Factor Graph-based Multipath-assisted Indoor Passive Localization with Inaccurate Receiver

Abstract

Passive wireless devices have increasing civilian and military applications, especially in the scenario with wearable devices and Internet of Things. In this paper, we study indoor localization of a target equipped with radio-frequency identification (RFID) device in ultra-wideband (UWB) wireless networks. With known room layout, deterministic multipath components, including the line-of-sight (LOS) signal and the reflected signals via multipath propagation, are employed to locate the target with one transmitter and a single inaccurate receiver. A factor graph corresponding to the joint posterior position distribution of target and receiver is constructed. However, due to the mixed distribution in the factor node of likelihood function, the expressions of messages are intractable by directly applying belief propagation on factor graph. To this end, we approximate the messages by Gaussian distribution via minimizing the Kullback-Leibler divergence (KLD) between them. Accordingly, a parametric message passing algorithm for indoor passive localization is derived, in which only the means and variances of Gaussian distributions have to be updated. Performance of the proposed algorithm and the impact of critical parameters are evaluated by Monte Carlo simulations, which demonstrate the superior performance in localization accuracy and the robustness to the statistics of multipath channels.

1. Introduction

Context-awareness has attracted enormous interest in wireless networks. This trendency motivates the demand for precise localization. Indoor passive localization awareness plays a crucial role in many fields such as life-saving, asset tracking, environmental monitoring and privacy security [1-3]. However, locating the target in indoor multipath environment is still quite challenging. Fortunately, owing to the fine time-resolution, Ultra-wideband (UWB) signals are potential candidates for localization in harsh multipath environments [4].

Localization based on time-of-arrival (TOA) have been widely investigated. Many existing algorithms utilize the line-of-sight (LOS) signal only, which leads to positioning error in non-line-of-sight (NLOS) environment. Generally speaking, NLOS mitigation techniques can be used to reduce the impact on location accuracy [5]. However, NLOS signals, e.g., strong single reflections, contain useful information which may benefit the passive localization. The reflections signals named deterministic multipath components (MPCs) coming from walls can be used to extract additional position information and realize the multipath assisted localization with the concept of virtual anchors (VAs). With the help of VAs, it is possible to reduce the number of transmitters and receivers to obtain high localization accuracy.

Different from the active localization, to use the deterministic MPCs between the target and the receiver, Time-reversal (TR) processing [6] has been used in backscatter channels [7], which can insure the signals coming from the transmitter arrive to the target at the same TOA by calibrating the delay parameters of signals. According to the energy focused on the target by TR processing, the passive targets scatter the signals using passive transponder, e.g., radio-frequency identification (RFID) tag. After that, the RFID modulated signals received by the receiver can be distinguished from other multipath signals at the receiver side [8]. However, to the best knowledge of the authors, multipath-assisted algorithms in passive localization scenario has not been studied. Moreover, all the above passive localization researches assume that the receivers’ locations are perfectly known. In practice, however, utilizing the inaccurate position information of the receiver will lead to positioning error.

The contributions of this paper can be summarized as follows:

We propose a factor graph-based [9] algorithm which can locate the target by one transmitter and a single inaccurate receiver in multipath indoor scenario. For messages on factor graph which are difficult to be expressed and updated, Kullback-Leibler divergence (KLD) minimization is employed to obtain a low complexity parametric message passing algorithm. Simulation results show that the proposed algorithm outperforms the particle-based algorithm and approximated maximum a posteriori probability (AMAP) algorithm.

The rest of this paper is organized as follows. The system model is given in Section 2. In Section 3, factor graph is constructed and the proposed parametric message passing algorithm is derived. The performance of the proposed algorithm and the impact of critical parameters are evaluated in Section 4. Finally, the conclusions are drawn in Section 5.

 

2. Related Work

Using UWB signals in passive localization, a TOA based two-step estimation (TSE) algorithm is proposed for passive localization in [10]. Passive localization in quasi-synchronous network is investigated in [11]. In [8], a novel network architecture which jointly localizes passive tags and moving passive objects through the discrimination of their backscattered responses is studied. Different from the above algorithms with LOS signals, it is shown in [12] that deterministic multipath components (MPCs) can be used to extract additional position information and realize the multipath assisted localization with the concept of VAs. In [13], position error bound based on the equivalent Fisher information matrix (EFIM) in indoor passive localization is derived, which obtained a lower bound with the help of TR process. However, in this scenario, multipath-assisted algorithm in passive localization scenario has not been considered. Moreover, the receiver’s position uncertainty is not taking into account.

 

3. System Model

We consider a passive indoor localization scenario illustrated in Fig. 1, which consists of a transmitter, a receiver and a target. For simplicity, we assume a perfect synchronous circumstance and the floorplan is known. At the beginning, the transmitter generates signal s(t) based on the TR processing, which is composed of an assembly of complex weighted gain and time delay components of pulse p(t) .

where a expresses the complex weight determined, denotes the corresponding delay of each VATX , and denotes the corresponding delay of each VATX is the transmission delay of the LOS signal. TR processing has the intrinsic attributes to overcome the degeneration of pinhole channels caused by multipath propagation in indoor scenario, because it concentrates the radiant energy onto the target [7] after the LOS propagation’s time between the transmitter and the target.

Fig. 1.Floorplan with a transmitter (TX) node, receiver (RX) node and a subset of Virtual Receivers (i.e. VARX,i , i ∈ 2,3,4,5 ).

We assume that the MPC parameters between the transmitter and the target used in TR process in (1) are perfectly estimated [13]. Then the signal is modulated by commercial passive ultra-high frequency (UHF) RFID tag in target. At the receiver side, signal can be distinguished from other multipath signals (clutters) [8].

Let’s assume the transmitter’s position is ϕ ≜ [0 0]T and the target’s position is φ ≜ [xφ yφ]T . In a time slot, the transmitter sends signal s(t) . Then the scattered signals after RFID modulation transmitted to the receiver consists of LOS signal and reflected signals. As TR processing calibrates the delay parameters of signals among the single reflections between the transmitter’s VAs and the target, the delay equals the LOS measurement between the transmitter and target. With the use of floorplan information, single reflections can be interpreted as LOS signals between the target and VARX,i in Fig. 1, which is useful for localization. Now we get a series of measurements from the transmitter to the receiver (or VARX,i ) via the target.

The position of the receiver is θ1 ≜ [xθ1 yθ1]T and that of the virtual receiver VARX,i is θi ≜ [xθi yθi]T,i = 2,⋯,NRX, which are stacked to the vector . Since we only consider the LOS component and the single reflections between the target and the receiver as deterministic MPCs, the maximum number of deterministic components in MPCs is NRX = 5 . We assume the deterministic MPCs are orthogonal, which means no overlapping and are convenient to detection. After detection, multiplying Δtm by the signal propagation speed c , the range measurements from transmitter to receiver and virtual receiver via target can be expressed as

where Δtm is signal propagation time, Rm is the range measurement from the m-th multipath, nRm is the range measurement noise, which is zero-mean Gaussian distributed with variance

Since the deterministic MPCs may be sheltered by obstacles and the clutters may exist in indoor scenario, we cannot associate each ranging measurement {Rm} with each VARX accurately. Therefore, a probatilistic approach is an appropriate alternative, where each measurement is received by a certain virtual receiver with probability. Accordingly, the likelihood function p(Rm|θ,φ) can be expressed as

where PVA is the probability that Rm belongs to deterministic MPCs, Pv,p / Pv is the probability that Rm is the m-th multipath component, normalization factor . The former part in (3) denotes that Rm is clutter with probability (1 - PVA ) , which is uniformly distributed over the range from zero to a maximum ranging value Rmax .

Based on the assumption that the range measurements in deterministic MPCs are independent, the joint likelihood distribution p(R|θ,φ) is given by

where R ≜ [R1 R2 ⋯ RNRX]T.

Besides the above range measurements R , the receiver can also obtain the range measurements from transmitter to receiver (or virtual receiver VARX,i ), which are expressed as

where Δτi is signal propagation time between ϕ and . As we assume the receiver has a minor ambiguity in indoor scenario, it is possible to associate Di with each receiver and virtual receiver VARX,i according to the Euclidean distance constraint.

Accordingly, the joint likelihood function of the range measurement between φ and θi are given by

where D ≜ [D1 D2 ⋯ DNRX]T and p(Di|θi) is

 

4. Message Passing Algorithm for Passive Localization

Since the range measurements R and D are independent, the joint posterior distribution of positions of the target and receiver can be expressed as

Using the Bayes’ rule, we have

where p(θi) = ∫ p(θi|θ1)p(θ1)dθ1 for i ∈ {2,3,4,5}.

The corresponding posterior distributions for the target and the receiver can be calculated by marginalizing the joint distribution in (9) and (10), i.e.

where ‘ θ ╲ θ1 ’ denotes the variables in θ except θ1 .The marginalization in (11) and (12) can be efficiently calculated by message passing algorithm on factor graph. Using the factorization in (9) and (10), factor graph of the joint posterior distribution in (8) is illustrated in Fig. 2. It is seen that the factor graph contains cycles, which leads to iterative message passing algorithm.

Fig. 2.Factor Graph for the passive localization network.

The factor nodes G(θ,φ) and H(θ) denote the joint likelihood functions p(R|θ,φ) and p(D|θ) , respectively, which can be further factorized to with gRm(θ,φ) = p(Rm|θ,φ) and hDi(θi) = p(Di|θi). The prior position distributions of target φ ≜ [xφ yφ]T and receiver θ1 ≜ [xθ1 yθ1]T are assumed to be

Gaussian distributions, i.e.

where are the true positions of the target and receiver, respectively, represent position uncertainties. Assuming x-axis and y-axis of the position coordinates are independent, with

Since VARX,i,i ∈ {2,⋯,NRX} is the symmetrical mirror image of the receiver, we use the indicator function Δ1,i(θ1,θi),i ∈ {2,⋯,NRX} to represent position constraints between θ1 and θi. Therefore, p(θi|θ1) = Δ1,i (θ1,θi) can be expressed as

where L × W describe the dimensions in this room and each virtual receiver VARX,i has the same position uncertainty . In fact, the variables are the target and receiver’s positions only, which means the mirror variables θi,i ∈ {2,⋯,NRX}, can be decided by (15) and (16). Nevertheless, we draw the mirror variables θi on factor graph to ease the derivation.

Then we calculate the corresponding messages on factor graph from the top to the bottom. Firstly, the messages from θ1 to Δ1,i are given by

where θ1 = (xθ1, yθ1).

Secondly, the messages from Δ1,i to θi,i ∈ {2,3,4,5} are given by

Based on obtained in (18)-(19), the messages from factor node gRm(θ,φ) in G(θ,φ) to variable nodes φ=(xφ, yφ) are given by

The details of message passing algorithm on are shown in Appendix A.

After having all the normalized messages to the variable φ , the belief of target position can be calculated by multiplying all the incoming messages, i.e.

Substituting (20) into (21), and taking into account the prior distribution given in (13), the beliefs can be calculated, which are quite complex to be expressed and intractable to be used in the next iteration of message passing on factor graph.

As it is shown in Fig. 3(a), the multiplication in (21) has a series of local maxima, which may cause large positioning error. With the constraint of Gaussian prior distribution, the belief is illustrated in Fig. 3(b). Note that since the prior is not accurate, may still contain multiple local maxima. Nevertheless, the result in Fig. 3(b) is close to Gaussian distribution.

Fig. 3.Illustration of the target’s likelihood function, original belief and the approximated belief via KLD minimization.

Based on this observation, we are able to approximate the belief of target position by a Gaussian distribution, i.e.,

A general metric of closeness between two distributions is the Kullback-Leibler divergence (KLD) [14], which is given by

Substituting (21) and (22) into (23), yields

Then, the goal is to find in a given class R of Gaussian distribution to minimize the KLD

The minimization in (25) can be easily solved by numerical method. Then, the beliefs are approximated by Gaussian expression

which is drawn in Fig. 3(c).

Given the Gaussian approximation of the beliefs, the messages are still intractable using the standard belief propagation rules. To this end, we resort to use the belief (φ) to approximate

Then we will calculate the messages on the factor graph from the bottom to the top.

Firstly, the message from node gRm to θi in factor G can be calculated by

the message from node hDi to θi in factor H can be calculated by

The details of message passing algorithm on are shown in Appendix B.

Based on the indicator function Δ1,i (θ1,θi),i ∈ {2,⋯,NRX}, for coordinate xθi in variable θi , turns to be the message from Δ1,i to xθ1, i.e.,

where

The former part means the component attributed by measurements R , the latter part in (29) represents the component attributed by Di . Similar operation and expression is straightforward to coordinate yθi in variable θi .

Thirdly, the message from G to θ1 is

Having all the messages transmitted from the neighboring factor nodes to the variable node θ1 , we are able to calculate the belief

With the help of the prior information, substituting (29)-(31) and (43)-(44) into (31), and rearranging the results yields

where is defined as

Obviously, the expression (32) cannot be directly employed as messages on factor graph due to the complicated structure in (33). Similar to the method to , we resort to approximate by Gaussian distributions via KLD minimization, which leads to

Substituting (34) into (32), we have

where

The beliefs of target and receiver are sent to the connected factor node for next iteration. The above message updating procedure repeats until the convergence or the number of iterations reaches the maximum.

Finally, given the expressions of , we can calculate the positions of the target and receiver by MMSE estimators. The proposed algorithm is summarized in Table 1. Remark that we may receive more than NRX range measurements in one time slot in practical scenarios. In this case, suitable amount of deterministic MPCs can be selected based on the method in [15] and the proposed algorithm is still applicable. If the number of range measurements is less than three, we can drop the measurements and wait for the next time slot.

Table 1.The Proposed Passive Localization Algorithm

 

5. Simulation Results

The proposed algorithm is evaluated by Monte Carlo simulations, with parameters shown in Table 2. Considering a room with known layout. Positions of the target and receiver are initialized based on standard deviations σφ and σθ1 . The standard deviations of the range measurement noise σRm and σDi of each link are uniformly distributed. For the proposed algorithm, NKLD samples and MKLD iterations are employed to minimize the KLD in (25).

Table 2.Simulation Parameters

The cumulative distribution function (CDF) of the target positioning error is illustrated in Fig. 4. For comparison purpose, particle-based algorithm and approximated maximum a posteriori probability (AMAP) algorithm, are also evaluated. Particle-based algorithm uses particles to represent messages, which avoid the Taylor expansion of nonlinear function and Gaussian approximation of messages employed in the proposed algorithm, at the cost of huge computational complexity which is proportional to the number of particles used [16]. In AMAP method, the position estimates of target and receiver instead of the distribution functions are employed in the estimation, which means the uncertainties of positions are ignored in localization. It is seen that all the three algorithms improve the prior localization accuracy. Increasing the number of particles Np will improve the localization accuracy of the target. However, the performance gain becomes negligible with large Np . The proposed algorithm outperforms the AMAP method and the particle-based algorithm with Np = 1000 , and performs very close to the latter with Np = 6000 .

Fig. 4.CDF of the positioning error of the target

Similar CDF results can be observed in Fig. 5, which shows the receiver’s positioning error.

Fig. 5.CDF of the positioning error of the receiver

The mean squared error (MSE) performance of the proposed algorithm, AMAP and particle-based algorithm versus the number of iterations are compared in Fig. 6. We can observe that particle-based algorithm converges very fast at the cost of computational complexity. The convergence speed of the proposed algorithm is faster than that of the AMAP estimator.

Fig. 6.MSE of target positioning error versus the number of iterations

The numbers of received deterministic MPCs and clutters are random variables that are determined by the parameters in Table 2. To evaluate the impact of the two signals, localization performance with different numbers of deterministic MPCs and clutters are evaluated in Fig. 7. We can observe that the appearance of clutters will significantly degrade the performance. Moreover, when the number of received deterministic MPCs increases, the localization accuracy can be improved.

Fig. 7.Positioning error of the target with different numbers of deterministic MPCs and clutters

One of the challenges in indoor wideband localization is the data association, i.e., to associate different paths with TOA observations. However, this model in (3) intrinsically allows the same Rm to be associated with different virtual receivers, which leads to multiple local maxima in the likelihood function. This problem can be alleviated by including the constraint of target’s prior distribution. We evaluate the impact of the employed model with different prior distributions in Fig. 8. It can be observed that the performance gap between ideal data association and the employed model which does not perform data association are small. When the standard deviation of prior distribution increases, the posterior distribution may contain more local maxima which degrade the localization performance.

Fig. 8.CDF of the positioning error of the target with different prior position information

In practice, receiver may not be able to perfectly know the statistics of the multipath channel, which means the values of PVA and Pv,p may be different from the real values. To evaluate the robustness of the proposed algorithm with inaccurate channel information, we compare the localization accuracy of the proposed algorithm by varying the value of PVA in the likelihood function in Fig. 9. This scenario means that the receiver may not have the perfect knowledge whether a range measurement belongs to deterministic MPCs. It can be seen that the best localization accuracy is obtained when the value of PVA matches the real value. We can observe that when PVA employed in the likelihood function is close to the real value, e.g., PVA = 0.95, 0.85 , the performance loss is negligible. However, when significant mismatch happens, e.g., PVA = 0.1 is employed, large performance degradation can be observed due to the increased possibility that mapping the deterministic MPC as clutters and the clutters vice versa.

Fig. 9.CDF of the positioning error of the target with different parameters PVA

The robustness of the proposed algorithm to the value of Pv,p , which represents the probability that the deterministic MPC is a LOS component or a single reflection, is also evaluated in Fig. 10. Note that although Pv,p = , 0.5 p ∈ {2,3,4,5} is illustrated, since Pv,p will be normalized by Pv in (3), the different value of Pv,1 will lead to the variation of Pv,p / Pv. As we expected, it is seen that the best localization performance is obtained when the parameter Pv,1 in the likelihood function matches the one used in the model to generate LOS component. However, the performance gaps by adopting different values of Pv,1 are small, which demonstrates that the performance is not sensitive to the value of Pv,1 . The reason for this phenomenon is that the local maxima in (3) determined by different values of Pv,1 has an exponential decay due to the Gaussian prior. The true maxima constrained by the prior will have higher weight than other local maxima and the belief of target will have a smaller variance than prior information after minimizing KLD. An exception is that Pv,1 = 0 , which results in a very poor performance. This is due to the fact that, the algorithm has to adjust the mapping mode to satisfy the assumption that there is no measurement coming from a certain receiver or virtual receiver, which significantly decreases the localization performance.

Fig. 10.CDF of the positioning error of the target with different parameters Pv,p

Table 3 illustrates the computational complexities of AMAP, particle-based algorithm and the proposed algorithm. Although AMAP does not need to minimize KLD to represent belief of variable, its computational complexity is still high due to the fine grids method employed. The computational complexity of particle-based algorithm depends on the number of particles Np , which is very large in order to obtain an acceptable performance. For the proposed algorithm, simulation results show that the product MKLDNKLD is much smaller than Np to obtain the similar localization accuracy, which makes the proposed algorithm attractive in practice.

Table 3.Computational Complexity of Three Algorithms

 

6. Conclusion

In this paper, we studied an indoor passive localization of the target equipped with RFID device in UWB wireless network. Due to the fine resolution of UWB signal, deterministic multipath components (MPCs) were used to locate the target by one transmitter and one inaccurate receiver. However, due to the mixed distribution in the likelihood function and the nonlinear terms included, it was intractable to derive the messages on factor graph using belief propagation directly. We employed two methods to solve this problem. First, the nonlinear terms were linearized by Taylor expansion around the previous position estimation. Second, intractable message was approximated by Gaussian distribution via minimizing the Kullback-Leibler divergence (KLD) between them. Since only the means and variances have to be updated, the proposed algorithm significantly reduced the computational complexity. Simulation results showed that the proposed algorithm outperformed the approximated maximum a posteriori probability (AMAP) algorithm and the particle-based algorithm when the number of particles is not very large. The impact of critical parameters of the proposed algorithm and the robustness to the statistics of multipath channels were also evaluated.