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RSSDbased 3D localization of an unknown radio transmitter using weighted least square and factor graph
EURASIP Journal on Wireless Communications and Networking volume 2019, Article number: 10 (2019)
Abstract
Realizing accurate detection of an unknown radio transmitter (URT) has become a challenging problem due to its unknown parameter information. A method based on received signal strength difference (RSSD) fingerprint positioning technique and using factor graph (FG) for 2D scenario has been developed. However, the URT positioning under 3D scenario is more difficult with the large number of unknown parameters and has greater practical significance. In addition, the previous RSSDbased FG model is not accurate enough to express the relationship between the RSSD and corresponding location coordinates since the RSSD variances of reference points are different in practice. This paper proposes a more accurate 3D FG model to reduce the influence of difference of RSSD measurement variances on positioning accuracy effectively by utilizing weighted least square (WLS). With the proposed RSSDbased 3D WLSFG model and sumproduct rule, positioning process of the proposed 3D RSSDWLSFG algorithm is derived. To verify the feasibility of the proposed method, we also explores the effects of different signal receiver numbers and grid distances on positioning accuracy. The simulation experiment results show that the proposed algorithm can obtain the best positioning performance compared with the conventional K nearest neighbor (KNN) algorithm and RSSDFG algorithm under different grid distances and signal receiver numbers.
Introduction
Currently, a variety of radio signals have been widely used and existed in our daily life. To strengthen the monitoring of illegal radio spectrum resources, protect the rights and interests of legitimate users, and combat the occupation of illegal signal resources are related to the security and privacy of the nation, enterprises and individuals. A challenging issue is that realizing the accurate localization of an unknown radio transmitter (URT) is an important work in radio management.
A variety of the positioning techniques for the URT have been developed based on measurements obtained from signal receiver, which is also denoted as access point (AP). The measurements mainly include time of arrival (TOA) [1], time difference of arrival (TDOA) [2], frequency different of arrival (FDOA) [3], received signal strength (RSS) [4, 5], angle of arrival (AOA) [6], and hybrid of them [7–9]. Among all the measurement parameters, the RSSbased positioning techniques are widely used for the advantages of no extra antenna arrays, no time synchronization limitation, and low cost. A large number of RSSbased techniques have been developed in recent years. Due to the RSS information that can be converted to distance estimates for constructing a set of linear equations, a wellknown method was proposed by utilizing least square (LS) algorithm to estimate the optimal location of the target by minimizing the sum of squares of geometric distance errors between the target and AP [10]. Since the distance between the target and each AP is different, the estimated geometric distance error is not the same when using the channel model to convert RSS to geometric distance. In order to improve the robustness to the errors in the estimation of the geometric distance, weighted least squares (WLS) estimators were proposed in [11, 12]. However, due to serious signal attenuation and multipath effect, the channel model used to convert RSS and geometric distance is not enough to accurately reflect the real distance between the target and AP in the actual scenario. Therefore, the positioning accuracy will be seriously affected. The RSSbased fingerprint positioning technique does not require accurate signal propagation model and estimation of parameters such as the geometric distance between target and APs. The process of RSSbased fingerprint positioning method is mainly divided into two phases: the offline training phase and the online positioning phase. In the offline training phase, RSS measurements are collected from different access points (APs) deployed in the positioning area. Then, RSS and corresponding location coordinates construct offline fingerprint database. In the online positioning phase, the positioning target location is estimated by collecting the realtime RSS measurements and matching the fingerprint database with appropriate positioning algorithm. Two typical RSSbased fingerprint positioning techniques are RADAR [13] and LANDMARC [14], which are based on K nearest neighbor (KNN) algorithm [15]. The basic principle of KNN approach is to calculate the location of positioning target by averaging locations of K reference points with the minimum RSS Euclidean distance found in the fingerprint database. But the techniques mentioned above do not fully take into account the stochastic properties of measurement errors, and the Euclidian distance cannot reflect the geometric distance exactly. To solve this issue, the famous Maximumlikelihood (ML) [16] positioning technique was proposed to process the measurement information in the form of probability. In this way, it not only directly reflects the stochastic properties of measurement information but also achieve higher accuracy compared with the deterministic positioning methods [17, 18]. Nevertheless, the major problem of ML approach is hard to implement in practice because the ML cost function is highly nonlinear and contains multiple local minima and maxima [19]. Furthermore, MLbased algorithm is not feasible because of its high computational complexity.
Among all positioning techniques, the technique base on factor graph (FG) is famous for low computational complexity and high positioning accuracy [20]. Many kinds of FG positioning techniques based on the various measurements described above have been developed. The TOAFG [21] technique used the signal propagation time between the radio transmitter and APs to estimate the distance, but this requires the clocks of both to be synchronous. In order to be free from the limitation of clock synchronization between the radio transmitter and APs, TDOAFG method was proposed in [22]. It should be noted that radio transmitter and APs also need to be synchronized with the clock of the reference node. However, TOAFG and TDOAFG are both more suitable for line of sight (LOS) positioning scenario. The RSSFG [23] technique overcomes the requirement of clock synchronization and establishes the mathematical relationship between RSS measurement and corresponding location coordinates. Moreover, RSS information contains the result information caused by the influence of environment and device hardware, and it is also suitable for LOS and nonline of sight (NLOS) positioning environment. Yet, the RSSFG algorithm has been proved to be unable to achieve the localization of the URT, because both transmitting frequency and power of the URT are unknown. After that, a robust fingerprint database based on received signal strength difference (RSSD) was first proposed to eliminate the influence caused by the difference between the testing devices and training devices in [24]. Besides, it also mitigates the influence of hardware variations between the testing devices and the training devices. Taking advantage of this characteristic of RSSD information, Aziz et al. [25] proposed an RSSDFG technique to realize the detection of an URT. However, the mathematical model established in RSSDFG algorithm is not accurate enough, because the model considers that the RSSD measurement variance of each selected reference point is the same and ignores the difference of the RSS measurement variances among the selected reference points. To attain higher positioning accuracy, a RSSDAOA FG method with simulation was proposed in [26]. However, AOA measurements of the URT is easily affected by NLOS positioning scenario in practice.
The above positioning methods based on factor graph are all focus on 2D scenario. To the best of our knowledge, there is no existing research in the literature to realize the localization of an URT in 3D scenario. In this paper, we combine the RSSDbased FG fingerprint positioning technique and WLS method to propose a new 3D RSSDWLSFG algorithm, and it is considered to be an effective method for detecting an URT in 3D space.
The following are the contributions of this paper:
1) A more accurate RSSDbased 3D WLSFG model was first constructed by combining the FG technique with WLS for an URT, which can effectively reduce the impact of differences in variance of RSSD at different reference points on positioning accuracy.
2) On the basis of the proposed model, a new 3D RSSDWLSFG algorithm was derived by using sumproduct theorem and softinformation calculation.
3) In addition, we explored the effects of different grid distances and AP numbers on positioning accuracy and conducted simulation and experimental to verify our proposed approach.
Section 2 introduces our positioning principle and framework. The proposed 3D RSSDWLSFG positioning algorithm is presented in Section 3. In Section 4 and Section 5, the results and discussion of simulation and experiment are presented respectively. Section 6 presents the conclusion and future directions for our research.
The principle and system model
Factor graph and sumproduct algorithm
In this subsection, we introduce the spirit of factor graph and sumproduct principle at first. The main idea of factor graph and sumproduct algorithm is briefly summarized as in [27]. A factor graph is a bidirectional graph that describes how to decompose a multivariate global function into the product of multiple local functions. On the basis of the factor graph, sumproduct algorithm is developed to calculate the edge distribution of global functions by softinformation transmission in the corresponding graph model. In general, a factor graph model consists of the variable nodes (g, w_{1},..., w_{M}) and factor nodes (F, P_{1},..., P_{M}) as shown in Fig. 1. The factor node represents the local function node in related to variable nodes. Considering the fragment of factor graphs shown in Fig. 1, the factor nodes can be separated with joint distribution into groups. Here, the problem of calculating local functions is solved by using sumproduct algorithm. The sumproduct algorithm works by collecting the softinformation transmission of local function product along the path in the factor graph. The softinformation transported between the variable nodes and factor nodes represents the stochastic properties of the associated variable nodes. With several iterative processes of softinformation transported between the variable nodes and factor nodes, the solution of variables to a problem expressed by a factor graph can be easily obtained.
In this paper, we denote SI(a,b) being the soft information transported from node a to node b, which represents the statistical properties of the variable nodes and measurement errors in the form of a Gaussian probability density function \(\left (SI(a,b) \sim N\left ({m_{a,b}},\sigma _{a,b}^{2}\right)\right)\). The mean and variance of SI(a,b) are m_{a,b} and \(\sigma _{a,b}^{2}\) respectively. For example, the soft information transported from g to F can be expressed by:
where P_{k} is kth factor node. The product of some independent Gaussian distributions is still a Gaussian distribution. The mean and variance of SI(g,F) can be obtained by:
and
The soft information transported from the factor node to a variable node can be calculated from the product of a local function associated with the factor node and all the soft information of the other variable nodes as:
where f(g,w_{1},...,w_{M}) is the local function of factor node F associated with all the variable nodes.
With the above sumproduct update rules of (1)–(4), the variable node can be estimated through all the softinformation conveyed by the corresponding factor nodes. Thus, we can calculate the whole soft information of g with this method as follows:
Positioning system
The proposed 3D RSSDbased FG positioning system for a radio transmitter is described in Fig. 2. First, we place APs in the positioning area to collect RSS measurements from the radio transmitter and the positioning area is divided into several cube subareas with equal side length. The vertex of the cube is denoted as the reference point, and the side of the cube is called grid distance represented by d. In the offline training phase, A radio transmitter with a known fixed emission strength and frequency is used to traverse each reference point. Meanwhile, the RSSD measurements obtained from the sampling RSS and location coordinates of each reference point are recorded and stored to establish the fingerprint database. In our work, RSSD characteristic parameter is applied to adapt to the diversity of unknown radio emitters, since RSSD is not affected by the emission strength and frequency [28]. In this way, it greatly reduces the workload of RSSbased fingerprint positioning technique in constructing corresponding databases to match different targets. More importantly, RSSbased parameter cannot realize the localization of the URE. After that, the RSSDFG model can be obtained as done in [25]. Combining the RSSD database and RSSDFG model, we can obtain the proposed RSSDbased 3D WLSFG model by using WLS method.
In the online positioning phase, realtime RSS measurements from the positioning target will be collected and used as the input values of the proposed RSSDbased 3D WLSFG model. Finally, the estimated location of positioning target can be calculated by the sumproduct algorithm and soft information transported back and forth in the model. The detailed process of our proposed algorithm will be introduced in the next section.
Proposed 3D RSSDWLSFG algorithm
RSSDbased 3D WLSFG model
The proposed RSSDbased 3D WLSFG model also consists two kinds of nodes as shown in Fig. 3, the factor nodes (A_{1}, A_{2}, A_{t}, D_{1}, D_{2}, D_{t}, P_{1}, P_{2}, P_{i} and P_{j}) and the variable nodes (p_{1}, p_{2}, p_{i}, p_{j}, R_{1}, R_{2}, R_{t}, x, y and z), where i and j are the index number of AP (i≠j and i,j = 1,2,...,N) and t is the index number of AP combination (t = 1,2,...,N). As shown in Fig. 2, the AP combination is “ij= 12, 23, 34, 41”. First, RSS measurements of different APs are used as the input measurements (\(\widehat {p}_{1}\), \(\widehat {p}_{2}\),..., \(\widehat {p}_{i}\)). When an URT enters the positioning area, AP will collect RSS measurements and they can be expressed as:
where \(\widetilde {p}_{w,i}\) is the RSS errorfree measurement of RSS in units of watt (W) and it can be obtained by averaging a certain number of sampling measurements. The number of samples in this paper is set to 100. e_{i} represents the measurement error of ith AP in units of watt (W), and it can be expressed by zeromean Gaussian distribution \(\left (e_{i} \sim N\left (0,\sigma _{i}^{2}\right)\right)\). In order to better reflect the local linearity characteristic of RSS, it is processed in logarithmic scale, where \(\widehat {p}_{i} = 10 \cdot \log _{10}\left (\widetilde {p}_{w,i} + e_{i}\right)\). The logarithmic RSS distribution is proved to be Gaussian approximation [23]. Factor node P_{i} is to utilize the logarithmic RSS and variance of RSS measurements to generate variable node p_{i} with gaussian distribution \(\left ({p_{i}} \sim N\left (\widetilde {p}_{i},\sigma _{p_{i}}^{2}\right)\right)\), where \(\widetilde {p}_{i}\) and \(\sigma _{p_{i}}^{2}\) are the mean and variance of sampling logarithmic RSS respectively. Factor node D_{t} represents the subtraction relationship of two different APs, and it can be expressed by:
where R_{t} represents the subtraction relationship of RSS between ith AP and jth AP. Second, factor nodes transport the soft information from the variable nodes by using the simple local functions. Finally, the root variable nodes x and y combine with the soft information of all the connected factor nodes based on the sumproduct algorithm. Thus, location of the URT will be estimated with a few iterative process among the source factor nodes and variable nodes.
As known from the lognormal shadowing model [29], RSS is related to the distance from AP. Thus, the relationship between the location coordinate (x,y,z) and logarithmic RSSD (R) can formulate a linear equation, which can be expressed by:
where k_{x}, k_{y}, k_{z}, and k_{r} are the coefficients and c is a nonzero constant usually set to one. Thus, we can utilize least square (LS) approach to obtain the coefficients of Eq. (8). In this paper, five reference points are selected by using patternrecognition technique [23] and the positioning area consisted of these five reference points is defined as the subpositioning area. Since the logarithmic RSSD and the location of the five reference points are known, five linear equations in matrix form for tth AP combination are given by:
where
where k_{x,t}, k_{y,t}, k_{z,t} and k_{r,t} are coefficients of the linear equation corresponding to ith AP combination, (x_{s},y_{s}) (s=1, 2, 3, 4, 5) is the location coordinate of sth reference point, and \(\widetilde {R}_{t,s}\) is the mean logarithmic RSSD of sth reference point from tth AP combination. Here, the mean logarithmic RSSD can be obtained by averaging 100 sampling logarithmic RSSD. According to Eq. (10), the coefficients can be calculated by K=(A^{T}·A)^{−1}·A^{T}·C. Thus, the relationship between the location coordinates (x,y,z) and mean logarithmic RSSD \(\left (\widetilde {R}_{t}\right)\) of tth AP combination within the selected subpositioning area can be expressed as:
Actually, the variances of RSSD measurements at different reference points are various due to the changing scenario, multipath effect, measuring position, etc. Therefore, we utilize LS method to obtain a more accurate linear relationship between location coordinates and logarithmic RSSD, since we have acquired the relationship between the coordinates (x_{s},y_{s},z_{s}) of sth reference point and mean logarithmic RSSD \(\left (\widetilde {R}_{t,s}\right)\) of sth reference point from tth AP combination, which is represented by:
Next, according to (12), we use (x_{s},y_{s},z_{s}) to calculate the estimated mean logarithmic RSSD \(\left (\widetilde {R}_{t,s}^{'}\right)\) of sth reference point from tth AP combination and the stochastic error (E_{t,s}) between the measured value and estimated value is given by:
Thus, the approximate estimator of the stochastic error is obtained by using LS method and construct the weighted matrix (W) given by:
where
If both sides of Eq. (9) are multiplied by D^{−1}, we can obtain a new linear equation expressed by D^{−1}·A·K=D^{−1}·C. In this way, the coefficients of the new linear equation can be obtained by:
where \(\mathbf {K}^{'} = {\left [ {\begin {array}{*{20}{c}}{k_{x,t}^{'}}&{k_{y,t}^{'}}&{k_{z,t}^{'}}&{k_{r,t}^{'}}\end {array}} \right ]^{T}}\) is the coefficient matrix of the new linear equation. Now that the subpositioning area of the URT has been determined, \({\widetilde R_{t}}\) can be replaced by the variable node R_{t} of target logarithmic RSSD. Thus, the expected relationship between the location coordinates variable nodes (x,y,z) and RSSD variable node R_{t} within the choosing subpositioning area is given by:
The other linear equations corresponding to different AP combinations can also be obtained by using the same process above.
Softinformation calculation and iteration process
According to the sumproduct algorithm, we introduces how to calculate soft information and deduce the iterative process of the proposed model in this subsection. The initial soft information passing from variable nodes x, y and z to factor node A_{t} should be calculated at first. Utilizing the sumproduct rules, SI(x,A_{t}), SI(y,A_{t}), and SI(z,A_{t}) are given by:
and
Taking an example of SI(x,A_{t}), the mean and variance can be calculated by:
and
In the same way, the soft information SI(y,A_{t}) and SI(z,A_{t}) can also be obtained. From (18), the factor node A_{t} to variable node x transporting the softinformation SI(A_{t},x) can be obtained by:
and
Then, the softinformation SI(A_{t},y) and SI(A_{t},z) can be calculated with the similar manner. The softinformation transported from variable node R_{t} to factor A_{t} is equal to factor node D_{t} to variable node R_{t}, where \({m_{{R_{t}},{A_{t}}}} = {m_{{D_{t}},{R_{t}}}}\) and \(\sigma _{{R_{t}},{A_{t}}}^{2} = \sigma _{{D_{t}},{R_{t}}}^{2}\). From (7), the soft information SI(D_{t},R_{t}) is calculated by:
and
The factor nodes P_{i} and P_{j} directly transports the softinformation to node D_{t}, where \({m_{{p_{i}},{D_{t}}}} = {m_{{P_{i}},{p_{i}}}}\), \(\sigma _{{p_{i}},{D_{t}}}^{2} = \sigma _{{P_{i}},{p_{i}}}^{2}\) and \({m_{{p_{j}},{D_{t}}}} = {m_{{P_{j}},{p_{j}}}}\), \(\sigma _{{p_{j}},{D_{t}}}^{2} = \sigma _{{P_{j}},{p_{j}}}^{2}\), respectively. According to (6), SI(P_{i},p_{i}) and SI(P_{j},p_{j}) can be directly obtained. In the same way, the softinformation of other AP combinations can also be calculated. As mentioned above, all the soft information has been calculated with the sumproduct algorithm and the entire iterative process will be repeated until the precise location of target is obtained. Finally, the soft information of SI(x), SI(y), and SI(z) can be updated by:
and
From Eqs. (27) to (29), the estimated location of the URT is determined by m_{x}, m_{y}, and m_{z}. Figure 4 shows the flow chart of the proposed algorithm. For better understanding, we summarize the entire iteration process as shown in Table 1. On the basis of the simulation experience, the soft information can converge with 10 iterations. Although there is no mathematical proof of convergence in this paper, the simulation experiment results can prove it. This may be because the proposed algorithm takes the stochastic properties of measurement errors into account. Besides, the initialization of the target location does not have a critical impact on convergence and can be set to arbitrary value.
Results and discussion
Simulation setup
To verify positioning performance of the proposed algorithm, computer simulations of different algorithms were conducted on the platform of MATLAB2014a. The simulation scenario consists of 100 random target locations, four APs, and a positioning area of length (100m), width (100m) and height (100m), respectively. Here, we chose a wellknown and easily implemented logarithmic shadow model [29] to generate logarithmic RSS for random measurements, which is expressed by:
where d_{i,s} is the distance between sth reference point and ith AP, d_{0} is the reference distance, P(d_{0}) is the RSS in decibel at the reference d_{0}, P(d_{i,s}) is the RSS of sth reference point from ith AP, α is path loss exponent, and χ_{s} represents the variance of RSS measurement obeying zeromean Gaussian distribution \(\left ({\chi _{s}} \sim N\left (0,\sigma _{{\chi _{s}}}^{2}\right)\right)\). Then, the random RSS measurement \(\widetilde {p}_{i,s}\) can be considered as Gaussian distribution \(\left ({\widetilde p_{i,s}} \sim N\left (P({d_{0}})  10 \cdot \alpha \cdot {\log _{10}}\left (\frac {{{d_{i,s}}}}{{{d_{0}}}}\right),\sigma _{{\chi _{s}}}^{2}\right)\right)\). Due to the multipath effect, smallscale fading, system hardware influence, and measurement error, \(\sigma _{{\chi _{s}}}^{2}\) is not the same at different reference points. We denote \(\left (\sigma _{{\chi _{1}}}^{2},\sigma _{{\chi _{2}}}^{2},...,\sigma _{{\chi _{s}}}^{2}\right) \in \sigma _{\chi }^{2}\), and the variable \(\sigma _{\chi }^{2}\) is assumed as Gaussian distribution \(\left (\sigma _{\chi }^{2} \sim N\left ({m_{\chi } },\sigma _{h}^{2}\right)\right)\). Here, we define that m_{χ} is the mean variance of all reference points, where \({m_{\chi }} = \left (\sigma _{{\chi _{1}}}^{2} + \sigma _{{\chi _{2}}}^{2} +... + \sigma _{{\chi _{q}}}^{2}\right)/q\), where q is the number of reference point. The typical values for d_{0}=1m, P(d_{0})=10dB, and α=1.8 in [23]. To reflect the positioning performance of proposed method, m_{χ} varies from 5 to 30 dB and \(\sigma _{h}^{2}\) is fixed at 5 dB in our simulation. The mimetic logarithmic RSS of offline database and online positioning target can be obtained with the method as in [23].
Performance comparison of different grid distances and AP numbers
At first, we utilize four APs, 1.5m grid distance, and one test target at (27, 45, 67)m to show the positioning trajectory of the proposed algorithm as shown in Fig. 5. Location coordinates of four APs identified with black solid “ Δ” mark are (25, 25, 0)m, (75, 25, 0)m, (25, 75, 0)m and (75, 75, 0)m, respectively. It can be clearly seen from the result that with the increase of iteration number, the estimated location quickly approaches the real location of the target. Figure 6 shows the root mean square error (RMSE) of the proposed algorithm and RSSDFG method. The 100 single test locations are randomly chosen from the positioning area. The RMSE rapidly decreases with the increasing number of the iterations. However, the proposed algorithm is more accurate than the conventional RSSDFG algorithm. When iteration number approaches 10, the RMSE of RSSDWLSFG tends to be stable at 1.47m. The proposed algorithm still has characteristic of fast convergence.
Next, taking the different variances of logarithmic RSS measurements into account, RSSDKNN (K=4) and RSSDFG algorithms are selected to compare with the proposed algorithm in the case of different grid distances and AP numbers. First, three algorithms are simulated with different grid distances to evaluate the positioning performance of these algorithms. In this simulation, four APs are selected and three grid distances are 1.5 m, 2 m, and 3 m as shown in Fig. 7. From the simulation results in Fig. 7, it can be seen that the proposed RSSDWLSFG algorithm has a higher positioning accuracy than RSSDFG algorithm and RSSDKNN algorithm respectively in the case of three different grid distances. Taking m_{χ} = 20 dB and 1.5 m grid distance as an example, the RMSE for each algorithm is 1.35 m for RSSDWLSFG, 1.47 m for RSSDFG, and 1.79 m for RSSDKNN. From the trend of the curves, it can be concluded that the smaller the grid distance is, the higher positioning accuracy is achieved. This is because the larger grid distance leads to fewer collected RSS information within the positioning area, and it degrades the positioning accuracy of the proposed algorithm. Although the increasing mean variance of all reference points leads to increasing error in small scale, the higher signal strength will generally improve the positioning accuracy. These two indicators are not in the same category. The results verify that the proposed RSSDWLSFG algorithm has the best positioning performance under the condition of different mean variances.
Second, we explore the influence of different AP numbers on the positioning accuracy of the three different algorithms. It is found that the RMSE becomes smaller as the number of APs increasing shown in Fig. 8. Comparing with the RSSDFG and RSSDKNN algorithms, the RMSE of proposed RSSDWLSFG algorithm with corresponding different APs is the smallest. However, with the increasing AP numbers, positioning accuracy is not unlimited to be enhanced. For example, the RMSE of proposed algorithm with four APs is 1.42 m, and it approaches that with five APs which is 1.35 m. Similarly, RMSE curves of four APs are also very close to RMSE curves of five APs. Conversely, the less number of APs results in lower positioning accuracy. When three APs are used, the positioning accuracy decreases so much that the positioning requirements cannot be met. Moreover, even with the use of four APs, the proposed algorithm still has a higher accuracy than the conventional RSSD4NN algorithm with five APs.
Computational complexity
Finally, we analyze and compare the computational complexity of different algorithms. Notation O(·) is defined as the computational complexity. As shown in Table 1, positioning result can be obtained only by simple arithmetic operations on each node of proposed algorithm. The computational complexity of the conventional 2D RSSDFG algorithm is linearly proportional to N (O(N)) as known in [25]. Although the proposed algorithm increases the dimension, it does not change the order of the local linear relationship and also only adds subtraction operation compared with RSSFG algorithm. Therefore, the computational complexity of the proposed algorithm is also linearly proportional to N (O(N)). The RSSDKNN algorithm needs to calculate Euclidean distance with each reference point in the database, and the computational complexity is proportional to the number of reference points (n). So the computational complexity of KNN method is O(n). The statistical test results of the three algorithms are shown in Table 2. It can be obtained that the proposed 3D RSSDWLSFG algorithm not only enjoys low time consumption the same as RSSDFG algorithm but also achieve higher accuracy compared with RSSDKNN algorithm.
Experimental
Finally, the proposed RSSDWLSFG algorithm is validated by field test, which is located on the first floor of the National Radio Monitoring Center, Beijing. The test field consists of an office and its adjacent corridor with an area of length (14.4 m), width (10.5 m), and height (4.8 m) respectively. Total plane area of the test field is 151.2 square meters. In addition, there are four windows on one wall of the office and two doors on the other wall adjacent to the corridor and there are no partitions or compartments. The main items in the office are six rows of desks, chairs, and computers, and human beings are free to enter and leave frequently throughout the whole test. In order to embody the internal structure of the positioning area more intuitively, Fig. 9 shows the plane layout of test field.
In the offline fingerprint database establishment phase, the positioning area is divided into two types of grid distances: 1.5 m and 2 m. The SA44B (Signal Hound Co. Ltd.) model signal receivers are used as APs to collect RSS information. The number and layout of APs are selected with three types, which labels are “1#”, “2#”, and “3#” as shown in Fig. 9. Three APs’ locations labeled “1#” are (2, 7, 0) m, (6, 3, 0) m, and (6, 9, 0) m, respectively. Four APs labeled “2#” are deployed in the office at (3, 2, 0) m, (8, 5, 0) m, (3, 9, 0) m, and (8, 12, 0) m. Five APs labeled “3#” are (3, 2, 0) m, (8, 5, 0) m, (3, 9, 0) m, (8, 12, 0) m, and (5, 6, 0) m, respectively. In this experiment, the radio transmitter TFG6300 (SUING Co. Ltd.) model used to establish the fingerprint database and be as the URT is adjustable in transmitting “frequency/strength”. In order to better prove the adaptability of the proposed RSSDWLSG algorithm for different frequency and strength, we choose “1GHz/20dB” offline database and “300MHz/13dB” URT in the test. Here, the stored RSS of each reference point in the fingerprint database are obtained by averaging 100 sampling RSS measurements from each AP. In the positioning area, 100 test locations are randomly selected for localization test.
The mean location error and cumulative distribution function (CDF) of location errors are used as the key evaluation indicators to compare the performance of RSSDWLSFG, RSSDFG, and RSSDKNN. The mean location errors are characterized by the average deviation value of all the positioning targets that compared with the real location. CDF represents the distribution of the location errors expressed as percentage. First, comparisons of different grid distances among three algorithms are conducted in the positioning area, which are using 1.5m grid distance and 2m grid distance, respectively. The mean location error of different algorithms with four APs are as shown in Table 3. When the grid distance is 1.5 m, the mean location error by proposed RSSDWLSFG algorithm is 1.18 m. In comparison, the mean location errors of RSSDFG and RSSDKNN are 1.57 m and 1.31 m, respectively. With the grid distance increasing to 2 m, the mean location errors of RSSDKNN, RSSDFG, and RSSDWLSFG are 1.79 m, 1.52 m, and 1.33 m, respectively. The CDF of RSSDKNN, RSSDFG, and RSSDWLSFG is as shown in Fig. 10. When the grid distance is 1.5 m or 2 m, the number of qualified testing points of the proposed algorithm within different location errors is larger than that of the other two algorithms. Considering the location error within 1.5 m, the CDF for each algorithm is 43% for RSSDKNN, 59% for RSSDFG, and 67% for RSSDWLSFG when the grid distance is 1.5 m. The CDF of RSSDKNN, RSSDFG, and RSSDWLSFG is 35%, 47%, and 58% respectively, when the the grid distance is 2 m and the location error is within 1.5 m. The results show that the positioning accuracy of proposed RSSDWLSFG algorithm is better than RSSDFG and RSSDKNN no matter the grid distance is 1.5 m or 2 m.
Next, we explore impact of the number of APs on the positioning accuracy through experiments. The 1.5m grid distance is selected to evaluate the positioning performance of different algorithms when the number of APs changes from three to five. Comparison of the mean location errors among different algorithms with 1.5m grid distance is as shown in Table 4. The mean location errors of RSSDKNN, RSSDFG, and RSSDWLSFG algorithms are 1.83 m, 1.71 m, and 1.56 m respectively when utilizing three APs. It can be observed from the comparison of experimental results that the mean location errors of three algorithms are 1.51 m, 1.35 m, and 1.12 m respectively with four APs. While using five APs, the mean location error of proposed is 1.05 m. In comparison, the mean location errors of RSSDKNN and RSSDFG are 1.43 m and 1.28 m. Figure 11 shows the CDF comparison of location errors with different numbers of APs. The CDF of RSSDKNN, RSSDLS, and RSSDWLSFG is 42%, 56%, and 62% respectively, when the number of APs is three and location error within 1.5 m. When the number of APs is four, the CDF for each algorithm is 62% for RSSDKNN, 68% for RSSDFG, and 72% for RSSDWLSFG when the location error is within 1.5 m. As the number increasing to five, CDF of the three algorithms are 55%, 67% and 77%, respectively. The results demonstrate that the increasing number of APs can improve the positioning accuracy. In the case of different number of APs, the positioning performance of proposed algorithm is superior to the other two algorithms. Considering hardware costs and precision requirements, a minimum number of APs is four, which is acceptable. The above experiment results show that the proposed RSSDWLSFG algorithm has a higher positioning accuracy than RSSDKNN and RSSDLS in different grid distances and AP numbers.
Conclusions
For localization requirement of the radio transmitter in 3D scenario, this paper proposed a new 3D RSSDWLSFG algorithm to achieve accurate detection of an URT. With the Gaussian assumption of RSS n, a novel RSSDbased 3D WLSFG model was established with WLS method to eliminate the influence from the variance diversity of reference points compared with the conventional 2D RSSDbased FG model. Utilizing the proposed weight calculation method, the relationship between RSSD measured value and location coordinates is more reasonable and accurate, which effectively mitigates the error caused by the reference point with larger variance of RSSD measurement and improves the positioning accuracy. The softinformation calculation and iterative process of the proposed algorithm were deduced by using the sumproduct algorithm. In addition, considering the main factors affecting the accuracy of fingerprint positioning technology in practical application, the positioning performance of the proposed algorithm under different grid distances and different AP numbers was explored respectively. Compared with the RSSDFG and RSSDKNN algorithms, numerical experiment results show that the proposed method effectively improves the positioning accuracy by about 22% and 13%, respectively. Hence, it not only meets the positioning requirements but also has a better application prospect. The effect of AP’s layout and localization of moving URT and multiple URTs will be involved in our future research.
Abbreviations
 AP:

Access point
 FG:

Factor graph
 LS:

Least square
 RSSD:

Received signal strength difference
 URT:

Unknown radio transmitter
 WLS:

Weighted least square
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Funding
This research was funded by the Ministry of Industry and Information Technology of the People’s Republic of China (CN) (No. 12MCKY14) and the Education Department of Hebei (No. ZD2017216).
Availability of data and materials
In our test, four SA44B (Signal Hound Co. Ltd.) measuring receivers as APs are utilized to collect the RSS measurements from the radio transmitter (TFG6300, SUING Co. Ltd.). The experimental environment is located on the first floor of National Radio Monitoring Center, Beijing.
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LZ proposed the main idea, derived the algorithm, and wrote the paper. TD review the work and versions. CJ wrote the simulation code, processed the experimental data, and revised the paper. All authors read and approved the final manuscript.
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Correspondence to Taihang Du.
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Zhang, L., Du, T. & Jiang, C. RSSDbased 3D localization of an unknown radio transmitter using weighted least square and factor graph. J Wireless Com Network 2019, 10 (2019). https://doi.org/10.1186/s1363801813295
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Keywords
 Radio transmitter
 3D Position location
 Received signal strength (RSSD)
 Weighted least square (WLS)
 Factor graph (FG)