research article channel-aware adaptive quantization ... · zation methods [, ] all require some...

Research ArticleChannel-Aware Adaptive Quantization Method forSource Localization in Wireless Sensor Networks

Guiyun Liu,1 Jing Yao,1 Yonggui Liu,2 Hongbin Chen,3 and Dong Tang1

1School of Mechanical and Electric Engineering, Guangzhou University, Guangzhou 510006, China2College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China3School of Information and Communication, Guilin University of Electronic Technology, Guilin 541004, China

Correspondence should be addressed to Guiyun Liu; [email protected]

Received 13 June 2015; Revised 21 August 2015; Accepted 1 September 2015

Academic Editor: Jianping He

Copyright © 2015 Guiyun Liu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers the problem of source localization using quantized observations in wireless sensor networks where, due tobandwidth constraint, each sensor’s observation is usually quantized into one bit of information. First, a channel-aware adaptivequantization scheme for target location estimation is proposed and local sensor nodes dynamically adjust their quantizationthresholds according to the position-based information sequence.Thenovelty of the proposed approach comes from the fact that thescheme not only adopts the distributed adaptive quantization instead of the conventional fixed quantization, but also incorporatesthe statistics of imperfect wireless channels between sensors and the fusion center (binary symmetric channels). Furthermore, theappropriate maximum likelihood estimator (MLE), the performance metric Cramer-Rao lower bound (CRLB), and a sufficientcondition for the Fisher information matrix being positive definite are derived, respectively. Simulation results are presented toshow that the appropriated CRLB is less than the fixed quantization channel-aware CRLB and the proposed MLE will approachtheir CRLB when the number of sensors is large enough.

1. Introduction

Wireless sensor networks (WSNs) is an emerging technologythat can provide an inexpensive solution for lots of applica-tions, including surveillance and urbanmonitoring [1], smartpower monitoring [2], and target localization or tracking [3–8]. Locating a target or object in an area of interest usinga wireless sensor network (WSN) is a typical application.TheWSN usually employs low-cost densely deployed sensorsthat have very limited energy and communication bandwidthresources. Its local sensors detect specific events and thenforward their sensed observation information to a fusioncenter (FC). Finally, the FC implements the task of targetlocation estimation [9].

Amultitude of studies have researched source localizationsince it has found lots of applications in radar, sonar, andmicrophone arrays [10, 11]. The localization problem is toestimate the coordinates of a source. Based on measurementmodels, the localization methods usually use the receivedsignal time of arrival (TOA), the distance measurement, the

received signal strength (RSS), the angle of arrival (AOA),the signal energy, and their combinations. It is worthwhileto mention that distance information is not directly avail-able and should be estimated based on other measurementmodels. On the other hand, techniques based on TOA andtime-delay of arrival (TDOA) estimation usually requirecomplicated timing or synchronization.

Compared with TOA, TDOA, distance measurement,and RSS, the localization problem based on acoustic energyhas practically attracted much attention over the past fewyears, due to its lower cost, lower complexity, and easierimplementation. In this paper, the source localization prob-lem with the use of the energy measurement model inWSNs is considered. These energy-based methods of sourcelocalization are classified into two major groups: analogmeasurement models [12–18] and quantized measurementmodels [19–24]. Due to bandwidth constraints in WSNs,it is naturally desirable that local sensors only transmittheir quantized observations to complete the estimation task.For example, consider a uniform quantizer with certain

Hindawi Publishing CorporationInternational Journal of Distributed Sensor NetworksVolume 2015, Article ID 214081, 13 pageshttp://dx.doi.org/10.1155/2015/214081

2 International Journal of Distributed Sensor Networks

quantization levels: (1) At each sensor, it observes a receivedsignal power following a power decay model, quantizes thesignal using the uniform quantizer, and then communicatesto the FC the quantized level; (2) At the FC, it reconstruct the“quantized” observations and estimates the coordinates of asource.

For these quantized measurement models, the maximumlikelihood target estimator and two heuristic design methodsfor optimizing quantization threshold have been proposed[20]. Within them, two heuristic design methods are provento be very robust under various situations. Additionally, thelocalization problem under nonideal channels between sen-sors and the FChas been investigated in [21]. For robust local-ization problems, compared against the classical maximumlikelihood estimators (MLEs), the proposed fault tolerantmaximum likelihood (FTML) estimator and the subtract onnegative add on positive (SNAP) have been shown to bemore fault tolerant in [22] and [23], respectively. Recently,Byzantine attacks for localization estimation in WSNs underbinary quantized data have been considered in [24] and anovel scheme in conjunction with the identification schemehas been proposed to make the Byzantines ineffective. How-ever, these quantized schemes only consider fixed thresholdsin designing local sensors’ quantization schemes, such asthe cases with the single threshold and a set of differentthresholds.

In other words, it is worth noting that the estimationperformance for the above quantized localization problemsdepends on the strategies of selecting sensors’ quantizationthresholds, that is, fixed quantization in local sensors. Forexample, the single threshold or a fixed set of differentthresholds have been studied in [19–21]. However, most ofthe works only considered the fixed quantization schemesin localization problems. Motivated by the observation thatadaptive quantization design schemes in distributed esti-mation problems [25, 26] can outperform fixed quantiza-tion schemes [27], we propose novel kinds of quantizationschemes that can adjust local sensor nodes’ quantizationthresholds adaptively according to sensor nodes’ positions ina region of interest, that is, adaptive quantization schemes forsource localization problems are adopted here.

As one of the early relatedworks in the distributed estima-tion problem, a common threshold 𝜏 is applied at all the sen-sors [27, 28], which is called the fixed quantization approach.It is of great challenge for the approach tomake a good choiceof the common threshold. To be close to the unknownparam-eter, a further improvement is to apply a set of thresholds withequal or unequal frequencies and hopes that one of thresholdscan be enough close [27, 28]. However, themultithresholdingis required to have the knowledge of the prior distribution ofthe parameter. Recently, being the most related work, a data-dependent distributed adaptive quantization approach isconsidered and sensors’ thresholds are adjusted dynamicallyto converge to the unknown parameter [25, 26].

As the early related work in the source localizationproblem, some prior knowledge about the region of interestis assumed and two kinds of intuitive quantization methodsare designed to obtain optimal quantization thresholds [20].Although the heuristic design methods require minimum

prior information about the system, identical thresholds aresimply employed at all the sensors and their performancesare constrained by the fixed common thresholds. In the latestwork [29], a dynamic nonidentical threshold design is pro-posed for making the Byzantines ineffective, where eachthreshold is calculated based on each sensor’s signal ampli-tude from the source. In other words, the proposed quanti-zation methods [20, 29] all require some prior distributionknowledge about the source’ unknown parameters.

The main contributions of this paper are as follows:(1) A novel adaptive and channel-aware quantization

scheme for target location estimation using one-bitdata is proposed.Within it, local sensors’ quantizationthresholds are adaptively adjusted according to sen-sors’ location coordinates and their received binaryinformation.

(2) The appropriate maximum likelihood estimator(MLE) is derived to seek the estimate of the locationof the target, and the metric Cramer-Rao lowerbound (CRLB) is derived theoretically to benchmarkthe performance of locating.

(3) Some additional simulations show that the proposedchannel-aware quantization schemehas better perfor-mance than the existing fixed schemes under severalcases.

The rest of the paper is organized as follows. Section 2introduces our system model. In Section 3, the position-based adaptive quantization scheme is proposed. Theircorresponding channel-aware MLE and the channel-awareCRLB both are derived in Section 4. Compared against theclassical MLE with the fixed quantization scheme [21], theproposedMLE is evaluated in simulation in Section 5. Finally,conclusions in Section 6 are presented.

2. Problem Formulation

Consider a wireless sensor network with a fusion center,where 𝑁 locally distributed sensors are densely deployed ina region of interest (ROI) [20]. As shown in Figure 1, theROI is a two-dimensional square region. Local sensors areassumed to share the communication channel on a time-sharing basis [25]. Additionally, we assume that each sensorand the FC have the prior knowledge of locations of all thesensors.Without loss of generality, each sensor is deployed ina grid for simplicity. The sensors are indexed as {1, 2, . . . , 𝑁}according to distances between each other. The sensor 𝑖, {𝑖 =1, . . . , 𝑁} is located at the coordinate point (𝑥

𝑖, 𝑦𝑖), where 𝑥

𝑖

and 𝑦𝑖denote the 𝑥-coordinate (m) and the 𝑦-coordinate (m)

of the 𝑖th sensor, respectively. Additionally, the coordinate ofthe target is denoted by (𝑥

𝑡, 𝑦𝑡). For example, (50, −50) and

(0, 0) are the coordinates of the 6th sensor and the target,respectively.

The target emits an acoustic attenuation signal, which isa function of the Euclidean distance between the target andlocal sensors [21]:

𝑎2

𝑖=

𝑃0𝑑𝑛

0

𝑑𝑛

𝑖

=

𝑃0

𝑑2

𝑖

, (1)

International Journal of Distributed Sensor Networks 3

where 𝑎𝑖, 𝑑𝑖, and 𝑃

0are the received signal amplitude at the

sensor, the Euclidean distance between the target and the 𝑖thsensor (i.e., √(𝑥

𝑖− 𝑥𝑡)2+ (𝑦𝑖− 𝑦𝑡)2), and the signal power

emitted by the target at a reference distance 𝑑0, respectively.

Without loss of generality, 𝑑0and 𝑛 are set to 1 and 2,

respectively [20]. It is assumed that the target is at least 𝑑0

meters away from any sensors at any times.The observation model for local sensors is considered as

given below:

𝑧𝑖= 𝑎𝑖+ 𝜔𝑖, (2)

where 𝜔𝑖denotes the measurement noise process at the

sensor 𝑖. {𝜔𝑖, 𝑖 = 1, . . . , 𝑁} are assumed to be independent

and identically distributed (i.i.d.) white Gaussian noises withmean zero and variance 𝜎2.

In this paper, (1) is shown to be a widely adopted modelfor the acoustic attenuation signal which is propagating infree space and (2) also is a reasonable model [20]. Withoutloss of generality, the same model in [20] is simply adoptedhere.Meanwhile, at each sensor, due to severe bandwidth lim-itations inWSNs, its received signal amplitude 𝑧

𝑖is quantized

and then transmitted to the FC. The quantization bit budgetfor each sensor is supposed to be one bit per sample.The one-bit quantization process for the 𝑖th sensor is such that

𝑏𝑖=

{

{

{

1, if 𝑧𝑖≥ 𝜏𝑖,

0, otherwise,(3)

where 𝜏𝑖denotes the quantization threshold for the 𝑖th sensor.

These binary data sequences {𝑏𝑖} are reported over wireless

channels to the receiving terminals (including subsequentsensors and the FC). There are unavoidable channel imper-fections which can not be neglected. The imperfect channelsare assumed to be orthogonal binary symmetric channels(BSCs). Here, without loss of generality, each BSC is assumedto be characterised by the same crossover probability 𝑝

𝑒.

Let {𝑏𝑖} denote the observation information at the receiving

terminals after transmission through BSCs. Thus, we have

𝑃 (𝑏𝑖= 1 | 𝑏

𝑖= 1) = 1 − 𝑝

𝑒,

𝑃 (𝑏𝑖= 0 | 𝑏

𝑖= 1) = 𝑝

𝑒,

𝑃 (𝑏𝑖= 0 | 𝑏

𝑖= 0) = 1 − 𝑝

𝑒,

𝑃 (𝑏𝑖= 1 | 𝑏

𝑖= 0) = 𝑝

𝑒,

(4)

where 𝑃(⋅) denotes the probability with which the eventwill occur. Based on the binary sequences {𝑏

𝑖} above, the FC

needs to estimate the parameter vector 𝜃 = [𝑃0, 𝑥𝑡, 𝑦𝑡].

3. Channel-Aware Position-BasedAdaptive Quantization

Generally, the thresholds {𝜏𝑖, 𝑖 = 1, . . . , 𝑁} are the same

for all the sensors [20–22]. It is well known that the CRLB

exponentially increases with |𝜏𝑖− 𝑎𝑖|2/𝜎2 [20] for target

location estimation to some extent. Thus, to obtain a betterestimation result, 𝜏

𝑖is required to be set close to its received

signal power 𝑎𝑖. However, the signal power 𝑎

𝑖is unknown

prior. It motivates us to propose an adaptive quantizationscheme that can adaptively decrease the distance ‖𝜏

𝑖− 𝑎𝑖‖.

Meanwhile, it is noted that the CRLB is also a function of thenoise parameter 𝜎. It is generally required that the distance‖𝜏𝑖− 𝑎𝑖‖ is in the effective range which is related to 𝜎, and the

CRLB is enough small [30, 31]. In other words, the locationis possibly retrieved from sensors’ binary observations onlywhen ‖𝜏

𝑖− 𝑎𝑖‖ is in the effective range. Thus, without loss

of generality, it is assumed that the distance ‖𝜏𝑖− 𝑎𝑖‖ is in

the effective range. Some more discussions will be given inSection 5.

It is noted that 𝑎𝑖is directly dependent on the parameter

𝑑𝑖from (1). Thus, it is considered that the close neighbors of

the 𝑖th sensor have roughly equal received signal amplitude𝑎𝑖. In other words, the close neighbors of the 𝑖th sensor

should be assigned almost the same threshold 𝜏𝑖.Thus, a novel

position-based and adaptive quantization scheme is proposedhere. Different from the adaptive quantization for distributedparameter estimation [25] being scheduled sequentially, ourproposed adaptive quantization scheme is scheduled by theposition-based binary sequence.

The key principle of the position-based informationsequence is that the close neighbors of the 𝑖th sensor areassigned almost the same threshold. That is, if the 𝑖th sensortransmits a binary data {1} to the fusion center, its subsequentneighbor will assume that the threshold of the precedingsensor is too small and then will increase the thresholdadaptively. Instead, if the 𝑖th sensor transmits a binary data{0} to the fusion center, its subsequent neighbor will assumethat the threshold of the preceding sensor is a little too largeand then will decrease the threshold adaptively.

The position-based and adaptive quantization schemeconsists of two stages: (1) Define the indexes of all localsensors according to their distance information; (2) Updatethresholds in order of indexes. For the first stage, it is notedthat local sensors are indexed as {1, 2, . . . , 𝑁} according todistances between each other as shown in Figure 1. Theflowchart of the first-stage operation specially is shown inFigure 2. The sensor locating in the coordinate (−50, −50) isindexed with 1. Then, the procedure needs to decide whichsensor will be the next sensor indexed with 2 according todistance information. Within it, it selects the closest sensorwith the minimum 𝑦-coordinate value, that is, the neighborlocating in the coordinate (−50, −30), and index it with 2.Similarly, the sensor locating in the coordinate (50, −30) isindexed with 7, and the sensor locating in the coordinate(30, −30) is indexed with 8, and so forth.

For the second stage, the position-based sequence oftransmission can be shown in Figure 3. It assumes that thecommunication channel on a time-sharing basis has a presetnumber of slots according to the sensors’ indexed order andeach sensor has received an acoustic attenuation signal. LetB𝑖= {𝑏𝑗: 𝑗 ∈ [0, 𝑖 − 1]} denote the received binary sequence

of the 𝑖th sensor, where 𝑗 is an integer and B1is empty.


−60 −40 −20 0 20 40 60−60

−40

−20

0

20

40

60

x-coordinate (m)

y-c

oord

inat

e (m

)

1

12

13

24

25

6

7

18

19

30

31

5

8

17

20

29

32

4

9

16

21

28

33

3

10

15

22

27

34

2

11

14

23

26

3536

SensorsTarget

Figure 1: The square ROI of target localization in WSN. The signalpower contours of a target are denoted by red solid circles.

Start

Select a sensor,index it with 1

Yes

Yes Yes

No

NoNo

i

Select the sensor,set as i + 1

Select the sensor withminimum y-coordinate

Is the closestsensor unique?

Is i + 1 equalto N?

Is i + 1 equalto N?

End

value, set as i + 1

Figure 2: Flow chart of the first stage of the proposed adaptivequantization scheme.

Based the fixed threshold 𝜏1, the 1st sensor generates a binary

observation 𝑏1:

𝑏1=

{

{

{

1, if 𝑧1≥ 𝜏1,

0, otherwise.(5)

𝑏1is transmitted to the fusion center and received by the

subsequent 2nd sensor over BSCs.The 2nd sensor will obtainthe binary sequence B

2and then calculate 𝜏

2as follows:

𝜏2=

{

{

{

𝜏1+ Δ, if 𝑏

1equals 1,

𝜏1− Δ, otherwise,

(6)

Start

Yes

No

i

Is i + 1 equalto N?

End

Init the first sensor’sthreshold

Broadcast its binaryobservation

Receive its binarysequences

Update its threshold

Broadcast its binaryobservation

Figure 3: Flow chart of the second stage of the proposed adaptivequantization scheme.

where Δ is a given step-size parameter. For the 𝑖th sensor,given the binary sequence B

𝑖, its binary observation 𝑏

𝑖is

generated as

𝑏𝑖=

{

{

{

1, if 𝑧𝑖≥ 𝜏𝑖,

0, otherwise,(7)

where {𝜏𝑖, 𝑖 ≥ 2} is expressed as

𝜏𝑖=

{

{

{

𝜏𝑖−1+ Δ, if 𝑏

𝑖−1equals 1,

𝜏𝑖−1− Δ, otherwise.

(8)

Note the signal power 𝑎𝑖is always positive. 𝜏

1is assumed

to be any positive value. Thus, after obtaining 𝜏𝑖+1

(7), it isassumed that the updating procedure is required; that is,

𝜏𝑖+1=

{

{

{

𝜏𝑖+1, if 𝜏

𝑖+1≥ 0,

𝜏𝑖, otherwise.

(9)

It is noted that formula (9) can guarantee 𝜏𝑖is always

nonnegative.Under the setting above, a set of binary observations

{𝑏(𝑛)} is generated, which can be denoted as B =

[𝑏(1) 𝑏(2) ⋅ ⋅ ⋅ 𝑏(𝑁)]. Effects of unreliable wirelesschannels between sensors and the FC can not be neglected[30]. Thus, the set of binary observations B is corrupted bythe imperfect wireless channels

B = [𝑏 (1) 𝑏 (2) ⋅ ⋅ ⋅ 𝑏 (𝑁)] , (10)

where𝑏(𝑖) is the corrupted observationwhich is sent from the𝑖th sensor to the FC. So based on B, estimating the parameter


5 10 15 20 25 30 35−1

−0.5

0

0.5

1

1.5

2

n (sensor index)

Sens

or o

utpu

t/thr

esho

ld

Sensor observationAdaptive quantization thresholdFixed quantization threshold

Figure 4: Dynamic evolution of the quantization threshold versussensor index. 𝑃

0= 64W, 𝑥

𝑡= 𝑦𝑡= 0m, 𝜎2 = 0.01W, 𝜏

𝐹= 𝜏1= 1,

Δ = 0.4, and 𝑝𝑒= 0.01.

vector 𝜃 = [𝑃0𝑥𝑡𝑦𝑡] is our following major goal. Addi-

tionally, as shown in Figure 4, the proposed adaptive quanti-zation scheme adaptively decreases the distance ‖𝜏

𝑖− 𝑎𝑖‖ and

converges faster to the observed acoustic attenuation signalsthan the fixed quantization scheme [21]. Within it, 𝜏

𝐹and 𝜏1

is the fixed quantization threshold and the 1st sensor’s initialthreshold for the proposed adaptive quantization scheme,respectively.

4. Analysis of MLE and CRLB

4.1. MLE. Using the model of BSCs (5), the probability of areceived binary observation {𝑏

1} taking a specific value 𝑏 (1

or 0), given the initialized threshold 𝜏1, is expressed as

𝑃 (𝑏1= 𝑏 | 𝜏

1; 𝜃) =

1

∑

𝑏1=0

𝑃 (𝑏1= 𝑏 | 𝑏

1; 𝜃) 𝑃 (𝑏

1| 𝜏1; 𝜃) , (11)

where 𝑃(𝑏1= 𝑏 | 𝑏

1; 𝜃) is equal to 𝑃(𝑏

1= 𝑏 | 𝑏

1) defined

in formula (4) and 𝑃(𝑏1| 𝜏1; 𝜃) is the probability of a quant-

ized measurement taking a specific value conditioned on theinitialized threshold 𝜏

1. According to the sensor observation

model in formula (2) and the proposed channel-aware quan-tization model in formula (5), 𝑃(𝑏

1| 𝜏1; 𝜃) is given by

𝑃 (𝑏1| 𝜏1; 𝜃) =

{

{

{

𝑞 (𝜏1− 𝑎1) , 𝑏

1= 1,

1 − 𝑞 (𝜏1− 𝑎1) , 𝑏1= 0,

(12)

where 𝑞(⋅) is the complementary distribution function of thezero mean, 𝜎2 variance Gaussian distribution defined as [21]

𝑞 (𝑥) = ∫

+∞

𝑥

1

√2𝜋𝜎

𝑒−𝑥2/2𝜎2

𝑑𝑥. (13)

In general, according to formula (11), we have

𝑃 (𝑏𝑖= 𝑏 | 𝜏

𝑖; 𝜃) =

1

∑

𝑏𝑖=0

𝑃 (𝑏𝑖= 𝑏 | 𝑏

𝑖; 𝜃) 𝑃 (𝑏

𝑖| 𝜏𝑖; 𝜃) , (14)

where 𝑃(𝑏𝑖| 𝜏𝑖; 𝜃) is the probability of a quantized mea-

surement taking a specific value (𝑏𝑖) conditioned on the

quantization threshold 𝜏𝑖of the 𝑖th sensor. It is given by

𝑃 (𝑏𝑖| 𝜏𝑖; 𝜃) =

{

{

{

𝑞 (𝜏𝑖− 𝑎𝑖) , 𝑏

𝑖= 1,

1 − 𝑞 (𝜏𝑖− 𝑎𝑖) , 𝑏𝑖= 0,

(15)

where 𝜏𝑖is defined in the above proposed channel-aware

adaptive quantization scheme.Thus, the likelihood function of the parameter 𝜃 at the FC

based on the received binary sequence B can be expressed as

𝐹 (𝜃) =

𝑁

∏

𝑖=1

𝑓 (𝑏𝑖| B𝑖; 𝜃) =

𝑁

∏

𝑖=1

𝑃 (𝑏𝑖| 𝜏𝑖; 𝜃) , (16)

where𝑓(𝑏𝑖| B𝑖; 𝜃)denotes the conditional probability density

function of 𝑏𝑖and is equal to 𝑃(𝑏

𝑖| 𝜏𝑖; 𝜃). It is noted that the

variable 𝑏𝑖being a Bernoulli random value has only 0 and 1

as possible values. According to formula (14), it is concludedthat 𝑃(𝑏

𝑖| 𝜏𝑖; 𝜃) is given as the following:

𝑃 (𝑏𝑖| 𝜏𝑖; 𝜃) = 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖; 𝜃)

��𝑖

𝑃 (𝑏𝑖= 0 | 𝜏

𝑖; 𝜃)

1−��𝑖

. (17)

From formulas (16) and (17), we have

𝐹 (𝜃) =

𝑁

∏

𝑖=1

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖; 𝜃)

��𝑖

𝑃 (𝑏𝑖= 0 | 𝜏

𝑖; 𝜃)

1−��𝑖

. (18)

Then its log-likelihood function is expressed as

𝐿 (𝜃) =

𝑁

∑

𝑖=1

[𝑏𝑖ln (𝑃 (𝑏

𝑖= 1 | 𝜏

𝑖; 𝜃))

+ (1 −𝑏𝑖) ln (𝑃 (𝑏

𝑖= 0 | 𝜏

𝑖; 𝜃))] .

(19)

According to the definition of theMLE [32], for the proposedchannel-aware adaptive quantization scheme, the MLE isexpressed as

𝜃 = arg max

𝜃

𝐿 (𝜃) . (20)

It is noted that a systematic grid search, such as the sequentialquadratic programming (SQP) [20], is employed to find anapproximate maximum point of the global optimal value inthe MLE above.


4.2. CRLB. The aforementioned section has presented thechannel-aware adaptive quantization scheme. In this section,its CRLB associated with the proposed scheme is derivedto determine the lower bound on the variance of anyunbiased estimator and provide the best performance bench-mark.

According to the CRLB for vector parameter [32], theCRLB for the channel-aware adaptive quantization schemecan be given as follows:

CRLB (𝜃) = 𝐼 (𝜃)−1

=((

(

−𝐸[

𝜕2𝐿 (𝜃)

𝜕2𝜃1

] −𝐸[

𝜕2𝐿 (𝜃)

𝜕𝜃1𝜕𝜃2

] −𝐸[

𝜕2𝐿 (𝜃)

𝜕𝜃1𝜕𝜃3

]

−𝐸[

𝜕2𝐿 (𝜃)

𝜕𝜃2𝜕𝜃1

] −𝐸[

𝜕2𝐿 (𝜃)

𝜕2𝜃2

] −𝐸[

𝜕2𝐿 (𝜃)

𝜕𝜃2𝜕𝜃3

]

−𝐸[

𝜕2𝐿 (𝜃)

𝜕𝜃3𝜕𝜃1

] −𝐸[

𝜕2𝐿 (𝜃)

𝜕𝜃3𝜕𝜃2

] −𝐸[

𝜕2𝐿 (𝜃)

𝜕2𝜃3

]

))

)

−1

,

(21)

where 𝐼(𝜃) is the Fisher information matrix (FIM) and 𝜃𝑖

denotes the 𝑖th element of 𝜃. According to formulas (19) and(21), we have

𝜕2𝐿 (𝜃)

𝜕𝜃𝑗𝜕𝜃𝑘

=

𝜕 (𝜕𝐿 (𝜃) /𝜕𝜃𝑗)

𝜕𝜃𝑘

= −

𝑁

∑

𝑖=1

{

{

{

𝑏𝑖(

(𝜕𝑃 (𝑏𝑖= 1 | 𝜏

𝑖; 𝜃) /𝜕𝜃

𝑗) (𝜕𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖; 𝜃) /𝜕𝜃

𝑘)

𝑃2(𝑏𝑖= 1 | 𝜏

𝑖; 𝜃)

−

𝜕2𝑃 (𝑏𝑖= 1 | 𝜏

𝑖; 𝜃) /𝜕𝜃

𝑗𝜕𝜃𝑘

𝑃 (𝑏i = 1 | 𝜏𝑖; 𝜃)

) + (1 −𝑏𝑖)

⋅ (

(𝜕 (1 − 𝑃 (𝑏𝑖= 1 | 𝜏

𝑖; 𝜃)) /𝜕𝜃

𝑗) (𝜕 (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖; 𝜃)) /𝜕𝜃

𝑘)

(1 − 𝑃 (𝑏𝑖= 1 | 𝜏

𝑖; 𝜃))

2−

(𝜕2(1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖; 𝜃)) /𝜕𝜃

𝑗𝜕𝜃𝑘)

1 − 𝑃 (𝑏𝑖= 1 | 𝜏

𝑖; 𝜃)

)

}

}

}

≜ −

𝑁

∑

𝑖=1

𝐴𝑗,𝑘(𝑏𝑖, 𝜏𝑖, 𝜃) , 𝑗, 𝑘 ∈ {1, 2, 3} .

(22)

Thus, [𝐼(𝜃)]𝑗,𝑘

can be derived as

[𝐼 (𝜃)]𝑗,𝑘= −𝐸[

𝜕2𝐿 (𝜃)

𝜕𝜃𝑗𝜕𝜃𝑘

] =

𝑁

∑

𝑖=1

𝐸 [𝐴𝑗,𝑘(𝑏𝑖, 𝜏𝑖, 𝜃)]

(a)=

𝑁

∑

𝑖=1

𝐸𝜏𝑖

{𝐸��𝑖|𝜏𝑖

[𝐴𝑗,𝑘(𝑏𝑖, 𝜏𝑖, 𝜃)]}

≜

𝑁

∑

𝑖=1

𝐸𝜏𝑖

𝐺𝑗,𝑘(𝜏𝑖, 𝜃) ,

(23)

where the notation ≜ means that 𝐺𝑗,𝑘(𝜏𝑖, 𝜃) = 𝐸

��𝑖|𝜏𝑖

[𝐴𝑗,𝑘(𝑏𝑖,

𝜏𝑖, 𝜃)]. 𝐸

𝜏𝑖

and 𝐸��𝑖|𝜏𝑖

denote the expectation with respectto (w.r.t.) the distribution 𝑃(𝜏

𝑖; 𝜃) and the expectation with

respect to the conditional distribution 𝑃(𝑏𝑖| 𝜏𝑖; 𝜃). It is

obvious that

𝑃 (𝑏𝑖, 𝜏𝑖; 𝜃) = 𝑃 (𝜏

𝑖; 𝜃) 𝑃 (

𝑏𝑖| 𝜏𝑖; 𝜃) , (24)

where 𝑃(𝑏𝑖| 𝜏𝑖; 𝜃) is defined in formula (14). It is noted

that 𝐴𝑗,𝑘(𝑏𝑖, 𝜏𝑖, 𝜃) is a function of three variables (𝑏

𝑖, 𝜏𝑖,

and 𝜃) and 𝐺𝑗,𝑘(𝜏𝑖, 𝜃) is a function of two variables (𝜏

𝑖

and 𝜃). According to the definition of the expectation of atwo-dimensional random variable, 𝐸[𝜕2𝐿(𝜃)/𝜕𝜃

𝑗𝜕𝜃𝑘] equals

𝐸𝜏𝑖,𝑏𝑖

[𝜕2𝐿(𝜃)/𝜕𝜃

𝑗𝜕𝜃𝑘]. Thus, (a)= follows from the fact that

𝐸𝜏𝑖,𝑏𝑖

[𝜕2𝐿(𝜃)/𝜕𝜃

𝑗𝜕𝜃𝑗] equals 𝐸

𝜏𝑖

[𝐸𝑏𝑖|𝜏𝑖

(𝜕2𝐿(𝜃)/𝜕𝜃

𝑗𝜕𝜃𝑘)].

To obtain [𝐼(𝜃)]𝑗,𝑘, the probability distribution of the

variable 𝜏𝑖which is actually a discrete random variable is

discussed as follows. It is clear that 𝜏𝑖is a positive and discrete

random variable because of the definition of thresholds inthe channel-aware position-based adaptive quantization [see(5)–(9)].There exists a unique integer𝐾 such that 𝜏

1−𝐾Δ ≥ 0

and 𝜏1− (𝐾 + 1)Δ < 0, where the initialized threshold 𝜏

1> 0

and the step size Δ > 0. The possible values of 𝜏𝑖can be

expressed as

𝜏𝑖∈ {𝜏(1), 𝜏(2), . . . , 𝜏

(𝑁+𝐾+1)} , 𝑖 ∈ {1, 2, . . . , 𝑁} , (25)

where 𝜏(𝑙) ≜ 𝜏1− 𝐾Δ + (𝑙 − 1)Δ = 𝜏

1− (𝐾 + 1 − 𝑙)Δ and

𝑙 ∈ {1, 2, . . . , 𝑁 + 𝐾 + 1}. Let 𝑃(𝜏𝑖= 𝜏(𝑙); 𝜃) denote the

probability withwhich the local sensor 𝑖 uses 𝜏(𝑙) as the quant-ization threshold. For notational convenience, let

𝑃𝑖,𝑙≜ 𝑃 (𝜏

𝑖= 𝜏(𝑙); 𝜃) , (26)

where the subscript 𝑖 ∈ {1, 2, . . . , 𝑁} and the subscript 𝑙 ∈{1, 2, . . . , 𝑁+𝐾+1}. According to formulas (25) and (26), forthe given initialized threshold 𝜏

1, 𝑃1,𝐾+1

= 𝑃(𝜏1= 𝜏(𝐾+1); 𝜃) =

𝑃(𝜏1= 𝜏1; 𝜃) = 1 and 𝑃

1,𝑙= 0 for 𝑙 = 𝐾 + 1.


Then, for the (𝑖 + 1)th sensor,

𝑃𝑖+1,𝑙=

{

{

{

𝑃𝑖,𝑙𝑃 (𝜏𝑖+1= 𝜏(𝑙)| 𝜏𝑖= 𝜏(𝑙)) + 𝑃𝑖,𝑙+1𝑃 (𝜏𝑖+1= 𝜏(𝑙)| 𝜏𝑖= 𝜏(𝑙+1)) , if 𝑙 = 1,

𝑃𝑖,𝑙−1𝑃 (𝜏𝑖+1= 𝜏(𝑙)| 𝜏𝑖= 𝜏(𝑙−1)) + 𝑃𝑖,𝑙+1𝑃 (𝜏𝑖+1= 𝜏(𝑙)| 𝜏𝑖= 𝜏(𝑙+1)) , otherwise,

(b)=

{

{

{

𝑃𝑖,𝑙𝑃 (𝑏𝑖= 0 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) + 𝑃

𝑖,𝑙+1𝑃 (𝑏𝑖= 0 | 𝜏

𝑖= 𝜏(𝑙+1); 𝜃) , if 𝑙 = 1,

𝑃𝑖,𝑙−1𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙−1); 𝜃) + 𝑃

𝑖,𝑙+1𝑃 (𝑏𝑖= 0 | 𝜏

𝑖= 𝜏(𝑙+1); 𝜃) , otherwise,

(27)

where 𝑙 ∈ {1, 2, . . . , 𝑁+𝐾}. Note (b)= follows from the fact thatthe transition probability is expressed as

𝑃 (𝜏𝑖+1= 𝜏(𝑘)| 𝜏𝑖= 𝜏(𝑙)) =

{{{{

{{{{

{

𝑃(𝑏𝑖= 0 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) , if 𝑘 − 𝑙 = 0, 𝑙 = 1, or if 𝑘 − 𝑙 = −1, 𝑙 = 1,

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) , if 𝑘 − 𝑙 = 1,

0, otherwise,

(28)

where 𝑃(𝜏𝑖+1= 𝜏(𝑘)| 𝜏𝑖= 𝜏(𝑙)) denotes the transition

probability from the state 𝜏(𝑙) to 𝜏(𝑘) when considering localsequential sensors 𝑖 and 𝑖 + 1. 𝑃(𝑏

𝑖| 𝜏𝑖= 𝜏(𝑙); 𝜃) can be

calculated by using formula (14).For example, if 𝜏

1= 1, Δ = 0.4, and𝑁 = 3, we get 𝐾 = 2

and 𝑁 + 𝐾 + 1 = 6. Additionally, 𝑙 ∈ {1, 2, . . . , 6} and 𝜏(𝑙) =𝜏1− (𝐾 + 1 − 𝑙)Δ = 1 − (3 − 𝑙) × 0.4 = 0.4𝑙 − 0.2. Obviously,

𝜏1= 𝜏(𝐾+1)

= 𝜏(3)= 1, 𝑃

1,𝐾+1= 𝑃1,3= 1 and 𝑃

1,𝑙= 0 for

𝑙 = 3. For the sequent local sensors, the values𝑃𝑖,𝑙can be easily

iteratively calculated by using formula (27). Its state diagramof the adaptive quantization threshold 𝜏

𝑖is shown in Figure 5.

Within it,𝑃𝑘;𝑗

denotes the transition probability from the state𝜏(𝑘) to 𝜏(𝑗) in formula (28). It is concluded that 𝑃

1;1+ 𝑃1;2=

𝑃2;1+ 𝑃2;3= 𝑃3;2+ 𝑃3;4= 𝑃4;3+ 𝑃4;5= 𝑃5;4+ 𝑃5;6= 1. Its

state-transition matrix is

(((((

(

𝑃1;1𝑃1;20 0 0 0

𝑃2;10 𝑃2;30 0 0

0 𝑃3;20 𝑃3;40 0

0 0 𝑃4;30 𝑃4;50

0 0 0 𝑃5;40 𝑃5;6

0 0 0 0 0 1

)))))

)

. (29)

Theorem 1. For any unbiased estimator 𝜃, the CRLB associ-ated with the proposed channel-aware position-based adaptivequantization scheme is given by

𝐶𝑅𝐿𝐵 (𝜃) = 𝐼 (𝜃)−1, (30)

in which 𝐼(𝜃) is defined in formula (21) and given as

[𝐼 (𝜃)]1,1=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1

𝑃 (𝜏𝑖= 𝜏(𝑙); 𝜃)

⋅ [

[

(1 − 2𝑝𝑒)2(𝑒−(𝜏(𝑙)−𝑎𝑖)2/𝜎2

/8𝜋𝜎2𝑎2

𝑖𝑑4

𝑖)

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]

]

,

[𝐼 (𝜃)]2,1= [𝐼 (𝜃)]1,2

=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1


⋅ [

[

(1 − 2𝑝𝑒)2((𝑥𝑖− 𝑥𝑡) /4𝜋𝜎

2𝑑4

𝑖) 𝑒−(𝜏(𝑙)−𝑎𝑖)2

/𝜎2

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]

]

,

[𝐼 (𝜃)]3,1= [𝐼 (𝜃)]1,3

=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1


⋅ [

[

(1 − 2𝑝𝑒)2((𝑦𝑖− 𝑦𝑡) /4𝜋𝜎

2𝑑4


/𝜎2

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]

]

,

[𝐼 (𝜃)]2,2=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1


⋅[

[

(1 − 2𝑝𝑒)2(𝑎2

𝑖(𝑥𝑖− 𝑥𝑡)2/2𝜋𝜎2𝑑4


/𝜎2

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]

]

,

[𝐼 (𝜃)]3,2= [𝐼 (𝜃)]2,3

=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1


⋅ [

[

(1 − 2𝑝𝑒)2(𝑎2

𝑖(𝑥𝑖− 𝑥𝑡) (𝑦𝑖− 𝑦𝑡) /2𝜋𝜎

2𝑑4


/𝜎2

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]

]

,

[𝐼 (𝜃)]3,3=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1


⋅[[

[

(1 − 2𝑝𝑒)2(𝑎2

𝑖(𝑦𝑖− 𝑦𝑡)

2

/2𝜋𝜎2𝑑4


/𝜎2

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]]

]

.

(31)


Proof. First, we derive the element [𝐼(𝜃)]1,1. According to

formulas (22) and (23), we have

[𝐼 (𝜃)]1,1=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1

𝑃 (𝜏𝑖= 𝜏(𝑙); 𝜃) 𝐺1,1(𝜏𝑖= 𝜏(𝑙), 𝜃)

=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1

𝑃 (𝜏𝑖= 𝜏(𝑙); 𝜃) 𝐸��𝑖|𝜏𝑖=𝜏(𝑙) [𝐴1,1

(𝑏𝑖, 𝜏𝑖

= 𝜏(𝑙), 𝜃)] =

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1

𝑃 (𝜏𝑖= 𝜏(𝑙); 𝜃) 𝐸��𝑖|𝜏𝑖=𝜏(𝑙)

⋅ [

[

𝑏𝑖(

(𝜕𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) /𝜕𝜃

1)

2

𝑃2(𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃)

−

𝜕2𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) /𝜕𝜃

2

1

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃)

) + (1 −𝑏𝑖)

⋅ (

(𝜕 (1 − 𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃)) /𝜕𝜃

1)

2

(1 − 𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

2

−

𝜕2(1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃)) /𝜕𝜃

2

1

1 − 𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃)

)]

]

.

(32)

We have

𝜕𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃)

𝜕𝜃1

=

𝜕 [𝑞 (𝜏(𝑙)− 𝑎𝑖) + 𝑝𝑒− 2𝑝𝑒𝑞 (𝜏(𝑙)− 𝑎𝑖)]

𝜕𝜃1

= (1 − 2𝑝𝑒)

𝑒−(𝜏(𝑙)−𝑎𝑖)2/2𝜎2

2√2𝜋𝑃0𝜎𝑑𝑖

.

(33)

According to formula (8), we have 𝐸��𝑖|𝜏𝑖=𝜏(𝑙)𝑏𝑖= 𝑃(𝑏𝑖= 1 |

𝜏𝑖= 𝜏(𝑙); 𝜃) and 𝐸

��𝑖|𝜏𝑖=𝜏(𝑙)[1 −

𝑏𝑖] = 1 − 𝑃(

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃).

Noting the fact above, we have

[𝐼 (𝜃)]1,1=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1


⋅ [

[

(𝜕𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) /𝜕𝜃

1)

2

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]

]

=

𝑁

∑

𝑖=1

𝑁+𝐾+1

∑

𝑙=1


⋅ [

[

(1 − 2𝑝𝑒)2(𝑒−(𝜏(𝑙)−𝑎𝑖)2

/𝜎28𝜋𝑃0𝜎2𝑑2

𝑖)

𝑃 (𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃) (1 − 𝑃 (

𝑏𝑖= 1 | 𝜏

𝑖= 𝜏(𝑙); 𝜃))

]

]

.

(34)

Similarly, the other elements of 𝐼(𝜃) can be derived.The proofof the theory is done.

P1;1

P1;2 P2;3 P3;4 P4;5 P5;6

P2;1 P3;2 P4;3 P5;4

𝜏1 − 2Δ 𝜏1 − Δ 𝜏1 𝜏1 + Δ 𝜏1 + 2Δ 𝜏1 + 3Δ

Figure 5: Example of a state diagram.

The theory above introduces CRLB(𝜃), and here anotherconclusion is obtained which can express its performancemore directly.Thus, to prove the effectiveness of the proposedadaptive scheme, a theoretical calculation is given throughusing the eigenvalues of matrix 𝐼(𝜃).

Lemma 2. If matrix 𝐴 is positive definite, then matrixes 𝐴−1,𝐴1, and𝐴

2are also positive definite, wherein𝐴−1 is the inverse

matrix of 𝐴 = (𝑎11𝑎12⋅⋅⋅ 𝑎1𝑛

𝑎21𝑎22⋅⋅⋅ 𝑎2𝑛

⋅⋅⋅ ⋅⋅⋅ d ⋅⋅⋅𝑎𝑛1𝑎𝑛2⋅⋅⋅ 𝑎𝑛𝑛

), 𝐴1= 𝐴+ diag(𝑡

1, 𝑡2, . . . , 𝑡

𝑛) for

𝑡𝑖> 0 (𝑖 ∈ {1, 2, . . . , 𝑛}), and 𝐴

2= (

𝜆1𝑎11𝑎12⋅⋅⋅ 𝑎

1𝑛

𝑎21𝜆2𝑎22⋅⋅⋅ 𝑎

2𝑛

⋅⋅⋅ ⋅⋅⋅ d ⋅⋅⋅

𝑎𝑛1𝑎𝑛2⋅⋅⋅ 𝜆𝑛𝑎𝑛𝑛

), for

𝜆𝑖> 1 (𝑖 ∈ {1, 2, . . . , 𝑛}) [33].

Theorem 3. Let the multiplicity of the positive definite matrix𝐴 be 1. Then, if the elements on the main diagonal of 𝐴increase, the corresponding eigenvalues of the inverse matrixof the variant are less than the corresponding eigenvalues of theinverse matrix of 𝐴.

Proof. By hypothesis, 𝐴 is positive definite and thereforethere exists an orthogonal matrix 𝑇 such that 𝑇−1𝐴𝑇 =diag(𝜆

1, 𝜆2, . . . , 𝜆

𝑛), and 𝜆

𝑖(𝑖 ∈ {1, 2, . . . , 𝑛}) is the eigenvalue

of 𝐴.Meanwhile, we have

𝑇−1𝐴1𝑇 = 𝑇

−1(𝐴 + diag (𝑡

1, 𝑡2, . . . , 𝑡

𝑛)) 𝑇

= diag (𝜆1, 𝜆2, . . . , 𝜆

𝑛) + diag (𝑡

1, 𝑡2, . . . , 𝑡

𝑛)

= diag (𝜆1+ 𝑡1, 𝜆2+ 𝑡2, . . . , 𝜆

𝑛+ 𝑡𝑛) .

(35)

By Lemma 2, we have that matrixes 𝐴−1 and 𝐴1both

are positive definite. Then, 𝐴−11

is positive definite. Let𝜂1, 𝜂2, . . . , 𝜂

𝑛and 𝜁1, 𝜁2, . . . , 𝜁

𝑛be the eigenvalues of 𝐴−1 and

𝐴−1

1, respectively.It is obvious that 𝑇−1𝐴

1𝐴−1

1𝑇 = 𝐼. It follows that

𝑇−1𝐴1𝐴−1

1𝑇 = 𝑇

−1𝐴1𝑇𝑇−1𝐴−1

1𝑇

= diag (𝜆1+ 𝑡1, 𝜆2+ 𝑡2, . . . , 𝜆

𝑛+ 𝑡𝑛)

⋅ diag (𝜁1, 𝜁2, . . . , 𝜁

𝑛) = 𝐼.

(36)

Hence,

(𝜆𝑖+ 𝑡𝑖) 𝜁𝑖= 1, 𝑖 ∈ {1, 2, . . . , 𝑛} . (37)

Similarly, we have 𝑇−1𝐴𝐴−1𝑇 = 𝐼. It follows that

𝑇−1𝐴𝐴−1𝑇 = 𝑇

−1𝐴𝑇𝑇−1𝐴−1𝑇

= diag (𝜆1, 𝜆2, . . . , 𝜆

𝑛) diag (𝜂

1, 𝜂2, . . . , 𝜂

𝑛) = 𝐼.

(38)


Hence,

𝜆𝑖𝜂𝑖= 1, 𝑖 ∈ {1, 2, . . . , 𝑛} . (39)

It therefore follows from (36) and (38) that (𝜆𝑖+

𝑡𝑖)𝜁𝑖/𝜆𝑖𝜂𝑖= 1 (𝑖 ∈ {1, 2, . . . , 𝑛}).

Hence

𝜁𝑖

𝜂𝑖

=

𝜆𝑖

(𝜆𝑖+ 𝑡𝑖)

, 𝑖 ∈ {1, 2, . . . , 𝑛} . (40)

Since 𝐴 is positive definite, 𝜆𝑖> 0. Note that 𝑡

𝑖> 0 and

then

𝜁𝑖

𝜂𝑖

=

𝜆𝑖

(𝜆𝑖+ 𝑡𝑖)

<

𝜆𝑖

𝜆𝑖

= 1, 𝑖 ∈ {1, 2, . . . , 𝑛} . (41)

Thus, 𝜁𝑖< 𝜂𝑖. Similarly, it is derived that the eigenvalues of

𝐴−1

2are less than the corresponding eigenvalues of the inverse

matrix of 𝐴. The proof is now complete.

Remarks.The proposed position-based adaptive quantizationscheme in Section 3 is easy to apply. According to (20),the MLE is found through a systematic grid search andthe lower bound on this time complexity is 𝑂(𝑁 log𝑁).However, to prove the effectiveness of the proposed adaptivescheme, obtaining the solution of CRLB(𝜃) by Theorem 1 isnot enough.More precisely, the Fisher informationmatrix 𝐼 isnot invertible when some parameters in (21) are not properlyset, such as these parameters: the number of sensors 𝑁, thecrossover probability𝑝

𝑒, sensors’ randompositions (or sensor

density), the step-size parameter Δ, and the initial threshold𝜏𝑖.TheMLE is influenced by all these parameters andhas been

shown in the following simulations.Nevertheless, Theorem 3 introduces the basic principle

for designing indexes of all local sensors of stage 1 inSection 3: 𝐼 is a positive definite matrix with the multiplicity1 when the indexes of all local sensors change. It is notedthat the elements on the main diagonal of the matrix 𝐼 arelarger than the corresponding Fisher information matrix 𝐼 ofthe fixed quantization MLE described in [21]. Additionally,Theorem 3 is a sufficient condition but not an equivalentcondition. A large number of simulations have shown thatthe indexes of all local sensors are defined according to theirdistance information, and then the corresponding matrix 𝐼 isusually positive definite with the multiplicity 1.

5. Performance Evaluation

In this section, some simulations are given to assess the per-formance of the proposed MLE in Section 4.1. Its CRLB pro-vides a benchmark (lower bound) for comparison. Our MLEis also compared with the fixed quantization MLE describedin [21]. In order to find the global optimal value in theproposed MLE, a systematic grid search, such as the sequen-tial quadratic programming (SQP), is employed to find anapproximatemaximumpoint. Additionally, it is assumed thatthe WSNs are uniformly deployed as shown in Figure 1.

Here, we consider five simulation scenarios. Scenario 1considers the performance of different quantization methods

6 12 18 24 30 36N

107

106

105

104

103

102

F-CRLBP

P-CRLBP

F-CRLBX

P-CRLBX

F-CRLBY

P-CRLBY

CRLB

forP

,X,a

ndY

Figure 6: CRLB versus the number of sensors 𝑁. 𝑃0= 64W, 𝑥

𝑡=

𝑦𝑡= 0m, 𝜎2 = 1W, 𝜏

𝐹= 𝜏1= 0.8, Δ = 0.8, and 𝑝

𝑒= 0.01.

versus the number of sensors𝑁. Scenario 2 considers CRLBsof different quantization methods versus different crossoverprobabilities 𝑝

𝑒. Scenario 3 examines the effect of the sensor

density and the initial position of the target on CRLBs ofdifferent quantizationmethods. Scenario 4 considers the per-formance of different quantization methods versus the step-size parameter Δ. Scenario 5 examines the effective range ofthe parameter (𝛼−𝜏

𝑖)2/𝜎2 and discusses the problem of initial

threshold designing. Note the prefix F and the prefix P denotethe performances for the fixed quantization scheme [21]and the proposed adaptive quantization scheme in Section 3,respectively. For example, the F-CRLB and the P-CRLBdenote the CRLBs for the fixed quantization scheme and theproposed adaptive quantization scheme, respectively. Addi-tionally, their subscripts𝑃,𝑋, and𝑌denote their correspond-ing performances for parameters 𝑃

0, 𝑥𝑡, and 𝑦

𝑡, respectively.

Scenario 1. The performances of fixed and adaptive channel-aware CRLBs are compared for different numbers of sensors.As shown in Figure 6, it is noted that the proposed channel-aware adaptive quantization scheme has a lower CRLB thanthe classical fixed quantization scheme [21]. As 𝑁 increases,both P-CRLB and F-CRLB become lower. Further, the pro-posed MLE will approach its appropriate CRLB when thenumber of sensors is large enough, as shown in Figures 6 and7.

Scenario 2. The performances of fixed and adaptive channel-aware CRLBs are compared for different crossover prob-abilities, as shown in Figure 8. It is noted that the pro-posed channel-aware adaptive quantization scheme has alower CRLB than the classical fixed quantization scheme[21] for different crossover probabilities 𝑝

𝑒. Additionally,

the performances of both fixed and adaptive channel-awarequantization schemes degrade as 𝑝

𝑒increases. However, if


6 12 18 24 30 36N

107

106

105

104

103

108

PMLEP

PMLEX

PMLEY

FMLEP

FMLEX

FMLEY

RMS

erro

rs fo

rP,X

,and

Y

Figure 7: RMS versus the number of sensors 𝑁. 𝑃0= 64W, 𝑥

𝑡=

𝑦𝑡= 0m, 𝜎2 = 1W, 𝜏

𝐹= 𝜏1= 0.8, Δ = 0.8, and 𝑝

𝑒= 0.01.

0.01 0.06 0.11 0.16 0.21 0.26 0.31 0.36 0.41 0.46

F-CRLBP

P-CRLBP

F-CRLBX

P-CRLBX

F-CRLBY

P-CRLBY

107

106

105

104

103

108

pe

CRLB

forP

,X,a

ndY

Figure 8: CRLB versus different crossover probabilities. 𝑃0= 64W,

𝑥𝑡= 𝑦𝑡= 0m, 𝜎2 = 1W, 𝜏

𝐹= 𝜏1= 0.8, Δ = 0.8, and𝑁 = 12.

parameter 𝑝𝑒is enough small, that is, the wireless channel

gets better, the proposed adaptive quantization scheme willbe quite robust to the uncertainty of quantization thresholds,similarly depicted in the distributed estimation problem [25].

Scenario 3. The performances of the adaptive channel-awareCRLB are examined for different sensor densities.The sensordensity of ROI in Figure 9 is 4 times than that in Figure 1.Within it, the suffix 2 denotes the performances for the high-density case. For example, the P-CRLB2 denote the CRLB forthe higher-density case in Figure 9, and the P-CRLB denotethe CRLB for the case in Figure 1. Figure 10 shows that a

−30 −20 −10 0 10 20 30−30

−20

−10

0

10

20

30

x-coordinate (m)

y-c

oord

inat

e (m

)

1

12

13

24

25

6

7

18

19

30

31

5

8

17

20

29

32

4

9

16

21

28

33

3

10

15

22

27

34

2

11

14

23

26

3536

SensorsTarget

Figure 9: The higher sensor-density square ROI of target localiza-tion in WSN.

6 12 18 24 30 36N

106

105

104

103

102

P-CRLBP

P-CRLB2PP-CRLBX

P-CRLB2XP-CRLBY

P-CRLB2Y

CRLB

forP

,X,a

ndY

Figure 10: CRLB versus the sensor density. 𝑃0= 64W, 𝑥

𝑡= 𝑦𝑡=

0m, 𝜎2 = 1W, 𝜏𝐹= 𝜏1= 0.8, Δ = 0.8, and 𝑝

𝑒= 0.01.

higher density can get a better performance under numbersof sensors. This is more due to the density effect. The higher-density sensors mean that the key principle in the proposedposition-based quantization scheme, that is, close neighborsof the 𝑖th sensor are assigned almost the same threshold, ismore reasonable and accurate.

To further examine the effect of the initial position of thetarget, we uniformly choose the initial position of the target as(0, 0), (10, 0), (20, 0), (30, 0), (40, 0), and (50, 0), respectively.Their performances of fixed and adaptive channel-awareCRLBs are shown in Figure 11. From this figure, we see the


104

103

102

101

F-CRLBP

P-CRLBP

F-CRLBX

P-CRLBX

F-CRLBY

P-CRLBY

0 10 20 30 40 50

xt(yt = 0)

CRLB

forP

,X,a

ndY

Figure 11: CRLB versus different locations of the target. 𝑃0= 64W,

𝑥𝑡= 𝑦𝑡= 0m, 𝜎2 = 1W, 𝜏

𝐹= 𝜏1= 0.8, Δ = 0.8, 𝑁 = 36, and

𝑝𝑒= 0.01.

104

103

105

F-CRLBP

P-CRLBP

F-CRLBX

P-CRLBX

F-CRLBY

P-CRLBY

Δ

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

CRLB

forP

,X,a

ndY

Figure 12: CRLB versus the step-size parameter Δ. 𝑃0= 64W, 𝑥

𝑡=

𝑦𝑡= 0m, 𝜎2 = 1W, 𝜏

𝐹= 𝜏1= 0.3,𝑁 = 12, and 𝑝

𝑒= 0.01.

target’s initial position has little effect on the localization per-formance of both fixed and adaptive channel-aware schemes.The target’s initial position, specially, has very little effect onthe performance for estimating 𝑦

𝑡. If the initial position of

(50, 0) is chosen, that is, the position is far from the 𝑦-axis,the performance for estimating 𝑥

𝑡will worsen slightly. The

observation is consistent withTheorem 1.

Scenario 4. The performances of fixed and adaptive channel-aware CRLBs are compared for different step-size parametersΔ, as shown in Figure 12. Firstly, the fixed channel-awareCRLB is constant under different step-size parameters. Then,

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

107

106

105

104

103

102

101

𝜎 = 1, PCRLBP

𝜎 = 0.5, PCRLBP

𝜎 = 1, PCRLBX

𝜎 = 0.5, PCRLBX

𝜎 = 1, PCRLBY

𝜎 = 0.5, PCRLBY

𝜏1

CRLB

forP

,X,a

ndP

Figure 13: CRLB as a function of 𝛼𝑖− 𝜏𝑖for different 𝜎. 𝑃

0= 128W,

𝑥𝑡= 𝑦𝑡= 0m, Δ = 0.3,𝑁 = 6, and 𝑝

𝑒= 0.01.

0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7

Valu

es

100

102

101

𝜏1

10−1

10−2

10−3

10−4

Figure 14:Themaximum andminimumof (𝛼𝑖−𝜏𝑖)2/𝜎2 for different

𝜏1. Δ and ∇ denote their values for 𝜎2 = 1W and 𝜎2 = 0.25W,

respectively. 𝑃0= 128W, 𝑥

𝑡= 𝑦𝑡= 0m, Δ = 0.3, 𝑁 = 6, and

𝑝𝑒= 0.01.

a larger Δ gets a better performance because the proposedposition-based scheme with a larger Δ can track 𝛼

𝑖more

quickly. Thus, an overestimated step size is generally chosenin practice.

Scenario 5. As claimed in Theorem 1, the adaptive channel-aware CRLB is a function of (𝛼

𝑖− 𝜏𝑖)2/𝜎2, which is one of the

key variables. As shown in Figure 13, for the given target, theCRLB increases dramaticallywhen 𝜏

𝑖is larger.Meanwhile, the

effective range of 𝜏𝑖is narrowed and the deterioration rate of

the CRLB is increased rapidly when 𝜎 is larger. Additionally,as shown in Figure 14, the curves of (𝛼

𝑖− 𝜏𝑖)2/𝜎2 show that


the theoretical variation for 𝜎 = 0.5 is larger and has moreobvious influence on the estimation performance.

6. Conclusion

In this paper, the quantization scheme in source localizationproblems under imperfect communication channels wasinvestigated. A novel adaptive quantization scheme usingone-bit data was developed and the appropriate channel-awareMLEwhich incorporates channel statistics informationat the FC was also proposed. Additionally, the CRLBs for theadaptive quantization scheme was derived to benchmark theestimation performance and then a sufficient condition forthe corresponding Fisher information matrix being positivedefinite was introduced. Finally, simulation results show thatthe MLE is effective and the crossover probability of BSCsgreatly influences the performance of the CRLB.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This work was supported by National Natural ScienceFoundation, China (61403089, 61162008, 61573153, 51205072,and 51575115), Program for Guangzhou Municipal Collegesand Universities (1201431034), Natural Science Foundationof Guangdong Province, China (2014A030310418), Founda-tion for Distinguished Young Talents in Higher Educationof Guangdong, China (2013LYM 0068), Project of DEGP(2014A010105053), and Guangzhou Science and TechnologyFoundation (nos. 2014J4100142, 2014J410023).

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Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


research article channel-aware adaptive quantization ... · zation methods [, ] all require some...

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