# RSSD-based 3-D localization of an unknown radio transmitter using weighted least square and factor graph

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## 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 2-D scenario has been developed. However, the URT positioning under 3-D scenario is more difficult with the large number of unknown parameters and has greater practical significance. In addition, the previous RSSD-based 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 3-D 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 RSSD-based 3-D WLSFG model and sum-product rule, positioning process of the proposed 3-D RSSD-WLSFG 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 RSSD-FG algorithm under different grid distances and signal receiver numbers.

## Keywords

Radio transmitter 3-D Position location Received signal strength (RSSD) Weighted least square (WLS) Factor graph (FG)## Abbreviations

- AP
Access point

- FG
Factor graph

- LS
Least square

- RSSD
Received signal strength difference

- URT
Unknown radio transmitter

- WLS
Weighted least square

## 1 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, 8, 9]. Among all the measurement parameters, the RSS-based positioning techniques are widely used for the advantages of no extra antenna arrays, no time synchronization limitation, and low cost. A large number of RSS-based 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 well-known 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 multi-path 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 RSS-based 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 RSS-based fingerprint positioning method is mainly divided into two phases: the off-line training phase and the on-line positioning phase. In the off-line training phase, RSS measurements are collected from different access points (APs) deployed in the positioning area. Then, RSS and corresponding location coordinates construct off-line fingerprint database. In the on-line positioning phase, the positioning target location is estimated by collecting the real-time RSS measurements and matching the fingerprint database with appropriate positioning algorithm. Two typical RSS-based 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 Maximum-likelihood (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, ML-based 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 TOA-FG [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, TDOA-FG 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, TOA-FG and TDOA-FG are both more suitable for line of sight (LOS) positioning scenario. The RSS-FG [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 non-line of sight (NLOS) positioning environment. Yet, the RSS-FG 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 RSSD-FG technique to realize the detection of an URT. However, the mathematical model established in RSSD-FG 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 RSSD-AOA 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 2-D scenario. To the best of our knowledge, there is no existing research in the literature to realize the localization of an URT in 3-D scenario. In this paper, we combine the RSSD-based FG fingerprint positioning technique and WLS method to propose a new 3-D RSSD-WLSFG algorithm, and it is considered to be an effective method for detecting an URT in 3-D space.

The following are the contributions of this paper:

1) A more accurate RSSD-based 3-D 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 3-D RSSD-WLSFG algorithm was derived by using sum-product theorem and soft-information 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 3-D RSSD-WLSFG 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.

## 2 The principle and system model

### 2.1 Factor graph and sum-product algorithm

*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 sum-product algorithm. The sum-product algorithm works by collecting the soft-information transmission of local function product along the path in the factor graph. The soft-information transported between the variable nodes and factor nodes represents the stochastic properties of the associated variable nodes. With several iterative processes of soft-information transported between the variable nodes and factor nodes, the solution of variables to a problem expressed by a factor graph can be easily obtained.

*S*

*I*(

*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

*S*

*I*(

*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:

*P*

_{k}is

*k*-th factor node. The product of some independent Gaussian distributions is still a Gaussian distribution. The mean and variance of

*S*

*I*(

*g*,

*F*) can be obtained by:

where *f*(*g*,*w*_{1},...,*w*_{M}) is the local function of factor node *F* associated with all the variable nodes.

*g*with this method as follows:

### 2.2 Positioning system

*d*. In the off-line 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 RSS-based fingerprint positioning technique in constructing corresponding databases to match different targets. More importantly, RSS-based parameter cannot realize the localization of the URE. After that, the RSSD-FG model can be obtained as done in [25]. Combining the RSSD database and RSSD-FG model, we can obtain the proposed RSSD-based 3-D WLSFG model by using WLS method.

In the on-line positioning phase, real-time RSS measurements from the positioning target will be collected and used as the input values of the proposed RSSD-based 3-D WLSFG model. Finally, the estimated location of positioning target can be calculated by the sum-product 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.

## 3 Proposed 3-D RSSD-WLSFG algorithm

### 3.1 RSSD-based 3-D WLSFG model

*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:

*e*

_{i}represents the measurement error of

*i*-th AP in units of watt (W), and it can be expressed by zero-mean 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 *i*-th AP and *j*-th 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 sum-product algorithm. Thus, location of the URT will be estimated with a few iterative process among the source factor nodes and variable nodes.

*x*,

*y*,

*z*) and logarithmic RSSD (

*R*) can formulate a linear equation, which can be expressed by:

*k*

_{x},

*k*

_{y},

*k*

_{z}, and

*k*

_{r}are the coefficients and

*c*is a non-zero 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 pattern-recognition technique [23] and the positioning area consisted of these five reference points is defined as the sub-positioning area. Since the logarithmic RSSD and the location of the five reference points are known, five linear equations in matrix form for

*t*-th AP combination are given by:

*k*

_{x,t},

*k*

_{y,t},

*k*

_{z,t}and

*k*

_{r,t}are coefficients of the linear equation corresponding to

*i*-th AP combination, (

*x*

_{s},

*y*

_{s}) (

*s*=1, 2, 3, 4, 5) is the location coordinate of

*s*-th reference point, and \(\widetilde {R}_{t,s}\) is the mean logarithmic RSSD of

*s*-th reference point from

*t*-th 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

*t*-th AP combination within the selected sub-positioning area can be expressed as:

*x*

_{s},

*y*

_{s},

*z*

_{s}) of

*s*-th reference point and mean logarithmic RSSD \(\left (\widetilde {R}_{t,s}\right)\) of

*s*-th reference point from

*t*-th AP combination, which is represented by:

*x*

_{s},

*y*

_{s},

*z*

_{s}) to calculate the estimated mean logarithmic RSSD \(\left (\widetilde {R}_{t,s}^{'}\right)\) of

*s*-th reference point from

*t*-th AP combination and the stochastic error (

*E*

_{t,s}) between the measured value and estimated value is given by:

*W*) given 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:

*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 sub-positioning area is given by:

The other linear equations corresponding to different AP combinations can also be obtained by using the same process above.

### 3.2 Soft-information calculation and iteration process

*x*,

*y*and

*z*to factor node

*A*

_{t}should be calculated at first. Utilizing the sum-product rules,

*S*

*I*(

*x*,

*A*

_{t}),

*S*

*I*(

*y*,

*A*

_{t}), and

*S*

*I*(

*z*,

*A*

_{t}) are given by:

*S*

*I*(

*x*,

*A*

_{t}), the mean and variance can be calculated by:

*S*

*I*(

*y*,

*A*

_{t}) and

*S*

*I*(

*z*,

*A*

_{t}) can also be obtained. From (18), the factor node

*A*

_{t}to variable node

*x*transporting the soft-information

*S*

*I*(

*A*

_{t},

*x*) can be obtained by:

*S*

*I*(

*A*

_{t},

*y*) and

*S*

*I*(

*A*

_{t},

*z*) can be calculated with the similar manner. The soft-information 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

*S*

*I*(

*D*

_{t},

*R*

_{t}) is calculated by:

*P*

_{i}and

*P*

_{j}directly transports the soft-information 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),

*S*

*I*(

*P*

_{i},

*p*

_{i}) and

*S*

*I*(

*P*

_{j},

*p*

_{j}) can be directly obtained. In the same way, the soft-information of other AP combinations can also be calculated. As mentioned above, all the soft information has been calculated with the sum-product algorithm and the entire iterative process will be repeated until the precise location of target is obtained. Finally, the soft information of

*S*

*I*(

*x*),

*S*

*I*(

*y*), and

*S*

*I*(

*z*) can be updated 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.

The soft information processing of each node in Fig. 3

| Inputs | Outputs |
---|---|---|

| (\(\widehat {p}_{i}\), 0) | \(\left ({m_{{P_{i}},{p_{i}}}},\sigma _{{P_{i}},{p_{i}}}^{2}\right)\) |

| \(\left ({m_{{P_{i}},{p_{i}}}},\sigma _{{P_{i}},{p_{i}}}^{2}\right)\) | \(\left ({m_{{p_{i}},{D_{t}}}},\sigma _{{p_{i}},{D_{t}}}^{2}\right)\) |

| \(\left ({m_{{p_{i}},{D_{t}}}},\sigma _{{p_{i}},{D_{t}}}^{2}\right)\), \(\left ({m_{{p_{j}},{D_{t}}}},\sigma _{{p_{j}},{D_{t}}}^{2}\right)\) | \({m_{{D_{t}},{R_{t}}}} = {m_{{p_{i}},{D_{t}}}} - {m_{{p_{j}},{D_{t}}}}\), \(\sigma _{{D_{t}},{R_{t}}}^{2} = \sigma _{{p_{i}},{D_{t}}}^{2} + \sigma _{{p_{j}},{D_{t}}}^{2}\) |

| \(\left ({m_{{D_{t}},{R_{t}}}}, \sigma _{{D_{t}},{R_{t}}}^{2}\right)\) | \({m_{{R_{t}},{A_{t}}}} = {m_{{D_{t}},{R_{t}}}}\), \(\sigma _{{R_{t}},{A_{t}}}^{2} = \sigma _{{D_{t}},{R_{t}}}^{2}\) |

| \(\left ({m_{{R_{t}},{A_{t}}}},\sigma _{{R_{t}},{A_{t}}}^{2}\right)\), \(({m_{{y},{A_{t}}}},\sigma _{{y},{A_{t}}}^{2})\phantom {\dot {i}\!}\) | \({m_{{A_{t}},x}} = \left (1- k_{y,t}^{'}{m_{y,{A_{t}}}} - k_{z,t}^{'}{m_{z,{A_{t}}}}- k_{r,t}^{'}{m_{{R_{t}},{A_{t}}}}\right)/k_{x,t}^{'}\), |

\(\sigma _{_{{A_{t}},x}}^{2} = \left (k_{y,t}^{'2}\sigma _{y,{A_{t}}}^{2} + k_{z,t}^{'2}\sigma _{z,{A_{t}}}^{2} + k_{r,t}^{'2}\sigma _{{R_{t}},{A_{t}}}^{2}\right)/k_{x,t}^{'2}\) | ||

| \(\left ({m_{{R_{t}},{A_{t}}}},\sigma _{{R_{t}},{A_{t}}}^{2}\right)\), \(\left ({m_{{x},{A_{t}}}},\sigma _{{x},{A_{t}}}^{2}\right)\phantom {\dot {i}\!}\) | \({m_{{A_{t}},y}} = \left (1- k_{x,t}^{'}{m_{x,{A_{t}}}} - k_{z,t}^{'}{m_{z,{A_{t}}}} - k_{r,t}^{'}{m_{{R_{t}},{A_{t}}}}\right)/k_{y,t}^{'}\), |

\(\sigma _{_{{A_{t}},y}}^{2} = \left (k_{x,t}^{'2}\sigma _{x,{A_{t}}}^{2} + k_{z,t}^{'2}\sigma _{z,{A_{t}}}^{2} + k_{r,t}^{'2}\sigma _{{R_{t}},{A_{t}}}^{2}\right)/k_{y,t}^{'2}\) | ||

| \(\left ({m_{{R_{t}},{A_{t}}}},\sigma _{{R_{t}},{A_{t}}}^{2}\right)\), \(\left ({m_{{x_{t}},{A_{t}}}},\sigma _{{x_{t}},{A_{t}}}^{2}\right)\) | \({m_{{A_{t}},z}} = \left (1- k_{x,t}^{'}{m_{x,{A_{t}}}} - k_{y,t}^{'}{m_{y,{A_{t}}}}- k_{r,t}^{'}{m_{{R_{t}},{A_{t}}}}\right)/k_{z,t}^{'}\), |

\(\sigma _{_{{A_{t}},z}}^{2} = \left (k_{x,t}^{'2}\sigma _{x,{A_{t}}}^{2} + k_{y,t}^{'2}\sigma _{y,{A_{t}}}^{2} + k_{r,t}^{'2}\sigma _{{R_{t}},{A_{t}}}^{2}\right)/k_{z,t}^{'2}\) | ||

| \(\left ({m_{{A_{l}},x}},\sigma _{{A_{l}},x}^{2}\right)\) | \({m_{x,{A_{t}}}} = \sigma _{_{x,{A_{t}}}}^{2}\left (\sum \limits _{l \ne t}^{n} {\frac {{{m_{{A_{l}},x}}}}{{\sigma _{{A_{l}},x}^{2}}}}\right)\), \(\sigma _{x,{A_{t}}}^{2} = 1/\left (\sum \limits _{l \ne t}^{n} {\frac {1}{{\sigma _{{A_{l}},x}^{2}}}} \right)\) |

| \(\left ({m_{{A_{l}},y}},\sigma _{{A_{l}},y}^{2}\right)\) | \({m_{y,{A_{t}}}} = \sigma _{_{y,{A_{t}}}}^{2}\left (\sum \limits _{l \ne t}^{n} {\frac {{{m_{{A_{l}},y}}}}{{\sigma _{{A_{l}},y}^{2}}}}\right)\), \(\sigma _{y,{A_{t}}}^{2} = 1/\left (\sum \limits _{l \ne t}^{n} {\frac {1}{{\sigma _{{A_{l}},y}^{2}}}}\right)\) |

| \(\left ({m_{{A_{l}},z}},\sigma _{{A_{l}},z}^{2}\right)\) | \({m_{z,{A_{t}}}} = \sigma _{_{z,{A_{t}}}}^{2}\left (\sum \limits _{l \ne t}^{n} {\frac {{{m_{{A_{l}},z}}}}{{\sigma _{{A_{l}},z}^{2}}}}\right)\), \(\sigma _{z,{A_{t}}}^{2} = 1/\left (\sum \limits _{l \ne t}^{n} {\frac {1}{{\sigma _{{A_{l}},z}^{2}}}}\right)\) |

| \(\left ({m_{{A_{t}},x}},\sigma _{{A_{t}},x}^{2}\right)\) | \({m_{x}} = \sigma _{x}^{2} \cdot \left (\sum \limits _{t = 1}^{n} {\frac {{{m_{{A_{t}},x}}}}{{\sigma _{{A_{t}},x}^{2}}}} \right)\), \(\sigma _{x}^{2} = 1/\left (\sum \limits _{t = 1}^{n} {\frac {1}{{\sigma _{{A_{t}},x}^{2}}}} \right)\) |

| \(\left ({m_{{A_{t}},y}},\sigma _{{A_{t}},y}^{2}\right)\) | \({m_{y}} = \sigma _{y}^{2} \cdot \left (\sum \limits _{t = 1}^{n} {\frac {{{m_{{A_{t}},y}}}}{{\sigma _{{A_{t}},y}^{2}}}} \right)\), \(\sigma _{y}^{2} = 1/\left (\sum \limits _{t = 1}^{n} {\frac {1}{{\sigma _{{A_{t}},y}^{2}}}} \right)\) |

| \(\left ({m_{{A_{t}},z}},\sigma _{{A_{t}},z}^{2}\right)\) | \({m_{z}} = \sigma _{z}^{2} \cdot \left (\sum \limits _{t = 1}^{n} {\frac {{{m_{{A_{t}},z}}}}{{\sigma _{{A_{t}},z}^{2}}}} \right)\), \(\sigma _{z}^{2} = 1/\left (\sum \limits _{t = 1}^{n} {\frac {1}{{\sigma _{{A_{t}},z}^{2}}}} \right)\) |

## 4 Results and discussion

### 4.1 Simulation setup

where *d*_{i,s} is the distance between *s*-th reference point and *i*-th 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 *s*-th reference point from *i*-th AP, *α* is path loss exponent, and *χ*_{s} represents the variance of RSS measurement obeying zero-mean 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 multi-path effect, small-scale 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}=1*m*, *P*(*d*_{0})=10*d**B*, 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 off-line database and on-line positioning target can be obtained with the method as in [23].

### 4.2 Performance comparison of different grid distances and AP numbers

*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 RSSD-FG 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 RSSD-FG algorithm. When iteration number approaches 10, the RMSE of RSSD-WLSFG tends to be stable at 1.47

*m*. The proposed algorithm still has characteristic of fast convergence.

*m*

_{χ}= 20 dB and 1.5 m grid distance as an example, the RMSE for each algorithm is 1.35 m for RSSD-WLSFG, 1.47 m for RSSD-FG, and 1.79 m for RSSD-KNN. 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 RSSD-WLSFG algorithm has the best positioning performance under the condition of different mean variances.

### 4.3 Computational complexity

*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 RSS-FG algorithm. Therefore, the computational complexity of the proposed algorithm is also linearly proportional to N (O(

*N*)). The RSSD-KNN 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 3-D RSSD-WLSFG algorithm not only enjoys low time consumption the same as RSSD-FG algorithm but also achieve higher accuracy compared with RSSD-KNN algorithm.

Computational complexity comparison of different algorithms

Algorithm | Mean location error | Running time (s) |
---|---|---|

RSSD-KNN | 1.62 m | 0.623 |

RSSD-FG | 1.37 m | 0.372 |

RSSD-WLSFG | 1.16 m | 0.372 |

## 5 Experimental

In the off-line 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 RSSD-WLSG algorithm for different frequency and strength, we choose “1GHz/20dB” off-line 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.

Mean location errors of different grid distances with four APs

Grid distance | RSSD-KNN | RSSD-FG | RSSD-WLSFG |
---|---|---|---|

1.5 m | 1.57 m | 1.31 m | 1.18 m |

2 m | 1.79 m | 1.52 m | 1.33 m |

Mean location errors of different APs with 1.5-m grid distance

Number of APs | RSSD-KNN | RSSD-FG | RSSD-WLSFG |
---|---|---|---|

3 APs | 1.83 m | 1.71 m | 1.56 m |

4 APs | 1.51 m | 1.35 m | 1.12 m |

5 APs | 1.43 m | 1.28 m | 1.05 m |

## 6 Conclusions

For localization requirement of the radio transmitter in 3-D scenario, this paper proposed a new 3-D RSSD-WLSFG algorithm to achieve accurate detection of an URT. With the Gaussian assumption of RSS n, a novel RSSD-based 3-D WLSFG model was established with WLS method to eliminate the influence from the variance diversity of reference points compared with the conventional 2-D RSSD-based 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 soft-information calculation and iterative process of the proposed algorithm were deduced by using the sum-product 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 RSSD-FG and RSSD-KNN 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.

## Notes

### Funding

This research was funded by the Ministry of Industry and Information Technology of the People’s Republic of China (CN) (No. 12-MC-KY-14) 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.

### Authors’ contributions

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.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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