Channel-Aware Adaptive Quantization Method for Source Localization in Wireless Sensor Networks

Abstract

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

1. Introduction

Wireless sensor networks (WSNs) is an emerging technology that can provide an inexpensive solution for lots of applications, including surveillance and urban monitoring [1], smart power monitoring [2], and target localization or tracking [3–8]. Locating a target or object in an area of interest using a wireless sensor network (WSN) is a typical application. The WSN usually employs low-cost densely deployed sensors that have very limited energy and communication bandwidth resources. Its local sensors detect specific events and then forward their sensed observation information to a fusion center (FC). Finally, the FC implements the task of target location estimation [9].

A multitude of studies have researched source localization since it has found lots of applications in radar, sonar, and microphone arrays [10, 11]. The localization problem is to estimate the coordinates of a source. Based on measurement models, the localization methods usually use the received signal 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 worthwhile to mention that distance information is not directly available and should be estimated based on other measurement models. On the other hand, techniques based on TOA and time-delay of arrival (TDOA) estimation usually require complicated timing or synchronization.

Compared with TOA, TDOA, distance measurement, and RSS, the localization problem based on acoustic energy has practically attracted much attention over the past few years, due to its lower cost, lower complexity, and easier implementation. In this paper, the source localization problem with the use of the energy measurement model in WSNs is considered. These energy-based methods of source localization are classified into two major groups: analog measurement models [12–18] and quantized measurement models [19–24]. Due to bandwidth constraints in WSNs, it is naturally desirable that local sensors only transmit their quantized observations to complete the estimation task. For example, consider a uniform quantizer with certain quantization levels: (1) At each sensor, it observes a received signal power following a power decay model, quantizes the signal using the uniform quantizer, and then communicates to the FC the quantized level; (2) At the FC, it reconstruct the “quantized” observations and estimates the coordinates of a source.

For these quantized measurement models, the maximum likelihood target estimator and two heuristic design methods for optimizing quantization threshold have been proposed [20]. Within them, two heuristic design methods are proven to be very robust under various situations. Additionally, the localization problem under nonideal channels between sensors and the FC has been investigated in [21]. For robust localization problems, compared against the classical maximum likelihood estimators (MLEs), the proposed fault tolerant maximum likelihood (FTML) estimator and the subtract on negative add on positive (SNAP) have been shown to be more fault tolerant in [22] and [23], respectively. Recently, Byzantine attacks for localization estimation in WSNs under binary quantized data have been considered in [24] and a novel scheme in conjunction with the identification scheme has been proposed to make the Byzantines ineffective. However, these quantized schemes only consider fixed thresholds in designing local sensors' quantization schemes, such as the cases with the single threshold and a set of different thresholds.

In other words, it is worth noting that the estimation performance for the above quantized localization problems depends on the strategies of selecting sensors' quantization thresholds, that is, fixed quantization in local sensors. For example, the single threshold or a fixed set of different thresholds have been studied in [19–21]. However, most of the works only considered the fixed quantization schemes in localization problems. Motivated by the observation that adaptive quantization design schemes in distributed estimation problems [25, 26] can outperform fixed quantization schemes [27], we propose novel kinds of quantization schemes that can adjust local sensor nodes' quantization thresholds adaptively according to sensor nodes' positions in a region of interest, that is, adaptive quantization schemes for source localization problems are adopted here.

As one of the early related works in the distributed estimation problem, a common threshold τ is applied at all the sensors [27, 28], which is called the fixed quantization approach. It is of great challenge for the approach to make a good choice of the common threshold. To be close to the unknown parameter, a further improvement is to apply a set of thresholds with equal or unequal frequencies and hopes that one of thresholds can be enough close [27, 28]. However, the multithresholding is required to have the knowledge of the prior distribution of the parameter. Recently, being the most related work, a data-dependent distributed adaptive quantization approach is considered and sensors' thresholds are adjusted dynamically to converge to the unknown parameter [25, 26].

As the early related work in the source localization problem, some prior knowledge about the region of interest is assumed and two kinds of intuitive quantization methods are designed to obtain optimal quantization thresholds [20]. Although the heuristic design methods require minimum prior information about the system, identical thresholds are simply employed at all the sensors and their performances are constrained by the fixed common thresholds. In the latest work [29], a dynamic nonidentical threshold design is proposed for making the Byzantines ineffective, where each threshold is calculated based on each sensor's signal amplitude from the source. In other words, the proposed quantization methods [20, 29] all require some prior distribution knowledge 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-bit data is proposed. Within it, local sensors' quantization thresholds are adaptively adjusted according to sensors' location coordinates and their received binary information.

(2)

The appropriate maximum likelihood estimator (MLE) is derived to seek the estimate of the location of the target, and the metric Cramér-Rao lower bound (CRLB) is derived theoretically to benchmark the performance of locating.

(3)

Some additional simulations show that the proposed channel-aware quantization scheme has better performance than the existing fixed schemes under several cases.

The rest of the paper is organized as follows. Section 2 introduces our system model. In Section 3, the position-based adaptive quantization scheme is proposed. Their corresponding channel-aware MLE and the channel-aware CRLB both are derived in Section 4. Compared against the classical MLE with the fixed quantization scheme [21], the proposed MLE 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 in a region of interest (ROI) [20]. As shown in Figure 1, the ROI is a two-dimensional square region. Local sensors are assumed to share the communication channel on a time-sharing basis [25]. Additionally, we assume that each sensor and the FC have the prior knowledge of locations of all the sensors. Without loss of generality, each sensor is deployed in a grid for simplicity. The sensors are indexed as $\{\mathrm{1,2},\dots ,N\}$ according to distances between each other. The sensor i, $\left\{i=\mathrm{1},\dots ,N\right\}$ is located at the coordinate point $\left(x_i,y_i\right)$ , where $x_i$ and $y_i$ denote the x-coordinate (m) and the y-coordinate (m) of the ith sensor, respectively. Additionally, the coordinate of the target is denoted by $\left(x_t,y_t\right)$ . For example, $\left(\mathrm{50},-\mathrm{50}\right)$ and $\left(\mathrm{0,0}\right)$ are the coordinates of the 6th sensor and the target, respectively.

Figure 1

The square ROI of target localization in WSN. The signal power contours of a target are denoted by red solid circles.

The target emits an acoustic attenuation signal, which is a function of the Euclidean distance between the target and local sensors [21]:

$$\begin{array}{cc}& a_i^{\mathrm{2}}=\frac{P_{\mathrm{0}}{d}_{\mathrm{0}}^n}{d_i^n}=\frac{P_{\mathrm{0}}}{d_i^{\mathrm{2}}},\end{array}$$

(1)

where $a_i$ , $d_i$ , and $P_{\mathrm{0}}$ are the received signal amplitude at the sensor, the Euclidean distance between the target and the ith sensor (i.e., $\sqrt{{\left(x_i-x_t\right)}^{\mathrm{2}}+{\left(y_i-y_t\right)}^{\mathrm{2}}}$ ), and the signal power emitted by the target at a reference distance $d_{\mathrm{0}}$ , respectively. Without loss of generality, $d_{\mathrm{0}}$ and n are set to $\mathrm{1}$ and $\mathrm{2}$ , respectively [20]. It is assumed that the target is at least $d_{\mathrm{0}}$ meters away from any sensors at any times.

The observation model for local sensors is considered as given below:

$$\begin{array}{cc}& z_i=a_i+{\omega }_i,\end{array}$$

(2)

where ${\omega }_i$ denotes the measurement noise process at the sensor i. $\{{\omega }_i,i=\mathrm{1},\dots ,N\}$ are assumed to be independent and identically distributed (i.i.d.) white Gaussian noises with mean zero and variance ${\sigma }^{\mathrm{2}}$ .

In this paper, (1) is shown to be a widely adopted model for the acoustic attenuation signal which is propagating in free space and (2) also is a reasonable model [20]. Without loss of generality, the same model in [20] is simply adopted here. Meanwhile, at each sensor, due to severe bandwidth limitations in WSNs, its received signal amplitude $z_i$ is quantized and then transmitted to the FC. The quantization bit budget for each sensor is supposed to be one bit per sample. The one-bit quantization process for the ith sensor is such that

$$\begin{array}{cc}& b_i=\left\{\begin{array}{cc}\mathrm{1},\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{z}_i\ge {\tau }_i,\hfill \\ \mathrm{0},\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(3)

where ${\tau }_i$ denotes the quantization threshold for the ith sensor. These binary data sequences $\left\{b_i\right\}$ are reported over wireless channels to the receiving terminals (including subsequent sensors and the FC). There are unavoidable channel imperfections which can not be neglected. The imperfect channels are assumed to be orthogonal binary symmetric channels (BSCs). Here, without loss of generality, each BSC is assumed to be characterised by the same crossover probability $p_e$ . Let $\left\{{\tilde{b}}_i\right\}$ denote the observation information at the receiving terminals after transmission through BSCs. Thus, we have

$$\begin{array}{cc}& P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{b}_i=\mathrm{1}\right)=\mathrm{1}-p_e,\\ P\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{b}_i=\mathrm{1}\right)=p_e,\\ P\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{b}_i=\mathrm{0}\right)=\mathrm{1}-p_e,\\ P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{b}_i=\mathrm{0}\right)=p_e,\end{array}$$

(4)

where $P(·)$ denotes the probability with which the event will occur. Based on the binary sequences $\{{\tilde{b}}_i\}$ above, the FC needs to estimate the parameter vector $\theta =[P_0,x_t,y_t]$ .

3. Channel-Aware Position-Based Adaptive Quantization

Generally, the thresholds $\{{\tau }_i,\hspace{0.17em}\hspace{0.17em}i=\mathrm{1},\dots ,N\}$ are the same for all the sensors [20–22]. It is well known that the CRLB exponentially increases with ${\left|{\tau }_i-a_i\right|}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ [20] for target location estimation to some extent. Thus, to obtain a better estimation result, ${\tau }_i$ is required to be set close to its received signal power $a_i$ . However, the signal power $a_i$ is unknown prior. It motivates us to propose an adaptive quantization scheme that can adaptively decrease the distance $‖{\tau }_i-a_i‖$ . Meanwhile, it is noted that the CRLB is also a function of the noise parameter σ. It is generally required that the distance $‖{\tau }_i-a_i‖$ is in the effective range which is related to σ, and the CRLB is enough small [30, 31]. In other words, the location is possibly retrieved from sensors' binary observations only when $‖{\tau }_i-a_i‖$ is in the effective range. Thus, without loss of generality, it is assumed that the distance $‖{\tau }_i-a_i‖$ is in the effective range. Some more discussions will be given in Section 5.

It is noted that $a_i$ is directly dependent on the parameter $d_i$ from (1). Thus, it is considered that the close neighbors of the ith sensor have roughly equal received signal amplitude $a_i$ . In other words, the close neighbors of the ith sensor should be assigned almost the same threshold ${\tau }_i$ . Thus, a novel position-based and adaptive quantization scheme is proposed here. Different from the adaptive quantization for distributed parameter estimation [25] being scheduled sequentially, our proposed adaptive quantization scheme is scheduled by the position-based binary sequence.

The key principle of the position-based information sequence is that the close neighbors of the ith sensor are assigned almost the same threshold. That is, if the ith sensor transmits a binary data $\{\mathrm{1}\}$ to the fusion center, its subsequent neighbor will assume that the threshold of the preceding sensor is too small and then will increase the threshold adaptively. Instead, if the ith sensor transmits a binary data $\{\mathrm{0}\}$ to the fusion center, its subsequent neighbor will assume that the threshold of the preceding sensor is a little too large and then will decrease the threshold adaptively.

The position-based and adaptive quantization scheme consists of two stages: (1) Define the indexes of all local sensors according to their distance information; (2) Update thresholds in order of indexes. For the first stage, it is noted that local sensors are indexed as $\{\mathrm{1,2},\dots ,N\}$ according to distances between each other as shown in Figure 1. The flowchart of the first-stage operation specially is shown in Figure 2. The sensor locating in the coordinate $(-\mathrm{50},-\mathrm{50})$ is indexed with 1. Then, the procedure needs to decide which sensor will be the next sensor indexed with 2 according to distance information. Within it, it selects the closest sensor with the minimum y-coordinate value, that is, the neighbor locating in the coordinate $(-\mathrm{50},-\mathrm{30})$ , and index it with 2. Similarly, the sensor locating in the coordinate $(\mathrm{50},-\mathrm{30})$ is indexed with $\mathrm{7}$ , and the sensor locating in the coordinate $(\mathrm{30},-\mathrm{30})$ is indexed with $\mathrm{8}$ , and so forth.

Figure 2

Flow chart of the first stage of the proposed adaptive quantization scheme.

For the second stage, the position-based sequence of transmission can be shown in Figure 3. It assumes that the communication channel on a time-sharing basis has a preset number of slots according to the sensors' indexed order and each sensor has received an acoustic attenuation signal. Let ${\tilde{\mathbf{B}}}_i=\left\{{\tilde{b}}_j:j\in \left[\mathrm{0},i-\mathrm{1}\right]\right\}$ denote the received binary sequence of the ith sensor, where j is an integer and ${\tilde{\mathbf{B}}}_{\mathrm{1}}$ is empty. Based the fixed threshold ${\tau }_{\mathrm{1}}$ , the 1st sensor generates a binary observation $b_{\mathrm{1}}$ :

$$\begin{array}{cc}& b_{\mathrm{1}}=\left\{\begin{array}{cc}\mathrm{1},\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{z}_{\mathrm{1}}\ge {\tau }_{\mathrm{1}},\hfill \\ \mathrm{0},\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

(5)

$b_{\mathrm{1}}$ is transmitted to the fusion center and received by the subsequent 2nd sensor over BSCs. The 2nd sensor will obtain the binary sequence ${\tilde{\mathbf{B}}}_{\mathrm{2}}$ and then calculate ${\tau }_{\mathrm{2}}$ as follows:

$$\begin{array}{cc}& {\tau }_{\mathrm{2}}=\left\{\begin{array}{cc}{\tau }_{\mathrm{1}}+\Delta ,\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}{\tilde{b}}_{\mathrm{1}}\text{\hspace{0.17em}\hspace{0.17em}equals\hspace{0.17em}\hspace{0.17em}}\mathrm{1},\hfill \\ {\tau }_{\mathrm{1}}-\Delta ,\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(6)

where Δ is a given step-size parameter. For the ith sensor, given the binary sequence ${\tilde{\mathbf{B}}}_i$ , its binary observation $b_i$ is generated as

$$\begin{array}{cc}& b_i=\left\{\begin{array}{cc}\mathrm{1},\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{z}_i\ge {\tau }_i,\hfill \\ \mathrm{0},\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(7)

where $\{{\tau }_i,\hspace{0.17em}\hspace{0.17em}i\ge \mathrm{2}\}$ is expressed as

$$\begin{array}{cc}& {\tau }_i=\left\{\begin{array}{cc}{\tau }_{i-\mathrm{1}}+\Delta ,\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{\tilde{b}}_{i-\mathrm{1}}\text{\hspace{0.17em}\hspace{0.17em}equals\hspace{0.17em}\hspace{0.17em}}\mathrm{1},\hfill \\ {\tau }_{i-\mathrm{1}}-\Delta ,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

(8)

Figure 3

Flow chart of the second stage of the proposed adaptive quantization scheme.

Note the signal power $a_i$ is always positive. ${\tau }_{\mathrm{1}}$ is assumed to be any positive value. Thus, after obtaining ${\tau }_{i+\mathrm{1}}$ (7), it is assumed that the updating procedure is required; that is,

$$\begin{array}{cc}& {\tau }_{i+\mathrm{1}}=\left\{\begin{array}{cc}{\tau }_{i+\mathrm{1}},\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}{\tau }_{i+\mathrm{1}}\ge \mathrm{0},\hfill \\ {\tau }_i,\hfill & \text{otherwise}.\hfill \end{array}\right.\end{array}$$

(9)

It is noted that formula (9) can guarantee ${\tau }_i$ is always nonnegative.

Under the setting above, a set of binary observations $\{b(n)\}$ is generated, which can be denoted as $\mathbf{B}=\left[b\left(\mathrm{1}\right)\hspace{1em}b\left(\mathrm{2}\right)\hspace{1em}\cdots \hspace{1em}b\left(N\right)\right]$ . Effects of unreliable wireless channels between sensors and the FC can not be neglected [30]. Thus, the set of binary observations B is corrupted by the imperfect wireless channels

$$\begin{array}{cc}& \tilde{\mathbf{B}}=\left[\begin{array}{cccc}\tilde{b}\left(\mathrm{1}\right)& \tilde{b}\left(\mathrm{2}\right)& \cdots & \tilde{b}\left(N\right)\end{array}\right],\end{array}$$

(10)

where $\tilde{b}(i)$ is the corrupted observation which is sent from the ith sensor to the FC. So based on $\tilde{\mathbf{B}}$ , estimating the parameter vector $\theta =[P_{\mathrm{0}}\hspace{1em}{x}_t\hspace{1em}{y}_t]$ is our following major goal. Additionally, as shown in Figure 4, the proposed adaptive quantization scheme adaptively decreases the distance $‖{\tau }_i-a_i‖$ and converges faster to the observed acoustic attenuation signals than the fixed quantization scheme [21]. Within it, ${\tau }_F$ and ${\tau }_{\mathrm{1}}$ is the fixed quantization threshold and the 1st sensor's initial threshold for the proposed adaptive quantization scheme, respectively.

Figure 4

Dynamic evolution of the quantization threshold versus sensor index. $P_{\mathrm{0}}=\mathrm{64}$  W, $x_t=y_t=\mathrm{0}$  m, ${\sigma }^{\mathrm{2}}=\mathrm{0.01}$  W, ${\tau }_F={\tau }_{\mathrm{1}}=\mathrm{1}$ , $\Delta =\mathrm{0.4}$ , and $p_e=\mathrm{0.01}$ .

4. Analysis of MLE and CRLB

4.1. MLE

Using the model of BSCs (5), the probability of a received binary observation $\{{\tilde{b}}_{\mathrm{1}}\}$ taking a specific value b ( $\mathrm{1}$ or $\mathrm{0}$ ), given the initialized threshold ${\tau }_{\mathrm{1}}$ , is expressed as

$$\begin{array}{cc}& P\left({\tilde{b}}_{\mathrm{1}}=b\hspace{0.17em}|\hspace{0.17em}{\tau }_{\mathrm{1}};\theta \right)={\displaystyle \sum }_{b_{\mathrm{1}}=\mathrm{0}}^{\mathrm{1}}P\left({\tilde{b}}_{\mathrm{1}}=b\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}};\theta \right)P\left(b_{\mathrm{1}}\hspace{0.17em}|\hspace{0.17em}{\tau }_{\mathrm{1}};\theta \right),\end{array}$$

(11)

where $P\left({\tilde{b}}_{\mathrm{1}}=b\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}};\theta \right)$ is equal to $P\left({\tilde{b}}_{\mathrm{1}}=b\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}\right)$ defined in formula (4) and $P\left(b_{\mathrm{1}}\hspace{0.17em}|\hspace{0.17em}{\tau }_{\mathrm{1}};\theta \right)$ is the probability of a quantized measurement taking a specific value conditioned on the initialized threshold ${\tau }_{\mathrm{1}}$ . According to the sensor observation model in formula (2) and the proposed channel-aware quantization model in formula (5), $P\left(b_{\mathrm{1}}\hspace{0.17em}|\hspace{0.17em}{\tau }_{\mathrm{1}};\theta \right)$ is given by

$$\begin{array}{cc}& P\left(b_{\mathrm{1}}\hspace{0.17em}|\hspace{0.17em}{\tau }_{\mathrm{1}};\theta \right)=\left\{\begin{array}{cc}q\left({\tau }_{\mathrm{1}}-a_{\mathrm{1}}\right),\hfill & b_{\mathrm{1}}=\mathrm{1},\hfill \\ \mathrm{1}-q\left({\tau }_{\mathrm{1}}-a_{\mathrm{1}}\right),\hfill & b_{\mathrm{1}}=\mathrm{0},\hfill \end{array}\right.\end{array}$$

(12)

where $q\left(·\right)$ is the complementary distribution function of the zero mean, ${\sigma }^{\mathrm{2}}$ variance Gaussian distribution defined as [21]

$$\begin{array}{cc}& q\left(x\right)={{\displaystyle \int }}_x^{+\infty }\frac{\mathrm{1}}{\sqrt{\mathrm{2}\pi }\sigma }{e}^{-x^{\mathrm{2}}/\mathrm{2}{\sigma }^{\mathrm{2}}}dx.\end{array}$$

(13)

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

$$\begin{array}{cc}& P\left({\tilde{b}}_i=b\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)={\displaystyle \sum }_{b_i=\mathrm{0}}^{\mathrm{1}}P\left({\tilde{b}}_i=b\hspace{0.17em}|\hspace{0.17em}{b}_i;\theta \right)P\left(b_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right),\end{array}$$

(14)

where $P\left(b_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)$ is the probability of a quantized measurement taking a specific value ( $b_i$ ) conditioned on the quantization threshold ${\tau }_i$ of the ith sensor. It is given by

$$\begin{array}{cc}& P\left(b_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)=\left\{\begin{array}{cc}q\left({\tau }_i-a_i\right),\hfill & b_i=\mathrm{1},\hfill \\ \mathrm{1}-q\left({\tau }_i-a_i\right),\hfill & b_i=\mathrm{0},\hfill \end{array}\right.\end{array}$$

(15)

where ${\tau }_i$ is 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 $\tilde{\mathbf{B}}$ can be expressed as

$$\begin{array}{cc}& F\left(\theta \right)={\displaystyle \prod }_{i=\mathrm{1}}^{N}f\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tilde{\mathbf{B}}}_i;\theta \right)={\displaystyle \prod }_{i=\mathrm{1}}^{N}P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right),\end{array}$$

(16)

where $f\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tilde{\mathbf{B}}}_i;\theta \right)$ denotes the conditional probability density function of ${\tilde{b}}_i$ and is equal to $P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)$ . It is noted that the variable ${\tilde{b}}_i$ being a Bernoulli random value has only $\mathrm{0}$ and $\mathrm{1}$ as possible values. According to formula (14), it is concluded that $P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)$ is given as the following:

$$\begin{array}{cc}& P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)=P{\left({\tilde{b}}_i=1\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)}^{{\tilde{b}}_i}P{\left({\tilde{b}}_i=0\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)}^{\mathrm{1}-{\tilde{b}}_i}.\end{array}$$

(17)

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

$$\begin{array}{cc}& F\left(\theta \right)={\displaystyle \prod }_{i=\mathrm{1}}^{N}P{\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)}^{{\tilde{b}}_i}P{\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)}^{\mathrm{1}-{\tilde{b}}_i}.\end{array}$$

(18)

Then its log-likelihood function is expressed as

$$\begin{array}{cc}& L\left(\theta \right)={\displaystyle \sum }_{i=\mathrm{1}}^N\left[{\tilde{b}}_i\ln \left(P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)\right)+\left(\mathrm{1}-{\tilde{b}}_i\right)\ln \left(P\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)\right)\right].\end{array}$$

(19)

According to the definition of the MLE [32], for the proposed channel-aware adaptive quantization scheme, the MLE is expressed as

$$\begin{array}{cc}& \widehat{\theta }=\arg \hspace{0.17em}\underset{\theta }{\max }\hspace{0.17em}L\left(\theta \right).\end{array}$$

(20)

It is noted that a systematic grid search, such as the sequential quadratic programming (SQP) [20], is employed to find an approximate maximum point of the global optimal value in the MLE above.

4.2. CRLB

The aforementioned section has presented the channel-aware adaptive quantization scheme. In this section, its CRLB associated with the proposed scheme is derived to determine the lower bound on the variance of any unbiased estimator and provide the best performance benchmark.

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

$$\begin{array}{cc}& \text{CRLB}\left(\theta \right)=I{\left(\theta \right)}^{-\mathrm{1}}={\left(\begin{array}{ccc}-E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{{\partial }^{\mathrm{2}}{\theta }_{\mathrm{1}}}\right]& -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_{\mathrm{1}}\partial {\theta }_{\mathrm{2}}}\right]& -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_{\mathrm{1}}\partial {\theta }_{\mathrm{3}}}\right]\\ -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_{\mathrm{2}}\partial {\theta }_{\mathrm{1}}}\right]& -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{{\partial }^{\mathrm{2}}{\theta }_{\mathrm{2}}}\right]& -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_{\mathrm{2}}\partial {\theta }_{\mathrm{3}}}\right]\\ -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_{\mathrm{3}}\partial {\theta }_{\mathrm{1}}}\right]& -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_{\mathrm{3}}\partial {\theta }_{\mathrm{2}}}\right]& -E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{{\partial }^{\mathrm{2}}{\theta }_{\mathrm{3}}}\right]\end{array}\right)}^{-\mathrm{1}},\end{array}$$

(21)

where $I(\theta )$ is the Fisher information matrix (FIM) and ${\theta }_i$ denotes the ith element of θ. According to formulas (19) and (21), we have

$$\begin{array}{cc}& \frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_j\partial {\theta }_k}=\frac{\partial \left(\partial L\left(\theta \right)/\partial {\theta }_j\right)}{\partial {\theta }_k}=-{\displaystyle \sum }_{i=\mathrm{1}}^N\left\{{\tilde{b}}_i\left(\frac{\left(\partial P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)/\partial {\theta }_j\right)\left(\partial P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)/\partial {\theta }_k\right)}{P^{\mathrm{2}}\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)}-\frac{{\partial }^{\mathrm{2}}P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)/\partial {\theta }_j\partial {\theta }_k}{P\left({\tilde{b}}_{\mathrm{i}}=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)}\right)+\left(\mathrm{1}-{\tilde{b}}_i\right)\left(\frac{\left(\partial \left(1-P\left({\tilde{b}}_i=1\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)\right)/\partial {\theta }_j\right)\left(\partial \left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)\right)/\partial {\theta }_k\right)}{{\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)\right)}^2}-\frac{\left({\partial }^{\mathrm{2}}\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)\right)/\partial {\theta }_j\partial {\theta }_k\right)}{\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)}\right)\right\}\triangleq -{\displaystyle \sum }_{i=\mathrm{1}}^N{A}_{j,k}\left({\tilde{b}}_i,{\tau }_i,\theta \right),\hspace{1em}j,k\in \left\{\mathrm{1,2},\mathrm{3}\right\}.\end{array}$$

(22)

Thus, $[I(\theta ){]}_{j,k}$ can be derived as

$$\begin{array}{cc}& {\left[I\left(\theta \right)\right]}_{j,k}=-E\left[\frac{{\partial }^{\mathrm{2}}L\left(\theta \right)}{\partial {\theta }_j\partial {\theta }_k}\right]={\displaystyle \sum }_{i=\mathrm{1}}^{N}E\left[A_{j,k}\left({\tilde{b}}_i,{\tau }_i,\theta \right)\right]\stackrel{\left(\mathrm{a}\right)}{=}{\displaystyle \sum }_{i=\mathrm{1}}^N{E}_{{\tau }_i}\left\{E_{{\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i}\left[A_{j,k}\left({\tilde{b}}_i,{\tau }_i,\theta \right)\right]\right\}\triangleq {\displaystyle \sum }_{i=\mathrm{1}}^N{E}_{{\tau }_i}{G}_{j,k}\left({\tau }_i,\theta \right),\end{array}$$

(23)

where the notation ≜ means that $G_{j,k}({\tau }_i,\theta )=E_{{\tilde{b}}_i|{\tau }_i}[A_{j,k}({\tilde{b}}_i,{\tau }_i,\theta )]$ . $E_{{\tau }_i}$ and $E_{{\tilde{b}}_i|{\tau }_i}$ denote the expectation with respect to (w.r.t.) the distribution $P\left({\tau }_i;\theta \right)$ and the expectation with respect to the conditional distribution $P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)$ . It is obvious that

$$\begin{array}{cc}& P\left({\tilde{b}}_i,{\tau }_i;\theta \right)=P\left({\tau }_i;\theta \right)P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right),\end{array}$$

(24)

where $P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i;\theta \right)$ is defined in formula (14). It is noted that $A_{j,k}({\tilde{b}}_i,{\tau }_i,\theta )$ is a function of three variables ( ${\tilde{b}}_i$ , ${\tau }_i$ , and θ) and $G_{j,k}({\tau }_i,\theta )$ is a function of two variables ( ${\tau }_i$ and θ). According to the definition of the expectation of a two-dimensional random variable, $E[{\partial }^{\mathrm{2}}L\left(\theta \right)/\partial {\theta }_j\partial {\theta }_k]$ equals $E_{{\tau }_i,b_i}[{\partial }^{\mathrm{2}}L\left(\theta \right)/{\partial \theta }_j{\partial \theta }_k]$ . Thus, $\stackrel{\left(\mathrm{a}\right)}{=}$ follows from the fact that $E_{{\tau }_i,b_i}[{\partial }^{\mathrm{2}}L\left(\theta \right)/\partial {\theta }_j\partial {\theta }_j]$ equals $E_{{\tau }_i}\left[E_{b_i|{\tau }_i}\left({\partial }^{\mathrm{2}}L\left(\theta \right)/\partial {\theta }_j\partial {\theta }_k\right)\right]$ .

To obtain ${\left[I\left(\theta \right)\right]}_{j,k}$ , the probability distribution of the variable ${\tau }_i$ which is actually a discrete random variable is discussed as follows. It is clear that ${\tau }_i$ is a positive and discrete random variable because of the definition of thresholds in the channel-aware position-based adaptive quantization [see (5)–(9)]. There exists a unique integer K such that ${\tau }_{\mathrm{1}}-K\Delta \ge \mathrm{0}$ and ${\tau }_{\mathrm{1}}-\left(K+\mathrm{1}\right)\Delta <\mathrm{0}$ , where the initialized threshold ${\tau }_{\mathrm{1}}>\mathrm{0}$ and the step size $\Delta >\mathrm{0}$ . The possible values of ${\tau }_i$ can be expressed as

$$\begin{array}{cc}& {\tau }_i\in \left\{{\tau }^{\left(\mathrm{1}\right)},{\tau }^{\left(\mathrm{2}\right)},\dots ,{\tau }^{\left(N+K+\mathrm{1}\right)}\right\},\hspace{1em}i\in \left\{\mathrm{1,2},\dots ,N\right\},\end{array}$$

(25)

where ${\tau }^{(l)}\triangleq {\tau }_{\mathrm{1}}-K\Delta +(l-\mathrm{1})\Delta ={\tau }_{\mathrm{1}}-(K+\mathrm{1}-l)\Delta$ and $l\in \{\mathrm{1,2},\dots ,N+K+\mathrm{1}\}$ . Let $P({\tau }_i={\tau }^{(l)};\theta )$ denote the probability with which the local sensor i uses ${\tau }^{(l)}$ as the quantization threshold. For notational convenience, let

$$\begin{array}{cc}& P_{i,l}\triangleq P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right),\end{array}$$

(26)

where the subscript $i\in \{\mathrm{1,2},\dots ,N\}$ and the subscript $l\in \{\mathrm{1,2},\dots ,N+K+\mathrm{1}\}$ . According to formulas (25) and (26), for the given initialized threshold ${\tau }_{\mathrm{1}}$ , $P_{\mathrm{1},K+\mathrm{1}}=P({\tau }_{\mathrm{1}}={\tau }^{(K+\mathrm{1})};\theta )=P({\tau }_{\mathrm{1}}={\tau }_{\mathrm{1}};\theta )=\mathrm{1}$ and $P_{\mathrm{1},l}=\mathrm{0}$ for $l\ne K+\mathrm{1}$ .

Then, for the $\left(i+\mathrm{1}\right)\mathrm{t}\mathrm{h}$ sensor,

$$\begin{array}{cc}& P_{i+\mathrm{1},l}=\left\{\begin{array}{cc}{P}_{i,l}P\left({\tau }_{i+\mathrm{1}}={\tau }^{\left(l\right)}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)}\right)+P_{i,l+\mathrm{1}}P\left({\tau }_{i+\mathrm{1}}={\tau }^{\left(l\right)}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l+\mathrm{1}\right)}\right),\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}l=\mathrm{1},\hfill \\ P_{i,l-\mathrm{1}}P\left({\tau }_{i+\mathrm{1}}={\tau }^{\left(l\right)}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l-\mathrm{1}\right)}\right)+P_{i,l+\mathrm{1}}P\left({\tau }_{i+\mathrm{1}}={\tau }^{\left(l\right)}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l+\mathrm{1}\right)}\right),\hfill & \text{otherwise},\hfill \end{array}\right.\\ \stackrel{\left(\mathrm{b}\right)}{=}\left\{\begin{array}{cc}{P}_{i,l}P\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)+P_{i,l+\mathrm{1}}P\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l+\mathrm{1}\right)};\theta \right),\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}l=\mathrm{1},\hfill \\ P_{i,l-\mathrm{1}}P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l-\mathrm{1}\right)};\theta \right)+P_{i,l+\mathrm{1}}P\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l+\mathrm{1}\right)};\theta \right),\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(27)

where $l\in \{\mathrm{1,2},\dots ,N+K\}$ . Note $\stackrel{\left(\mathrm{b}\right)}{=}$ follows from the fact that the transition probability is expressed as

$$\begin{array}{cc}& P\left({\tau }_{i+\mathrm{1}}={\tau }^{\left(k\right)}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)}\right)=\left\{\begin{array}{cc}P\left({\tilde{b}}_i=\mathrm{0}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right),\hfill & \text{if\hspace{0.17em}\hspace{0.17em}}k-l=\mathrm{0},l=\mathrm{1},\text{or if\hspace{0.17em}\hspace{0.17em}}k-l=-\mathrm{1},\hspace{0.17em}l\ne \mathrm{1},\hfill \\ P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right),\hfill & \text{if}k-l=\mathrm{1},\hfill \\ \mathrm{0},\hfill & \text{otherwise},\hfill \end{array}\right.\end{array}$$

(28)

where $P\left({\tau }_{i+\mathrm{1}}={\tau }^{(k)}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{(l)}\right)$ denotes the transition probability from the state ${\tau }^{(l)}$ to ${\tau }^{(k)}$ when considering local sequential sensors i and $i+\mathrm{1}$ . $P\left({\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{(l)};\theta \right)$ can be calculated by using formula (14).

For example, if ${\tau }_{\mathrm{1}}=\mathrm{1}$ , $\Delta =\mathrm{0.4}$ , and $N=\mathrm{3}$ , we get $K=\mathrm{2}$ and $N+K+\mathrm{1}=\mathrm{6}$ . Additionally, $l\in \{\mathrm{1,2},\dots ,\mathrm{6}\}$ and ${\tau }^{(l)}={\tau }_{\mathrm{1}}-(K+\mathrm{1}-l)\Delta =\mathrm{1}-(\mathrm{3}-l)\times \mathrm{0.4}=\mathrm{0.4}l-\mathrm{0.2}$ . Obviously, ${\tau }_{\mathrm{1}}={\tau }^{(K+\mathrm{1})}={\tau }^{(\mathrm{3})}=\mathrm{1}$ , $P_{\mathrm{1},K+\mathrm{1}}=P_{\mathrm{1,3}}=\mathrm{1}$ and $P_{\mathrm{1},l}=\mathrm{0}$ for $l\ne \mathrm{3}$ . For the sequent local sensors, the values $P_{i,l}$ can be easily iteratively calculated by using formula (27). Its state diagram of the adaptive quantization threshold ${\tau }_i$ is shown in Figure 5. Within it, $P_{k;j}$ denotes the transition probability from the state ${\tau }^{(k)}$ to ${\tau }^{(j)}$ in formula (28). It is concluded that $P_{\mathrm{1};\mathrm{1}}+P_{\mathrm{1};\mathrm{2}}=P_{\mathrm{2};\mathrm{1}}+P_{\mathrm{2};\mathrm{3}}=P_{\mathrm{3};\mathrm{2}}+P_{\mathrm{3};\mathrm{4}}=P_{\mathrm{4};\mathrm{3}}+P_{\mathrm{4};\mathrm{5}}=P_{\mathrm{5};\mathrm{4}}+P_{\mathrm{5};\mathrm{6}}=\mathrm{1}$ . Its state-transition matrix is

$$\begin{array}{cc}& \left(\begin{array}{cccccc}{P}_{\mathrm{1};\mathrm{1}}& P_{\mathrm{1};\mathrm{2}}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ P_{\mathrm{2};\mathrm{1}}& \mathrm{0}& P_{\mathrm{2};\mathrm{3}}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& P_{\mathrm{3};\mathrm{2}}& \mathrm{0}& P_{\mathrm{3};\mathrm{4}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& P_{\mathrm{4};\mathrm{3}}& \mathrm{0}& P_{\mathrm{4};\mathrm{5}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& P_{\mathrm{5};\mathrm{4}}& \mathrm{0}& P_{\mathrm{5};\mathrm{6}}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right).\end{array}$$

(29)

Figure 5

Example of a state diagram.

Theorem 1.

For any unbiased estimator $\widehat{\theta }$ , the CRLB associated with the proposed channel-aware position-based adaptive quantization scheme is given by

$$\begin{array}{cc}& CRLB\left(\theta \right)={I\left(\theta \right)}^{-\mathrm{1}},\end{array}$$

(30)

in which $I(\theta )$ is defined in formula (21) and given as

$$\begin{array}{cc}& {\left[I\left(\theta \right)\right]}_{\mathrm{1,1}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\mathrm{1}-\mathrm{2}{p}_e\right)}^2\left(e^{-{\left({\tau }^{\left(l\right)}-a_i\right)}^2/{\sigma }^{\mathrm{2}}}/\mathrm{8}\pi {\sigma }^{\mathrm{2}}{a}_i^{\mathrm{2}}{d}_i^{\mathrm{4}}\right)}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right],\\ {\left[I\left(\theta \right)\right]}_{\mathrm{2,1}}={\left[I\left(\theta \right)\right]}_{\mathrm{1,2}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\mathrm{1}-\mathrm{2}{p}_e\right)}^{\mathrm{2}}\left(\left(x_i-x_t\right)/\mathrm{4}\pi {\sigma }^{\mathrm{2}}{d}_i^{\mathrm{4}}\right)e^{{-\left({\tau }^{\left(l\right)}-a_i\right)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}}}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right],\\ {\left[I\left(\theta \right)\right]}_{\mathrm{3,1}}={\left[I\left(\theta \right)\right]}_{\mathrm{1,3}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\mathrm{1}-\mathrm{2}{p}_e\right)}^{\mathrm{2}}\left(\left(y_i-y_t\right)/\mathrm{4}\pi {\sigma }^{\mathrm{2}}{d}_i^{\mathrm{4}}\right)e^{{-\left({\tau }^{\left(l\right)}-a_i\right)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}}}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right],\\ {\left[I\left(\theta \right)\right]}_{\mathrm{2,2}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\mathrm{1}-\mathrm{2}{p}_e\right)}^{\mathrm{2}}\left(a_i^{\mathrm{2}}{\left(x_i-x_t\right)}^{\mathrm{2}}/\mathrm{2}\pi {\sigma }^{\mathrm{2}}{d}_i^{\mathrm{4}}\right)e^{{-\left({\tau }^{\left(l\right)}-a_i\right)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}}}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right],\\ {\left[I\left(\theta \right)\right]}_{\mathrm{3,2}}={\left[I\left(\theta \right)\right]}_{\mathrm{2,3}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\mathrm{1}-\mathrm{2}{p}_e\right)}^{\mathrm{2}}\left(a_i^{\mathrm{2}}\left(x_i-x_t\right)\left(y_i-y_t\right)/\mathrm{2}\pi {\sigma }^{\mathrm{2}}{d}_i^{\mathrm{4}}\right)e^{{-\left({\tau }^{\left(l\right)}-a_i\right)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}}}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right],\\ {\left[I\left(\theta \right)\right]}_{\mathrm{3,3}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\mathrm{1}-\mathrm{2}{p}_e\right)}^{\mathrm{2}}\left({a_i^{\mathrm{2}}\left(y_i-y_t\right)}^{\mathrm{2}}/\mathrm{2}\pi {\sigma }^{\mathrm{2}}{d}_i^{\mathrm{4}}\right)e^{{-\left({\tau }^{\left(l\right)}-a_i\right)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}}}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right].\end{array}$$

(31)

Proof.

First, we derive the element ${\left[I\left(\theta \right)\right]}_{\mathrm{1,1}}$ . According to formulas (22) and (23), we have

$$\begin{array}{cc}& {\left[I\left(\theta \right)\right]}_{\mathrm{1,1}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)G_{\mathrm{1,1}}\left({\tau }_i={\tau }^{\left(l\right)},\theta \right)={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)E_{{\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)}}\left[A_{\mathrm{1,1}}\left({\tilde{b}}_i,{\tau }_i={\tau }^{\left(l\right)},\theta \right)\right]={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)E_{{\tilde{b}}_i\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)}}\left[{\tilde{b}}_i\left(\frac{{\left(\partial P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)/\partial {\theta }_{\mathrm{1}}\right)}^{\mathrm{2}}}{P^{\mathrm{2}}\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)}-\frac{{\partial }^{\mathrm{2}}P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)/\partial {\theta }_{\mathrm{1}}^{\mathrm{2}}}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)}\right)+\left(\mathrm{1}-{\tilde{b}}_i\right)\left(\frac{{\left(\partial \left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)/\partial {\theta }_{\mathrm{1}}\right)}^{\mathrm{2}}}{{\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}^{\mathrm{2}}}-\frac{{\partial }^{\mathrm{2}}\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)/\partial {\theta }_{\mathrm{1}}^{\mathrm{2}}}{\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)}\right)\right].\end{array}$$

(32)

We have

$$\begin{array}{cc}& \frac{\partial P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)}{\partial {\theta }_{\mathrm{1}}}=\frac{\partial \left[q\left({\tau }^{\left(l\right)}-a_i\right)+p_e-\mathrm{2}{p}_{e}q\left({\tau }^{\left(l\right)}-a_i\right)\right]}{\partial {\theta }_{\mathrm{1}}}=\left(\mathrm{1}-\mathrm{2}{p}_e\right)\frac{e^{-{\left({\tau }^{\left(l\right)}-a_i\right)}^{\mathrm{2}}/\mathrm{2}{\sigma }^{\mathrm{2}}}}{\mathrm{2}\sqrt{\mathrm{2}\pi P_{\mathrm{0}}}\sigma d_i}.\end{array}$$

(33)

According to formula (8), we have $E_{{\tilde{b}}_i|{\tau }_i={\tau }^{\left(l\right)}}{\tilde{b}}_i=P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)$ and $E_{{\tilde{b}}_i|{\tau }_i={\tau }^{\left(l\right)}}\left[\mathrm{1}-{\tilde{b}}_i\right]=\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{(l)};\theta \right)$ . Noting the fact above, we have

$$\begin{array}{cc}& {\left[I\left(\theta \right)\right]}_{\mathrm{1,1}}={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\partial P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)/\partial {\theta }_{\mathrm{1}}\right)}^{\mathrm{2}}}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right]={\displaystyle \sum }_{i=\mathrm{1}}^N{\displaystyle \sum }_{l=\mathrm{1}}^{N+K+\mathrm{1}}P\left({\tau }_i={\tau }^{\left(l\right)};\theta \right)\left[\frac{{\left(\mathrm{1}-\mathrm{2}{p}_e\right)}^{\mathrm{2}}\left(e^{-{\left({\tau }^{\left(l\right)}-a_i\right)}^{\mathrm{2}}}/{\sigma }^{\mathrm{2}}\mathrm{8}\pi P_{\mathrm{0}}{\sigma }^{\mathrm{2}}{d}_i^{\mathrm{2}}\right)}{P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\left(\mathrm{1}-P\left({\tilde{b}}_i=\mathrm{1}\hspace{0.17em}|\hspace{0.17em}{\tau }_i={\tau }^{\left(l\right)};\theta \right)\right)}\right].\end{array}$$

(34)

Similarly, the other elements of $I(\theta )$ can be derived. The proof of the theory is done.

The theory above introduces $\mathrm{C}\mathrm{R}\mathrm{L}\mathrm{B}(\theta )$ , and here another conclusion is obtained which can express its performance more directly. Thus, to prove the effectiveness of the proposed adaptive scheme, a theoretical calculation is given through using the eigenvalues of matrix $I(\theta )$ .

Lemma 2.

If matrix A is positive definite, then matrixes $A^{-\mathrm{1}}$ , $A_{\mathrm{1}}$ , and $A_{\mathrm{2}}$ are also positive definite, wherein $A^{-\mathrm{1}}$ is the inverse matrix of $A=\left(\begin{array}{cccc}{a}_{\mathrm{11}}& a_{\mathrm{12}}& \cdots & a_{\mathrm{1}n}\\ a_{\mathrm{21}}& a_{\mathrm{22}}& \cdots & a_{\mathrm{2}n}\\ \cdots & \cdots & \ddots & \cdots \\ a_{n\mathrm{1}}& a_{n\mathrm{2}}& \cdots & a_{nn}\end{array}\right)$ , $A_{\mathrm{1}}=A+\mathrm{diag}\left(t_{\mathrm{1}},t_{\mathrm{2}},\dots ,t_n\right)$ for $t_i>\mathrm{0}\hspace{0.17em}\hspace{0.17em}\left(i\in \left\{\mathrm{1,2},\dots ,n\right\}\right)$ , and $A_{\mathrm{2}}=\left(\begin{array}{cccc}{\lambda }_{\mathrm{1}}{a}_{\mathrm{11}}& a_{\mathrm{12}}& \cdots & a_{\mathrm{1}n}\\ a_{\mathrm{21}}& {\lambda }_{\mathrm{2}}{a}_{\mathrm{22}}& \cdots & a_{\mathrm{2}n}\\ \cdots & \cdots & \ddots & \cdots \\ a_{n\mathrm{1}}& a_{n\mathrm{2}}& \cdots & {\lambda }_n{a}_{nn}\end{array}\right)$ , for ${\lambda }_i>\mathrm{1}\hspace{0.17em}\hspace{0.17em}\left(i\in \{\mathrm{1,2},\dots ,n\}\right)$ [33].

Theorem 3.

Let the multiplicity of the positive definite matrix A be $\mathrm{1}$ . Then, if the elements on the main diagonal of A increase, the corresponding eigenvalues of the inverse matrix of the variant are less than the corresponding eigenvalues of the inverse matrix of A.

Proof.

By hypothesis, A is positive definite and therefore there exists an orthogonal matrix T such that $T^{-\mathrm{1}}AT=\mathrm{diag}\left({\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}},\dots ,{\lambda }_n\right)$ , and ${\lambda }_i$ ( $i\in \left\{\mathrm{1,2},\dots ,n\right\}$ ) is the eigenvalue of A.

Meanwhile, we have

$$\begin{array}{cc}& T^{-\mathrm{1}}{A}_{\mathrm{1}}T=T^{-\mathrm{1}}\left(A+\mathrm{diag}\left(t_{\mathrm{1}},t_{\mathrm{2}},\dots ,t_n\right)\right)T=\mathrm{diag}\left({\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}},\dots ,{\lambda }_n\right)+\mathrm{diag}\left(t_{\mathrm{1}},t_{\mathrm{2}},\dots ,t_n\right)=\mathrm{diag}\left({\lambda }_{\mathrm{1}}+t_{\mathrm{1}},{\lambda }_{\mathrm{2}}+t_{\mathrm{2}},\dots ,{\lambda }_n+t_n\right).\end{array}$$

(35)

By Lemma 2, we have that matrixes $A^{-\mathrm{1}}$ and $A_{\mathrm{1}}$ both are positive definite. Then, $A_{\mathrm{1}}^{-\mathrm{1}}$ is positive definite. Let ${\eta }_{\mathrm{1}},{\eta }_{\mathrm{2}},\dots ,{\eta }_n$ and ${\zeta }_{\mathrm{1}},{\zeta }_{\mathrm{2}},\dots ,{\zeta }_n$ be the eigenvalues of $A^{-\mathrm{1}}$ and $A_{\mathrm{1}}^{-\mathrm{1}}$ , respectively.

It is obvious that $T^{-\mathrm{1}}{A}_{\mathrm{1}}{A}_{\mathrm{1}}^{-\mathrm{1}}T=I$ . It follows that

$$\begin{array}{cc}& T^{-\mathrm{1}}{A}_{\mathrm{1}}{A}_{\mathrm{1}}^{-\mathrm{1}}T=T^{-\mathrm{1}}{A}_{\mathrm{1}}T{T}^{-\mathrm{1}}{A}_{\mathrm{1}}^{-\mathrm{1}}T=\mathrm{diag}\left({\lambda }_{\mathrm{1}}+t_{\mathrm{1}},{\lambda }_{\mathrm{2}}+t_{\mathrm{2}},\dots ,{\lambda }_n+t_n\right)\mathrm{diag}\left({\zeta }_{\mathrm{1}},{\zeta }_{\mathrm{2}},\dots ,{\zeta }_n\right)=I.\end{array}$$

(36)

Hence,

$$\begin{array}{cc}& \left({\lambda }_i+t_i\right){\zeta }_i=\mathrm{1},\hspace{1em}i\in \left\{\mathrm{1,2},\dots ,n\right\}.\end{array}$$

(37)

Similarly, we have $T^{-\mathrm{1}}A{A}^{-\mathrm{1}}T=I$ . It follows that

$$\begin{array}{cc}& T^{-\mathrm{1}}A{A}^{-\mathrm{1}}T=T^{-\mathrm{1}}AT{T}^{-\mathrm{1}}{A}^{-\mathrm{1}}T=\mathrm{diag}\left({\lambda }_{\mathrm{1}},{\lambda }_{\mathrm{2}},\dots ,{\lambda }_n\right)\mathrm{diag}\left({\eta }_{\mathrm{1}},{\eta }_{\mathrm{2}},\dots ,{\eta }_n\right)=I.\end{array}$$

(38)

Hence,

$$\begin{array}{cc}& {\lambda }_i{\eta }_i=\mathrm{1},\hspace{1em}i\in \left\{\mathrm{1,2},\dots ,n\right\}.\end{array}$$

(39)

It therefore follows from (36) and (38) that $\left({\lambda }_i+t_i\right){\zeta }_i/{\lambda }_i{\eta }_i=\mathrm{1}\hspace{0.17em}\hspace{0.17em}\left(i\in \left\{\mathrm{1,2},\dots ,n\right\}\right)$ .

Hence

$$\begin{array}{cc}& \frac{{\zeta }_i}{{\eta }_i}=\frac{{\lambda }_i}{\left({\lambda }_i+t_i\right)},\hspace{1em}i\in \left\{\mathrm{1,2},\dots ,n\right\}.\end{array}$$

(40)

Since A is positive definite, ${\lambda }_i>\mathrm{0}$ . Note that $t_i>\mathrm{0}$ and then

$$\begin{array}{cc}& \frac{{\zeta }_i}{{\eta }_i}=\frac{{\lambda }_i}{\left({\lambda }_i+t_i\right)}<\frac{{\lambda }_i}{{\lambda }_i}=1,\hspace{1em}i\in \left\{\mathrm{1,2},\dots ,n\right\}.\end{array}$$

(41)

Thus, ${\zeta }_i<{\eta }_i$ . Similarly, it is derived that the eigenvalues of $A_{\mathrm{2}}^{-\mathrm{1}}$ are less than the corresponding eigenvalues of the inverse matrix of A. The proof is now complete.

Remarks. The proposed position-based adaptive quantization scheme in Section 3 is easy to apply. According to (20), the MLE is found through a systematic grid search and the lower bound on this time complexity is $O(N\log N)$ . However, to prove the effectiveness of the proposed adaptive scheme, obtaining the solution of $\mathrm{C}\mathrm{R}\mathrm{L}\mathrm{B}(\theta )$ by Theorem 1 is not enough. More precisely, the Fisher information matrix I is not invertible when some parameters in (21) are not properly set, such as these parameters: the number of sensors N, the crossover probability $p_e$ , sensors' random positions (or sensor density), the step-size parameter Δ, and the initial threshold ${\tau }_i$ . The MLE is influenced by all these parameters and has been shown in the following simulations.

Nevertheless, Theorem 3 introduces the basic principle for designing indexes of all local sensors of stage $\mathrm{1}$ in Section 3: I is a positive definite matrix with the multiplicity $\mathrm{1}$ when the indexes of all local sensors change. It is noted that the elements on the main diagonal of the matrix I are larger than the corresponding Fisher information matrix I of the fixed quantization MLE described in [21]. Additionally, Theorem 3 is a sufficient condition but not an equivalent condition. A large number of simulations have shown that the indexes of all local sensors are defined according to their distance information, and then the corresponding matrix I is usually positive definite with the multiplicity $\mathrm{1}$ .

5. Performance Evaluation

In this section, some simulations are given to assess the performance of the proposed MLE in Section 4.1. Its CRLB provides a benchmark (lower bound) for comparison. Our MLE is also compared with the fixed quantization MLE described in [21]. In order to find the global optimal value in the proposed MLE, a systematic grid search, such as the sequential quadratic programming (SQP), is employed to find an approximate maximum point. Additionally, it is assumed that the WSNs are uniformly deployed as shown in Figure 1.

Here, we consider five simulation scenarios. Scenario 1 considers the performance of different quantization methods versus the number of sensors N. Scenario 2 considers CRLBs of different quantization methods versus different crossover probabilities $p_e$ . Scenario 3 examines the effect of the sensor density and the initial position of the target on CRLBs of different quantization methods. Scenario 4 considers the performance of different quantization methods versus the step-size parameter Δ. Scenario 5 examines the effective range of the parameter ${\left(\alpha -{\tau }_i\right)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ and discusses the problem of initial threshold designing. Note the prefix F and the prefix P denote the 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-CRLB denote the CRLBs for the fixed quantization scheme and the proposed adaptive quantization scheme, respectively. Additionally, their subscripts P, X, and Y denote their corresponding performances for parameters $P_{\mathrm{0}}$ , $x_t$ , 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 than the classical fixed quantization scheme [21]. As N increases, both P-CRLB and F-CRLB become lower. Further, the proposed MLE will approach its appropriate CRLB when the number of sensors is large enough, as shown in Figures 6 and 7.

Figure 6

CRLB versus the number of sensors N. $P_{\mathrm{0}}=\mathrm{64}$  W, $x_t=y_t=\mathrm{0}$  m, ${\sigma }^{\mathrm{2}}=\mathrm{1}$  W, ${\tau }_F={\tau }_{\mathrm{1}}=\mathrm{0.8}$ , $\Delta =\mathrm{0.8}$ , and $p_e=\mathrm{0.01}$ .

Figure 7

RMS versus the number of sensors N. $P_{\mathrm{0}}=\mathrm{64}$  W, $x_t=y_t=\mathrm{0}$  m, ${\sigma }^{\mathrm{2}}=\mathrm{1}$  W, ${\tau }_F={\tau }_{\mathrm{1}}=\mathrm{0.8}$ , $\Delta =\mathrm{0.8}$ , and $p_e=\mathrm{0.01}$ .

Scenario 2.

The performances of fixed and adaptive channel-aware CRLBs are compared for different crossover probabilities, as shown in Figure 8. It is noted that the proposed channel-aware adaptive quantization scheme has a lower CRLB than the classical fixed quantization scheme [21] for different crossover probabilities $p_e$ . Additionally, the performances of both fixed and adaptive channel-aware quantization schemes degrade as $p_e$ increases. However, if parameter $p_e$ is enough small, that is, the wireless channel gets better, the proposed adaptive quantization scheme will be quite robust to the uncertainty of quantization thresholds, similarly depicted in the distributed estimation problem [25].

Figure 8

CRLB versus different crossover probabilities. $P_{\mathrm{0}}=\mathrm{64}$  W, $x_t=y_t=\mathrm{0}$  m, ${\sigma }^{\mathrm{2}}=\mathrm{1}$  W, ${\tau }_F={\tau }_{\mathrm{1}}=\mathrm{0.8}$ , $\Delta =\mathrm{0.8}$ , and $N=\mathrm{12}$ .

Scenario 3.

The performances of the adaptive channel-aware CRLB are examined for different sensor densities. The sensor density of ROI in Figure 9 is $\mathrm{4}$ 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 for the higher-density case in Figure 9, and the P-CRLB denote the CRLB for the case in Figure 1. Figure 10 shows that a higher density can get a better performance under numbers of sensors. This is more due to the density effect. The higher-density sensors mean that the key principle in the proposed position-based quantization scheme, that is, close neighbors of the ith sensor are assigned almost the same threshold, is more reasonable and accurate.

Figure 9

The higher sensor-density square ROI of target localization in WSN.

Figure 10

CRLB versus the sensor density. $P_{\mathrm{0}}=\mathrm{64}$  W, $x_t=y_t=\mathrm{0}$  m, ${\sigma }^{\mathrm{2}}=\mathrm{1}$  W, ${\tau }_F={\tau }_{\mathrm{1}}=\mathrm{0.8}$ , $\Delta =\mathrm{0.8}$ , and $p_e=\mathrm{0.01}$ .

To further examine the effect of the initial position of the target, we uniformly choose the initial position of the target as $(\mathrm{0,0})$ , $(\mathrm{10,0})$ , $(\mathrm{20,0})$ , $(\mathrm{30,0})$ , $(\mathrm{40,0})$ , and $(\mathrm{50,0})$ , respectively. Their performances of fixed and adaptive channel-aware CRLBs are shown in Figure 11. From this figure, we see the target's initial position has little effect on the localization performance of both fixed and adaptive channel-aware schemes. The target's initial position, specially, has very little effect on the performance for estimating $y_t$ . If the initial position of $(\mathrm{50,0})$ is chosen, that is, the position is far from the y-axis, the performance for estimating $x_t$ will worsen slightly. The observation is consistent with Theorem 1.

Figure 11

CRLB versus different locations of the target. $P_{\mathrm{0}}=\mathrm{64}$  W, $x_t=y_t=\mathrm{0}$  m, ${\sigma }^{\mathrm{2}}=\mathrm{1}$  W, ${\tau }_F={\tau }_{\mathrm{1}}=\mathrm{0.8}$ , $\Delta =\mathrm{0.8}$ , $N=\mathrm{36}$ , and $p_e=\mathrm{0.01}$ .

Scenario 4.

The performances of fixed and adaptive channel-aware CRLBs are compared for different step-size parameters Δ, as shown in Figure 12. Firstly, the fixed channel-aware CRLB is constant under different step-size parameters. Then, a larger Δ gets a better performance because the proposed position-based scheme with a larger Δ can track ${\alpha }_i$ more quickly. Thus, an overestimated step size is generally chosen in practice.

Figure 12

CRLB versus the step-size parameter Δ. $P_{\mathrm{0}}=\mathrm{64}$  W, $x_t=y_t=\mathrm{0}$  m, ${\sigma }^{\mathrm{2}}=\mathrm{1}$  W, ${\tau }_F={\tau }_{\mathrm{1}}=\mathrm{0.3}$ , $N=\mathrm{12}$ , and $p_e=\mathrm{0.01}$ .

Scenario 5.

As claimed in Theorem 1, the adaptive channel-aware CRLB is a function of $({\alpha }_i-{\tau }_i{)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ , which is one of the key variables. As shown in Figure 13, for the given target, the CRLB increases dramatically when ${\tau }_i$ is larger. Meanwhile, the effective range of ${\tau }_i$ is narrowed and the deterioration rate of the CRLB is increased rapidly when σ is larger. Additionally, as shown in Figure 14, the curves of $({\alpha }_i-{\tau }_i{)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ show that the theoretical variation for $\sigma =\mathrm{0.5}$ is larger and has more obvious influence on the estimation performance.

Figure 13

CRLB as a function of ${\alpha }_i-{\tau }_i$ for different σ. $P_{\mathrm{0}}=\mathrm{128}$  W, $x_t=y_t=\mathrm{0}$  m, $\Delta =\mathrm{0.3}$ , $N=\mathrm{6}$ , and $p_e=\mathrm{0.01}$ .

Figure 14

The maximum and minimum of $({\alpha }_i-{\tau }_i{)}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ for different ${\tau }_{\mathrm{1}}$ . Δ and ∇ denote their values for ${\sigma }^{\mathrm{2}}=\mathrm{1}$  W and ${\sigma }^{\mathrm{2}}=\mathrm{0.25}$  W, respectively. $P_{\mathrm{0}}=\mathrm{128}$  W, $x_t=y_t=\mathrm{0}$  m, $\Delta =\mathrm{0.3}$ , $N=\mathrm{6}$ , and $p_e=\mathrm{0.01}$ .

6. Conclusion

In this paper, the quantization scheme in source localization problems under imperfect communication channels was investigated. A novel adaptive quantization scheme using one-bit data was developed and the appropriate channel-aware MLE which incorporates channel statistics information at the FC was also proposed. Additionally, the CRLBs for the adaptive quantization scheme was derived to benchmark the estimation performance and then a sufficient condition for the corresponding Fisher information matrix being positive definite was introduced. Finally, simulation results show that the MLE is effective and the crossover probability of BSCs greatly influences the performance of the CRLB.