Performance Comparison Between Maximum Likelihood and Heuristic Weighted Average Estimation Methods for Energy-Based Target Localization in Wireless Sensor Networks Zhenxing Luo

Thomas C. Jannett

Dept. of Electrical and Computer Engineering The University of Alabama at Birmingham Birmingham, AL 35294-4461, USA [email protected]

Dept. of Electrical and Computer Engineering The University of Alabama at Birmingham Birmingham, AL 35294-4461, USA [email protected]

Abstract—This paper compares maximum likelihood estimation (MLE) and weighted average (WA) estimation methods for energybased target localization using a wireless sensor network (WSN) in which the communication channels between the sensors and fusion center are imperfect. We consider two different models for the communication channel between the sensors and the fusion center: a Binary Symmetric Channel (BSC) and a Rayleigh channel with coherent receiver and hard decoder. We compared the estimation performance and the computation time for both methods in simulations. The root mean square (RMS) errors given by the WA method were generally higher than the RMS errors given by the MLE method. The computation time for the WA method was much lower than the computation time for the MLE method. Keywords-wireless sensor networks; target localization; maximum likelihood estimation; Cramer-Rao lower bound; estimation performance

I. INTRODUCTION Wireless sensor networks (WSNs) have drawn a lot of attention recently due to their vast applications [1-6]. Typical application examples can be found in [7-9]. Usually, a WSN consists of a fusion center and a large number of low-cost and resource-constrained sensors [10]. Sensors make observations about the environment and send the information to the fusion center for further processing and analysis. In the energy-based target localization problem, sensors measure target signal strength and send the measurements to the fusion center. Based on these measurements, the fusion center estimates target power and target position [3], [10]. If sensors are deployed in a noisy environment, the decisions sent by the fusion center will be disturbed in the communication channels between the fusion center and sensors. Sensors having limited resources, such as energy and computation capacity, cannot use sophisticated communication and coding schemes [11] to counter this problem. In [11], this problem was addressed on the fusion center side by incorporating models of the communication channel into a maximum likelihood estimation

(MLE) scheme. Simulations were run to compare the channel aware MLE method and the channel unaware MLE method [11]. In the presence of communication channel errors, both the channel aware and channel unaware MLE methods require more computation time due to the impact of the additional communication channel errors on the iterative search required by the MLE method. In contrast, the heuristic weighted average (WA) estimation method presented in [10] dramatically reduces the computation time at the expense of reduced estimation performance. In this paper, we will compare the estimation performance and computation time of MLE and WA methods. The main contribution of this paper is the comparison of the MLE method and the WA method in the presence of communication channel errors. Results showed that the root mean square (RMS) errors of the MLE methods were lower than the RMS errors of the WA method. However, the computation time of the WA method was much less than the computation time of the MLE method. We also found that under certain circumstances, the WA method may outperform the channel unaware MLE method and give lower RMS errors. Section II presents the energy-based MLE target localization method. The WA method is described in Section III. In Section IV, we model the communication channel using the binary symmetric channel (BSC) and the Rayleigh channel with coherent receiver and hard decoder. The simulation setup is provided in Section V and simulation results are provided in Section VI. Discussion and analysis are given in Section VII, followed by concluding remarks in Section VIII. II.

ENERGY-BASED MAXIMUM LIKELIHOOD TARGET LOCALIZATION

Fig. 1 shows a typical sensor field, and the MLE and WA estimation methods are compared using this field. Similar to the setup in [10], a target emits a signal that decays according to the model

Gi P0'

 di

d0 

2

100 Sensors Estimated target location True target location

80

where P0' is the power of the target measured at a reference distance d 0 , the gain of the ith sensor is Gi , and the signal strength at the ith sensor is ai . The distance between the ith sensor located at (xi , yi ) and a target located at (xt , yt ) is defined as

di = (xi - xt )2 + (yi - yt )2 .

(2)

If we assume Gi  1 and d0  1 , then the model (1) can be simplified as

ai2 =

Sensor Field

(1)

P0 . di2

(3)

In this paper, the target is assumed to be at least d0 meters away from any sensor. Due to the presence of noise, the signal received at sensor i is (4)

si = ai + wi where wi is a Gaussian noise following distribution wi ~ N  0,σ 2  .

60

Sensor Field Y-coordinate (m)

ai2 =

40 20 0 -20 -40 -60 -80 -100 -100

-80

-60

-40 -20 0 20 40 Sensor Field X-coordinate (m)

60

80

100

Figure 1. Sensor field.

Next, the transition probability p(mi mi ) is incorporated into the MLE framework. The probability that mi is equal to m is

p(mi  m  ) =



mi 1,1

p(mi  m mi )p(mi  )

(9)

(5)

where p(mi  m mi ) denotes the transition probability of the

According to a pre-determined threshold  i , each sensor

communication channel. The communication channel between the sensor and the fusion center can be modeled using this approach if the transition probability p(mi mi ) is known.

quantizes the signal si to a value mi . The process is denoted by:

   si   i

1 mi   1

 i  si  

.

(6)

The maximum likelihood estimator is (7)

1

x

2

Q  x  

e



t2 2

dt .

(11)

For an unbiased estimate of  , the Cramer-Rao lower bound (CRLB) is (8)

The decision mi is transmitted to the fusion center through a communication channel. The decision received at the fusion center is denoted by mi . The values of mi and mi can be either 1 or -1. The transition probability between mi and mi is denoted by p(mi mi ) .

ˆ  max lnp  M   . 

where Q  x  is defined as 

fusion center estimates   [ P0 xt yt ]T by maximizing N   ln p(M  ) =  ln   p(mi  m mi )p(mi  ) . (10) i=1   mi 1,1

The probability that mi takes value m is

 i  ai  1  Q(  ) (m  1) p(mi  m  )    Q(  i  ai ) (m  1)  

For Μ  [m1 m2 ... mN ]T received at the fusion center, the

E{[ˆ(M)   ][ˆ(M)   ]T }  J 1

(12)

J   E  T lnp  M   .

(13)

The derivation of the CRLB matrix is presented in [11]. We used the normalized estimation error squared (NEES) as a criterion to evaluate the performance of the MLE methods [10], [12]. The NEES is defined as

  (  ˆ)T J(  ˆ)

(14)

-1

where J is the fisher information matrix defined in (13). The NEES follows a Chi-Square distribution and the average NEES falls into a confidence region determined by the number of parameters to be estimated and the number of Monte Carlo simulations. III.

An alternative method is the WA estimation method presented in [10]. In this method, the calculation is much simpler. This method can be expressed as

t 

 m

Di 1

i

Di 1

mi

(15)

COMMUNICATION CHANNEL MODELS

We consider two types of communication channels for the channel aware MLE approach used in this paper. The first model is the BSC. The transition relation of a BSC is shown in Fig. 2 where the crossover probability is p and the probability of correct transmission is q . The second communication channel we consider is the Rayleigh fading channel with coherent receiver and hard decoder (Fig. 3). If binary phase-shift keying (BPSK) signals are used, the Rayleigh fading channel with coherent receiver and hard decoder can be modeled as a BSC [11]. The relation between the channel SNR and the error probability Pe is defined as [11] Pe 

mi

p

1

1 q  1 p Figure 2. Binary symmetric channel.

        

  

  

  

Rayleigh fading channel

Coherent receiver and hard decoder in fusion center

  

Figure 3. Rayleigh fading channel with coherent receiver and hard decoder

i

where  i denotes the position of the ith sensor and  t denotes the estimated target position. If the ith sensor makes decision 1, the corresponding mi value is added to the denominator of (15) and the mi i value is added to the numerator of (15). The WA method is simple and straightforward. However, the fusion center does not use information about the communication channels between sensors and the fusion center. IV.

-1

  

i

m

q  1 p

p

HEURISTIC WEIGHTED AVERAGE ESTIMATION METHOD

The MLE method uses a search to find the minimum or maximum value. However, this process is time-consuming and computationally intensive. Furthermore, for difficult problems such as [11], iterative methods typically employ strategies to avoid becoming trapped by local minima. One idea is to use a two-stage method in which an approximate solution is determined for use as the starting point for the iteration. Another idea is to repeat the search several times, with each search beginning at a different starting point.

Fusion Center

Sensor

1 SNR (1  ) 2 1  SNR .

(16)

Therefore, we can set the crossover probability in the BSC model to the error probability in (16) and use the BSC model to investigate the relation between SNR and estimation performance. V.

SIMULATION SETUP

Monte Carlo simulations (1000 runs) were used to compare the estimation performance and computation times of the MLE and WA methods. In all simulations,  i  4 for all sensors, (xt , yt )=(12, 13) , and P0  10,000 . All simulations involving computation time were run using one node and 2GB memory in Cheaha server at The University of Alabama at Birmingham. One node is one core in the Quad-Core E5450 3.0 GHz CPU. VI.

SIMULATION RESULTS

For the BSC model, NEES values for the channel aware MLE method were less than the NEES values for the channel unaware MLE method (Table I). Similarly, for the Rayleigh channel with coherent receiver and hard decoder, NEES values for the channel aware MLE method were less than the NEES values for the channel unaware MLE method (Table II). The RMS estimation error is an important indicator of estimation performance. For the BSC model, RMS errors given by the channel aware MLE method were lower than the RMS errors given by the channel unaware MLE method (Fig. 4). The RMS errors given by the channel aware MLE method were close to the CRLB of channel aware MLE method (Fig. 4). Similarly,

for the Rayleigh channel with coherent receiver and hard decoder, RMS errors given by the channel aware MLE method were lower than the RMS errors given by the channel unaware MLE method (Fig. 5). The RMS errors given by the channel aware MLE method were close to the CRLB of channel aware MLE method (Fig. 5).

low crossover probability, NEES values were within the confidence interval (Table II). For low SNR values, NEES values were outside the confidence interval. Moreover, as expected, the channel unaware MLE method gave higher NEES values than the channel aware MLE method. TABLE II.

Computation time is an indicator of computational complexity. For BSC model, the computation times for the channel aware MLE method were slightly greater than the computation times for the channel unaware MLE method (Table III). The computation times for the channel unaware MLE method were much greater than for the WA method (Table III). Similarly, for the Rayleigh channel with coherent receiver and hard decoder, the computation times for the channel aware MLE method were slightly higher than for the channel unaware MLE method (Table IV). The computation times for the channel unaware MLE method were much greater than for the WA method (Table IV). In order to avoid traps in local minima and to reduce the computation time for the channel aware and channel unaware MLE methods, the starting point of the search was provided by the WA method [13].

NEES VALUES FOR CHANNEL AWARE AND CHANNEL UNAWARE MLE METHODS (BSC MODEL)

NEES

Crossover probability, p 0

0.1

0.2

0.3

channel aware

3.11

3.76

5.48

7.05

channel unaware

3.11

55.92

127.78

194.60

As mentioned earlier, the Rayleigh channel with coherent receiver and hard decoder can be modeled using a BSC. For each SNR value, there is a unique crossover probability in the corresponding BSC model. For high SNR values, which gave

10

15

20

channel aware

4.45

3.88

3.47

3.12

3.10

channel unaware

88.01

32.86

9.34

4.60

3.48

0

5

RMS error - y, m RMS error - x, m

RMS error - Po

4

2

x 10

1 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0

0.05

0.1

0.15 0.2 0.25 Crossover probability

0.3

0.35

20 10 0 20 10 0

RMS error - Po

Figure 4. RMS estimation errors (solid line: channel aware MLE, solid line+star: WA method, solid line+circle: CRLB of channel aware MLE, solid line+square: channel unaware MLE) 10000 5000 0

RMS error - y, m RMS error - x, m

TABLE I.

SNR (dB)

NEES

VII. ANALYSIS The two sided 95% confidence interval for the average NEES corresponding to 1000 Monte Carlo simulations and estimation of three parameters is [2.85 3.15]. When the crossover probability was low in the BSC model, the NEES values were within the confidence interval (Table I). When the crossover probability was high, the NEES values were outside the confidence interval. As the crossover probability increased, the NEES values also increased. The NEES values for the channel aware MLE method were much higher than the NEES values for the channel unaware MLE method except when the crossover probability was zero. The reason is obvious: channel unaware MLE methods did not account for the communication channel errors while channel aware MLE methods accounted for the communication channel errors. Therefore, in the presence of communication channel errors, channel unaware MLE methods did not perform as well as channel aware MLE methods. However, if no communication channel errors were present, both methods provided the same results.

NEES VALUES FOR CHANNEL AWARE AND CHANNEL UNAWARE MLE METHODS (RAYLEIGH MODEL)

0

5

10

15

20

0

5

10

15

20

0

5

10 SNR (dB)

15

20

20 10 0 20 10 0

Figure 5. RMS estimation errors (solid line: channel aware MLE, solid line+star: WA method, solid line+circle: CRLB of channel aware MLE, solid line+square: channel unaware MLE)

TABLE III. COMPUTATION TIMES FOR CHANNEL AWARE MLE METHOD, CHANNEL UNAWARE MLE METHOD, AND WA METHOD (BSC MODEL) Time (seconds)

Crossover probability, p 0

0.1

0.2

0.3

0.35

channel aware

1.2692 e+004

1.5438 e+004

1.6181 e+004

1.6142 e+004

1.6287 e+004

channel unaware

1.1961 e+004

1.1961 e+004

1.1912 e+004

1.1845 e+004

1.1749 e+004

WA

41.8992

40.5641

33.6484

38.9429

36.0093

channel unaware method. The channel aware method utilizes additional channel probability information in (9). Similarly, for the Rayleigh channel with coherent receiver and hard decoder, the computation times for the WA method were much lower than the computation times for the channel aware MLE and channel unaware MLE methods (Table IV). Moreover, the computation times for the channel aware MLE method were a little higher than for the channel unaware MLE method. VIII. CONCLUSION

TABLE IV. COMPUTATION TIMES FOR CHANNEL AWARE MLE METHOD, CHANNEL UNAWARE MLE METHOD, AND WA METHOD (RAYLEIGH CHANNEL WITH COHERENT RECEIVER AND HARD DECODER) Time (seconds)

SNR (dB) 0

5

10

15

20

channel aware

1.4705 e+004

1.3524 e+004

1.2866 e+004

1.2361 e+004

1.2457 e+004

channel unaware

1.2145 e+004

1.1728 e+004

1.0156 e+004

1.0250 e+004

9.9557 e+003

WA

43.8871

41.5238

43.5330

34.5891

33.0832

The RMS estimation errors increased as the crossover probability in the BSC model increased (Fig. 4). Moreover, RMS errors for the channel unaware MLE method were higher than for the channel aware MLE method. The RMS errors for the channel aware MLE method were close to the CRLB. It is interesting to see that the WA method outperformed the channel unaware MLE method, but not the channel aware MLE method, at high crossover probabilities although the WA method gave higher RMS errors at low crossover probabilities. Therefore, WA methods may be a better choice than channel unaware methods for high crossover probabilities. For a Rayleigh channel with coherent receiver and hard decoder, results were similar to those for the BSC model. The RMS errors for the channel aware MLE method were lower than for the channel unaware MLE method (Fig. 5). The channel aware MLE method took into account the channel information. For the channel aware method, RMS estimation errors were close to the CRLB. The results validated the CRLB. In the category of estimation error performance, the WA method could not compete with channel aware and channel unaware MLE methods. Computation times for the WA method for the BSC model were much less than the computation times for the channel aware and the channel unaware MLE methods (Table III). The MLE methods use a time-consuming iterative search process, but the WA method involves much simpler calculation and therefore takes much less time. Moreover, the computation time for the channel aware MLE method was a little higher than for the

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