Decentralized Kalman Filtering with Multilevel Quantized Innovation in Wireless Sensor Networks

Abstract

Because of low power consumption and limited power supply significance in wireless sensor networks (WSNs), this paper studies the multilevel quantized innovation Kalman filtering (MQI-KF) for decentralized state estimation in WSNs since the MQI-KF can help to save power. In the first place, the common features of the practical low energy consumption WSNs are explored. On this basis, the new quantization scheme is presented. Besides, this paper explores the quantization state estimation by adopting the Bayesian method rather than the traditional iterated conditional expectation method. After that, this paper proposes a new decentralized state estimation algorithm (MQI-KF) for WSNs. Information entropy is analyzed to evaluate the performance of the quantization scheme. Performance analysis and simulations show that the MQI-KF is more efficient than the other decentralized Kalman filtering (KF) algorithms, and the accuracy of its estimation is close to that of the standard KF based on nonquantized measurements. Since the new quantization scheme and algorithm take into consideration the features of real WSNs which are based on the universal network protocol IEEE 802.15.4 standard, they can almost be applied into all practical WSNs with low energy consumption.

1. Introduction

Decentralized state estimation is of great importance in studying WSNs. Against this background, this paper mainly discusses the decentralized state estimation of a Gaussian Markov stochastic process in WSNs. In general, a WSN is composed of a fusion center (FC) and lots of sensors. The computation capability and energy of these sensors are quite limited. Each one of them firstly acquires noisy measurements within its sensing range. Then the measurements will be transmitted to the FC which processes them with the KF to get the state estimation with minimum mean square error (MMSE). To save energy and the communication bandwidth, the measurements should be minimized through quantizing before the transmission. This is because the quantization helps to reduce the amount of bits and the energy consumption in transmission. Hence, the focus of the research is the decentralized state estimation with quantized measurements.

1.1. Related Work

To save energy and reduce bandwidth in WSNs, several decentralized state estimation approaches with quantized measurements were put forward in [1–5]. However, one disadvantage of these algorithms is that quantizing measurements directly may lead to different levels of accuracy. This means that when the measurement value is large, there appears large quantization noise.

In order to solve this problem, the KF for quantizing innovation at multiple levels was developed in [6]; an efficient quantization scheme was presented in [7, 8]; on the basis of the quantization scheme in [7, 8], the transmission strategy and the multilevel quantization state estimation were combined together in [9], leading to better performance and less bandwidth consumption.

However, on the one hand, these quantization schemes are theoretical ones which were put forward without considering the features of the practical low power consumption WSNs. So in this paper, a new quantization scheme is proposed on the basis of analyzing these features.

On the other hand, these decentralized state estimations in [6, 9] were derived by adopting the iterated conditional expectation method directly on the condition that the prior probability density function (pdf) of the state vector is Gaussian. However, these researches did not analyze the case of the posterior pdf of the state based on the quantized innovation. By contrast, this paper studies the situation of the posterior pdf explicitly to prepare for the further study of the state estimation with quantized innovation. It innovatively utilizes a new approach for the state estimation which is a new perspective for the decentralized state estimation. By taking all these factors into consideration, a novel multilevel quantized innovation KF algorithm is proposed on the basis of the new quantization scheme and the decentralized state estimation.

1.2. Contributions of This Paper

The main contributions of this paper are listed as follows:

• (1)

It puts forward an efficient quantization scheme by flexibly utilizing different communication modes.

(2)

This paper adopts the Bayesian method for the MQI-KF instead of the iterated conditional expectation method used in [6, 9].

(3)

It puts forward a decentralized state estimation algorithm based on the new quantization scheme and innovative KF.

Model and preliminary in WSNs are presented in Section 2. Section 3 introduces the details of MQI-KF. Performance analysis and experiments are explored in Sections 4 and 5, respectively. Finally, Section 6 concludes the research in this paper.

Notation. For a discrete random variable ψ, the probability mass function is presented as $\mathrm{P}\mathrm{r}\{\psi \}$ . When a random variable x is conditioned on d, the pdf is denoted by $p(x\hspace{0.17em}|\hspace{0.17em}d)$ . This paper utilizes $p\left(x\right)=\mathcal{N}[x;\mu ,P]$ to denote the Gaussian pdf with mean μ and covariance P. $\Phi (·)$ is the standard normal distribution function; namely, $\Phi (z)={\int }_{-\infty }^z\mathcal{N}[x;\mathrm{0,1}]dx$ .

2. Model and Preliminary

This paper analyzes a discrete-time linear stochastic system in the WSN which consists of FC and N sensors. The system can be denoted by

$$\begin{array}{cc}& X_k=F_{k-\mathrm{1}}{X}_{k-\mathrm{1}}+w_{k-\mathrm{1}},\end{array}$$

(1)

$$\begin{array}{cc}& y_k^i=h_k^i{X}_k+v_k^i,\end{array}$$

(2)

where $k=\mathrm{1,2},\dots$ and $i=\mathrm{1,2},\dots ,N$ . $F_{k-\mathrm{1}}\in {\mathbf{R}}^{n\times n}$ refers to the dynamic model of the system; $X_k,w_{k-\mathrm{1}}\in {\mathbf{R}}^n$ , $y_k^i,v_k^i\in \mathbf{R}$ , and $h_k^i\in {\mathbf{R}}^{\mathrm{1}\times n}$ is the measurement model of the ith sensor which is activated at time step k. Both $w_{k-\mathrm{1}}$ and $v_k^i$ are zero mean, white, and Gaussian noises independent of each other; their covariance matrixes are $Q_{k-\mathrm{1}}\in {\mathbf{R}}^{n\times n}$ and $R_k^i\in {\mathbf{R}}^n$ , respectively.

In order to save communication bandwidth and energy, the ith activated sensor obtains the measurement $y_k^i$ and then gets the innovation ${\tilde{y}}_k^i$ through computation within the sensor. This can be computed by

$$\begin{array}{cc}& {\tilde{y}}_k^i=y_k^i-{\widehat{y}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}^i,\end{array}$$

(3)

where ${\widehat{y}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}^i$ is the one-step predicted measurement. It is transmitted from FC, which is the same with the situations in [6–9]. The ${\tilde{y}}_k^i$ is quantized in the ith activated sensor before it is sent to the FC. The quantization model is like this:

$$\begin{array}{cc}& b_k^i=q_i\left[{\tilde{y}}_k^i\right],\end{array}$$

(4)

where $q_i[·]$ is the quantization scheme; $b_k^i$ is the quantized innovation of the ith activated sensor at time step k. The ${\tilde{y}}_k^i$ can be quantized and encoded into binary data.

The quantization schemes and transmission strategies in [6–9] are only applicable for the quantized innovation $b_k^i$ . However, the real situation is that, except for the quantized innovation $b_k^i$ , the wireless data packet transmitted from the activated sensors to the FC also contains destination address, source address, Frame Check Sequence (FCS), and so forth. Thus, by considering the features in the low energy consumption WSNs in practice, this paper designs an energy-saving, efficient decentralized state estimation algorithm.

The low energy consumption WSN designed for target tracking experiments is demonstrated in Figures 1 and 2. It is constructed on the basis of the universal network protocol IEEE 802.15.4 standard [10]. This protocol is widely acknowledged as a standard technology in the WSNs with low energy consumption [11]. Figure 3 shows the format of the data packet defined by this protocol. Two remarks are listed here for introducing the decentralized state estimation algorithm in Section 3.

Figure 1

The WSN for target tracking.

Figure 2

The components of the WSN.

Figure 3

The structure of data packet in IEEE 802.15.4.

Remark 1.

This paper utilizes the round-robin mode in the protocol to avoid jamming of FC's receiver, since this mode only allows the activated sensor to transmit data packet per time step. This means that the sensor index corresponds to the time step. For instance, at time step k, $h_k^i=h_k$ , $b_k^i=b_k$ , $y_k^i=y_k$ , and ${\tilde{y}}_k^i={\tilde{y}}_k$ .

Remark 2.

Two different modes can be adopted in the transmission of the data packet from the sensors to the FC, namely, broadcast communication and peer to peer communication. Which one to choose depends on the parameter in the MHR. With the same energy management mechanism in the PHR [10], the amounts of energy they consume are the same.

3. Kalman Filtering Based on Multilevel Quantized Innovation

In this section, firstly a more efficient quantization scheme is designed for normalized innovation; secondly, an innovative quantization KF is derived by adopting the Bayesian approach. Finally, a decentralized state estimation algorithm is developed based on this quantization scheme and the new quantization KF.

3.1. Quantization Scheme Design

To quantize the scalar innovation ${\gamma }_k$ at time step k, the FC transmits the one-step prediction ${\widehat{y}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}$ and the square root of the covariance matrix of innovation ${\Omega }_k$ to the ith activated sensor

$$\begin{array}{cc}& {\widehat{y}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}=h_k{\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}},\end{array}$$

(5)

$$\begin{array}{cc}& {\Omega }_k=\sqrt{h_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T+R_k}.\end{array}$$

(6)

For the ith activated sensor, the normalized innovation ${\gamma }_k$ can be computed by

$$\begin{array}{cc}& {\gamma }_k=\frac{\left(y_k-{\widehat{y}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}\right)}{{\Omega }_k}.\end{array}$$

(7)

The quantization scheme is demonstrated as follows:

$$\begin{array}{cc}& b_k=q\left[{\gamma }_k\right]=\left\{\begin{array}{cc}-{\eta }_l^b\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{1}}\left(k\right),{\tau }_{\mathrm{2}}\left(k\right)\right]\hfill \\ -{\eta }_l^p\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{2}}\left(k\right),{\tau }_{\mathrm{3}}\left(k\right)\right]\hfill \\ ⋮\hfill & ⋮\hfill \\ -{\eta }_{\mathrm{1}}^b\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{2}l-\mathrm{1}}\left(k\right),{\tau }_{\mathrm{2}l}\left(k\right)\right]\hfill \\ -{\eta }_{\mathrm{1}}^p\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{2}l}\left(k\right),{\tau }_{\mathrm{2}l+\mathrm{1}}\left(k\right)\right]\hfill \\ \mathrm{0}\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{2}l+\mathrm{1}}\left(k\right),{\tau }_{\mathrm{2}l+\mathrm{2}}\left(k\right)\right]\hfill \\ {\eta }_{\mathrm{1}}^p\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{2}l+\mathrm{2}}\left(k\right),{\tau }_{\mathrm{2}l+\mathrm{3}}\left(k\right)\right]\hfill \\ {\eta }_{\mathrm{1}}^b\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{2}l+\mathrm{3}}\left(k\right),{\tau }_{\mathrm{2}l+\mathrm{4}}\left(k\right)\right]\hfill \\ ⋮\hfill & ⋮\hfill \\ {\eta }_l^p\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{4}l}\left(k\right),{\tau }_{\mathrm{4}l+\mathrm{1}}\left(k\right)\right]\hfill \\ {\eta }_l^b\hfill & \mathrm{i}\mathrm{f}\hspace{0.17em}\hspace{0.17em}{\gamma }_k\in \left({\tau }_{\mathrm{4}l+\mathrm{1}}\left(k\right),{\tau }_{\mathrm{4}l+\mathrm{2}}\left(k\right)\right),\hfill \end{array}\right.\end{array}$$

(8)

where $\{{\tau }_{\mathrm{1}}(k),{\tau }_{\mathrm{2}}(k),\dots ,{\tau }_{\mathrm{4}l+\mathrm{1}}(k),{\tau }_{\mathrm{4}l+\mathrm{2}}(k)\}$ are the thresholds of the scheme; ${\tau }_{\mathrm{4}l+\mathrm{2}}(k)=+\infty$ , ${\tau }_{\mathrm{1}}(k)=-\infty$ ; L is the number of the quantization levels, and $L=\mathrm{4}l+\mathrm{1}$ . When $b_k=\pm {\eta }_m^p$ ( $\mathrm{1}\le m\le l$ ), the data packet will be transmitted to FC in the peer to peer manner. And if $b_k=\pm {\eta }_m^b$ ( $\mathrm{1}\le m\le l$ ), the broadcast manner will be chosen. In the case of $b_k=\mathrm{0}$ , there will be no transmission of the data packet. After receiving the data packet from the activated sensor, the FC can determine the level of the ${\gamma }_k$ by reading data payload of the packet and the transmission mode parameter in the MHR.

For example, when $b_k\in \{{\eta }_{\mathrm{1}}^b,-{\eta }_{\mathrm{1}}^b\}$ , $b_k$ will be transmitted by one bit through broadcast mode, and if $b_k\in \{{\eta }_{\mathrm{1}}^p,-{\eta }_{\mathrm{1}}^p\}$ , it can be transmitted by one bit through peer to peer mode. Thus, when $b_k\in \{\mathrm{0},\pm {\eta }_{\mathrm{1}}^p,\pm {\eta }_{\mathrm{1}}^b\}$ , $b_k$ will be transmitted using binary data no more than one bit. Therefore, quantizing the normalized innovation $b_k$ using one bit in the ith activated sensor, there are five quantization levels using the new quantization scheme. On the other hand, there are three levels in [7–9] and two levels in [6].

It is obvious that more quantization levels contribute to more estimation accuracy. Thus, the performance of the quantization scheme proposed in this paper is better than the others, which will be seen in the performance analysis.

3.2. State Estimation by Adopting the Bayesian Approach

The state estimations in [6, 9] are based on the method of iterated conditional expectation, which can be expressed as follows:

$$\begin{array}{cc}& {\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k}=E\left[X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k\right]=E\left[E\left[X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^{k-\mathrm{1}},y_k\right]\hspace{0.17em}|\hspace{0.17em}{b}_k\right].\end{array}$$

(9)

Different from this method, this subsection studies the posterior probability density of the state $X_k$ conditioned on the quantized innovation $b_{\mathrm{1}}^k$ . Once the posterior density $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k)$ is described explicitly, the MMSE of the state estimation ${\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k}$ can be determined.

Naturally, adopting the Bayesian method, the $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k)$ can be derived through two steps.

(1) Prediction Step. From $p(X_{k-\mathrm{1}}\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^{k-\mathrm{1}})$ to $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^{k-\mathrm{1}})$ , we can obtain $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^{k-\mathrm{1}})$ by

$$\begin{array}{l}p\left(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^{k-\mathrm{1}}\right)={\left[{\left(\mathrm{2}\pi \right)}^{n/\mathrm{2}}{\left|P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}\right|}^{\mathrm{1}/\mathrm{2}}\right]}^{-\mathrm{1}}\hfill \\ ·\exp \left\{-\frac{{\left(X_k-{\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}\right)}^T{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}^{-\mathrm{1}}\left(X_k-{\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}\right)}{\mathrm{2}}\right\}.\hfill \end{array}$$

(10)

Proof.

See Appendix A.

(2) Correction Step. From $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^{k-\mathrm{1}})$ to $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k)$ , $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k)$ can be calculated as follows:

$$\begin{array}{l}p\left(X_k\hspace{0.17em}|\hspace{0.17em}{b}_1^k\right)\\ ={\left[{\left(2\pi \right)}^{\left(n+1\right)/2}{\left|\frac{P_{k\hspace{0.17em}|\hspace{0.17em}k-1}{R}_k}{{\Omega }_k^2}\right|}^{1/2}\right]}^{-1}\\ ·\frac{{\int }_{{\tau }_j}^{{\tau }_{j+1}}{e}^{-{\left(\gamma -{\widehat{\gamma }}^X\right)}^{T}P{\left({\widehat{\gamma }}^X\right)}^{-1}\left(\gamma -{\widehat{\gamma }}^X\right)/2}d\gamma \times e^{-{\tilde{X}}^T{P}_{k\hspace{0.17em}|\hspace{0.17em}k-1}^{-1}\tilde{X}/2}}{\Phi \left({\tau }_{j+1}\right)-\Phi \left({\tau }_j\right)},\end{array}$$

(11)

where $\tilde{X}=X_k-{\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}$ , ${\widehat{\gamma }}^X=h\tilde{X}/{\Omega }_k$ , and $P({\widehat{\gamma }}^X)=R_k/{\Omega }_k^{\mathrm{2}}$ . γ and ${\Omega }_k$ are defined in (6) and (7), respectively. On the basis of $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k)$ , the conditional mean ${\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k}$ can be computed by

$$\begin{array}{cc}& {\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k}={\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}+{\omega }_k\frac{P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T}{\sqrt{h_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T+R_k}},\end{array}$$

(12)

where

$$\begin{array}{l}{\omega }_k\hfill \\ ={\left[{\left(2\pi \right)}^{1/2}\right]}^{-1}\frac{\exp \left[-{\tau }_j^2\left(k\right)/2\right]-\exp \left[-{\tau }_{j+1}^2\left(k\right)/2\right]}{\Phi \left[{\tau }_{j+1}\left(k\right)\right]-\Phi \left[{\tau }_j\left(k\right)\right]}.\hfill \end{array}$$

(13)

Proof.

See Appendix B.

Beside, its conditional covariance matrix $P_{k\hspace{0.17em}|\hspace{0.17em}k}$ is demonstrated as follows:

$$\begin{array}{cc}& P_{k\hspace{0.17em}|\hspace{0.17em}k}=P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}-{\xi }_k\frac{P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T{h}_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}}{h_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T+R_k},\end{array}$$

(14)

where

$$\begin{array}{l}{\xi }_k={\omega }_k^{\mathrm{2}}-{\left[{\left(\mathrm{2}\pi \right)}^{\mathrm{1}/\mathrm{2}}\right]}^{-\mathrm{1}}\hfill \\ ·\frac{{\tau }_j\left(k\right)\exp \left[-{\tau }_j^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]-{\tau }_{j+\mathrm{1}}\left(k\right)\exp \left[-{\tau }_{j+\mathrm{1}}^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]}{\Phi \left[{\tau }_{j+\mathrm{1}}\left(k\right)\right]-\Phi \left[{\tau }_j\left(k\right)\right]}.\hfill \end{array}$$

(15)

Proof.

See Appendix C.

The multilevel quantized innovation KF is demonstrated in (12), (13), (14), and (15). In this subsection, the posterior probability density $p(X_k\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k)$ is studied explicitly. On the basis of this, the multilevel quantized innovation KF is derived. The approach adopted here is different from the iterated conditional expectation method in [6, 9], and this provides a new method for the future study of the decentralized state estimation.

3.3. Algorithm

This subsection focuses on a new decentralized state estimation algorithm. It is proposed on the basis of the new quantization scheme, the multilevel quantized innovation KF, and transmission strategy in [9] (see the following).

Decentralized state estimation algorithm is as follows.

(1) Initialization. ${\widehat{X}}_{\mathrm{0}\hspace{0.17em}|\hspace{0.17em}\mathrm{0}}$ , $P_{\mathrm{0}\hspace{0.17em}|\hspace{0.17em}\mathrm{0}}$ , L and $\{{\tau }_{\mathrm{1}}(k),{\tau }_{\mathrm{2}}(k),\dots ,{\tau }_{\mathrm{4}l+\mathrm{1}}(k),{\tau }_{\mathrm{4}l+\mathrm{2}}(k)\}$ are set in the FC.

For $i=\mathrm{1},\dots ,N$ , L and $\{{\tau }_{\mathrm{1}}(k),{\tau }_{\mathrm{2}}(k),\dots ,{\tau }_{\mathrm{4}l+\mathrm{1}}(k),{\tau }_{\mathrm{4}l+\mathrm{2}}(k)\}$ are set in the ith sensor.

(2) Estimation. When $k\ge \mathrm{1}$ one-step state prediction ${\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}$ and $P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}$ can be computed in the FC.

(2.1) Prediction Step. According to (A.3) and (A.4)

$$\begin{array}{cc}& {\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}=F_{k-\mathrm{1}}{\widehat{X}}_{k-\mathrm{1}\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}},\\ P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}=F_{k-\mathrm{1}}{P}_{k-\mathrm{1}\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{F}_{k-\mathrm{1}}^T+Q_{k-\mathrm{1}}.\end{array}$$

(16)

(2.2) Correction Step. The computations of ${\widehat{y}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}$ and ${\Omega }_k$ are conducted in the FC which transmits them to the activated sensor. Then in this sensor ${\gamma }_k$ can be computed. If $b_k\ne \mathrm{0}$ after quantization, the data packet will be transmitted from the sensor to the FC by applying the transmission strategy in [9]; if $b_k=\mathrm{0}$ , there will be no transmission of the data packet. At last, by utilizing (12), (13), (14), and (15) the decentralized state estimation can be computed

$$\begin{array}{cc}& {\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k}={\widehat{X}}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}+{\omega }_k\frac{P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T}{\sqrt{h_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T+R_k}},\\ P_{k\hspace{0.17em}|\hspace{0.17em}k}=P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}-{\xi }_k\frac{P_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T{h}_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}}{h_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-\mathrm{1}}{h}_k^T+R_k},\end{array}$$

(17)

where

$$\begin{array}{l}{\omega }_k={\left[{\left(\mathrm{2}\pi \right)}^{\mathrm{1}/\mathrm{2}}\right]}^{-\mathrm{1}}\frac{\exp \left[-{\tau }_j^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]-\exp \left[-{\tau }_{j+\mathrm{1}}^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]}{\Phi \left[{\tau }_{j+\mathrm{1}}\left(k\right)\right]-\Phi \left[{\tau }_j\left(k\right)\right]},\hfill \\ {\xi }_k={\omega }_k^{\mathrm{2}}-{\left[{\left(\mathrm{2}\pi \right)}^{\mathrm{1}/\mathrm{2}}\right]}^{-\mathrm{1}}\hfill \\ ·\frac{{\tau }_j\left(k\right)\exp \left[-{\tau }_j^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]-{\tau }_{j+\mathrm{1}}\left(k\right)\exp \left[-{\tau }_{j+\mathrm{1}}^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]}{\Phi \left[{\tau }_{j+\mathrm{1}}\left(k\right)\right]-\Phi \left[{\tau }_j\left(k\right)\right]}.\hfill \end{array}$$

(18)

Both the broadcast and peer to peer communication modes are characteristics of the WSNs which are based on the IEEE 802.15.4. Therefore, the quantization scheme and the algorithm put forward on the basis of these modes are applicable in the majority of the actual WSNs which consume less energy.

4. Performance Analysis

In this section, the focus is the performance of the MQI-KF.

4.1. Optimal Thresholds of the Quantization Scheme

The error covariance matrix correction is computed by

$$\begin{array}{cc}& \Delta P_k=P_{k\hspace{0.17em}|\hspace{0.17em}k}-P_{k\hspace{0.17em}|\hspace{0.17em}k-1}={\rho }_k\frac{P_{k\hspace{0.17em}|\hspace{0.17em}k-1}{h}_k^T{h}_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-1}}{h_k{P}_{k\hspace{0.17em}|\hspace{0.17em}k-1}{h}_k^T+R_k},\end{array}$$

(19)

where ${\rho }_k$ is the coefficient of $\Delta P_k$ . When the value of $\Delta P_k$ is maximum, the thresholds are optimal. There is a famous solution for obtaining the optimal thresholds and it is proposed in the Lloyd-Max quantization scheme [12].

4.2. Performance Analysis of the Quantization Scheme

The comparison of the performances of different quantization schemes are made on the condition that the transmission bandwidth is one bit. In this way, the comparison will not be affected by different transmission strategies.

The quantization scheme is regarded as an information source and quantized innovation $b_k$ is its output. Hence, the value of the information entropy can be the standard to evaluate different quantization schemes. The bigger the value is, the more information the $b_k$ contains. The information entropy is defined in [13]

$$\begin{array}{cc}& H={\displaystyle \sum }_{j=\mathrm{1}}^L\mathrm{Pr}\left(j\right){\log }_{\mathrm{2}}\left[\frac{\mathrm{1}}{\mathrm{Pr}\left(j\right)}\right].\end{array}$$

(20)

Since the normalized innovation ${\gamma }_k$ is a standard normal distribution ( $p(\gamma \hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^{k-\mathrm{1}})=\mathcal{N}(\gamma ;\mathrm{0,1})$ in Appendix B) and the quantization scheme in (8), its information entropy can be obtained by

$$\begin{array}{l}H=\sum _{i=1}^L{\int }_{{\tau }_i(k)}^{{\tau }_{i+1}}\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}dx\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}.{log}_2\left[{\left({\int }_{{\tau }_i(k)}^{{\tau }_{i+1}(k)}\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}dx\right)}^{-1}\right].\hfill \end{array}$$

(21)

Therefore in the case of one-bit transmission bandwidth, the information entropies of MQI-KF and [6, 7, 9] can be seen in Table 1.

Table 1

Information entropies.

	Zhang	Xu	You	Msechu
H (bit)	2.203	1.536	1.536	1

From Table 1 we can see that, compared with quantization schemes in [6, 7, 9], the quantization scheme in this paper has greater information entropy. In other words, the quantized innovation of this quantization scheme contains more information. Thus, this scheme is more efficient because the number of its quantization levels is the biggest when the bandwidth is one bit.

4.3. Performance Analysis of the State Estimation Algorithm

The coefficient of the error covariance matrix correction ${\rho }_k$ is adopted as the criterion to evaluate different quantization state estimation algorithms. For standard KF, ${\rho }_k=\mathrm{1}$ ; for the MQI-KF, ${\rho }_k$ can be obtained by referring to [6]

$$\begin{array}{l}{\rho }_k=E\left[{\rho }_k^j\hspace{0.17em}|\hspace{0.17em}{b}_{\mathrm{1}}^k\right]\hfill \\ =\frac{\mathrm{1}}{\mathrm{2}\pi }{\displaystyle \sum }_{j=\mathrm{1}}^L\frac{{\left\{\exp \left[-{\tau }_j^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]-\exp \left[-{\tau }_{j+\mathrm{1}}^{\mathrm{2}}\left(k\right)/\mathrm{2}\right]\right\}}^{\mathrm{2}}}{\Phi \left[{\tau }_{j+\mathrm{1}}\left(k\right)\right]-\Phi \left[{\tau }_j\left(k\right)\right]}.\hfill \end{array}$$

(22)

Therefore, when the c (the amount of transmission bandwidth's bytes) are the same, the values of ${\rho }_k$ of different algorithms can be seen in Table 2.

Table 2

Values of $${\rho }_k$$ of different algorithms.

c	$${\rho }_k$$
	Zhang	Xu	You	Msechu
1 bit	0.9201	0.8098	0.8098	0.6366
2 bits	0.9859	0.9560	0.9201	0.8825

According to Table 2, the algorithm put forward in this paper is the most accurate one.

5. Results of Simulation Experiments

In order to demonstrate the performance of the MQI-KF, two simulation experiments are performed. To evaluate the performances of the quantization schemes of MQI-KF and those in [6, 7, 9], the experiment is conducted with one-bit transmission bandwidth. The second experiment aims to evaluate the efficiency of MQI-KF, and the comparison is made on the basis of the same transmission bandwidth.

A simulation scenario for target tracking in the WSN is designed. The following items are detailed information about it:

• (1)

The WSN consists of a FC and N ( $N=\mathrm{50}$ ) sensors which are randomly arranged over a region of 100 m × 100 m. The coordinate of each sensor is known.

(2)

The single target in this region is regarded as a point object. The state of the target at time k is defined as $X_k={\left[\begin{array}{cccccc}{x}_k& {\dot{x}}_k& {\ddot{x}}_k& y_k& {\dot{y}}_k& {\ddot{y}}_k\end{array}\right]}^T$ . In this formula, $x_k$ and $y_k$ are the positions at the X and Y coordinates, respectively; ${\dot{x}}_k$ and ${\dot{y}}_k$ are the velocities at the two axes; ${\ddot{x}}_k$ and ${\ddot{y}}_k$ are the acceleration of the target at the X and Y coordinates, respectively.

(3)

The time step of the FC is a constant ( $T=1$  s). During this time step, the three sensors which are the nearest of the moving target will be activated. Each activated sensor measures the distance between itself and the moving target and quantizes the innovation locally. Then the data packet will be transmitted to the FC. Finally, the algorithms under this scenario are over Monte-Carlo 1000 runs in the experiments.

This paper assumes that the target moves in the 2D Cartesian coordinate system, and the dynamic model is assumed as the constant acceleration (CA) model for the state vector.

For the ith activated sensor, its coordinate is $(x_i,y_i)$ , and the measurements are modeled as

$$\begin{array}{cc}& Z_k^i=h_k^i\left(X_k\right)+v_k=\sqrt{{\left(x_k-x_i\right)}^{\mathrm{2}}+{\left(y_k-y_i\right)}^{\mathrm{2}}}+v_k,\end{array}$$

(23)

where the white Gaussian noise sequences $w_{k-\mathrm{1}}$ and $v_k$ are independent of each other and their variances are $\mathrm{v}\mathrm{a}\mathrm{r}(w_{k-\mathrm{1}})=\mathrm{0.0}{\mathrm{1}}^{\mathrm{2}}$ and $\mathrm{v}\mathrm{a}\mathrm{r}(v_k)=\mathrm{1}{\mathrm{0}}^{\mathrm{2}}$ . The measurement equation is nonlinear and is linearized in the simulation experiments. A comparison is made among the MQI-KF in this paper, the other algorithms in [6, 7, 9], and the standard KF (EKF) based on nonquantized innovation. This paper assumes that no communication loss appears in this process.

5.1. Simulation without Considering the Transmission Strategy

Figures 4 and 5 show the root mean square error (RMSE) of the positions at the X and Y coordinates, respectively. It can be seen that with the one-bit transmission bandwidth, the performance of the MQI-KF is obviously better than the algorithms in [6, 7, 9]. The reason is that the quantization scheme in this paper has more quantization levels. Therefore, it is more efficient.

Figure 4

RMSE of the positions along the x-axis for the estimators.

Figure 5

RMSE of the positions along the y-axis for the estimators.

5.2. Simulation with Transmission Strategy

Figures 6 and 7 show that the performance of the MQI-KF is obviously more accurate than those in [6, 7, 9], on the basis of the same two-bit transmission bandwidth. Besides, it is close to the standard KF (EKF) which is based on nonquantized measurements. This means that, using the same bandwidth, the MQI-KF has higher accuracy since it is based on efficient quantization scheme.

Figure 6

RMSE of the positions along the x-axis for the estimators.

Figure 7

RMSE of the positions along the y-axis for the estimators.

6. Conclusion

This paper studies an innovative decentralized state estimation algorithm MQI-KF. It is based on the quantization scheme proposed by considering the characteristics of the practical WSNs with low energy consumption. Through studying the posterior pdf, the decentralized KF is derived by adopting the Bayesian approach. On this basis, a comprehensive decentralized state estimation algorithm is put forward. The performance analysis and simulation experiments show that the MQI-KF is better than those in [6, 7, 9]. Moreover, the quantization scheme and algorithm proposed in this paper can be applied into most of the WSNs which cost less energy in practice.