efficiency in Distributed Sens-ing, IEEE Transactions on Signal Processing, vol. 55, no. 9,pp. 46834695, September 2007.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

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