# A 3D Indoor Positioning Method of Wireless Network with Single Base Station in Multipath Environment

Authors: Youqing Wang; Kun Zhao; Zhengqi Zheng
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

The proliferation of indoor location-based services has increased the demand of indoor positioning technology. Severe multipath and coherence effects are the difference between signal propagation indoors and outdoors. Most existing indoor localization methods build their models in 2-dimensional space and try to avoid the influence of multipath. We propose a method to realize 3-dimensional indoor positioning with single base station by using multipath channel. The angles of multipath coherent signals are estimated by MIMO antenna and the delays are estimated by OFDM signal. To avoid the complicated calculation in joint estimation of angles and delays in 3-dimensional space, the angles and delays are estimated separately and matched by the proposed algorithm. The line-of-sight channel is differentiated by time delay, and the reflection paths for non-line-of-sight channels are established with angle information and indoor maps. Finally, combine the angle information of the reflection paths and the line-of-sight path to obtain the target position in 3-dimensional indoor space. We verified the method through simulation in an indoor space of6m×8m×4.5m. The positioning errors are submeter level in 95% cases and less than 0.4m in 60% cases. The proposed method requires only one base station and can be applied in most wireless networks. Compared with existing indoor localization methods, it has lower computational complexity and higher application potential.


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## Body

## 1. Introduction

Location information plays an important role in communications, navigation, and other fields. The expanding of indoor space of modern building facilities has derived many new services and applications based on location information, which puts forward new requirements and challenges for indoor positioning technology. Common indoor positioning technologies mainly include Inertial Navigation [1], Infrared [2], Ultrasonic [3], Radio Frequency IDentification [4], Bluetooth [5], WiFi [6], Ultra WideBand [7], Visible Light [8], and Wireless Communication Network [9]. Researchers have developed many indoor positioning methods based on these technologies [10], but most of these technologies cannot balance the positioning accuracy, coverage, and cost. Wireless communication network is one of the signals with the widest indoor coverage and the most equipment. In order to increase the transmission performance, the new technologies such as Orthogonal Frequency Division Multiplexing (OFDM) and Multiple Input Multiple Output (MIMO) are introduced, and the technical specifications also divide the higher frequency band and wider bandwidth to it, such as millimeter wave band in 5G mobile network [11], that also brings new opportunities for indoor positioning. In addition, 3GPP also puts forward the requirement that the positioning error of the User Equipment (UE) in future wireless communication should be less than 3 m for 80% of indoor environment [12]. Therefore, it is of great significance to study for indoor positioning technology based on wireless communication networks.The biggest difference between signal propagation in indoor and outdoor environment is that there has serious multipath interference in complex indoor environment. Researchers have proposed various solutions to this problem. Garcia et al. proposed a direct localization approach for Massive MIMO [13], which is based on the framework of a novel compressed sensing. They distinguished the Line-Of-Sight (LOS) and Non-Line-of-Sight (NLOS) signal paths by exploiting the fact that LOS components must originate from a common location, whereas NLOS components have arbitrary AOA. But the model they used is a 2-dimensional (2D) rectangle plane, and four circular array antennas are distributed in four corners. Ma et al. proposed an indoor localization method based on Angle of Arrival (AOA) and Phase Difference Of Arrival (PDOA) using virtual stations for passive ultrahigh frequency radio frequency identification [4]. They choose the two strongest paths according to the received signal strength and use the reflection principle of multipath to establish virtual BS to perform localization. But they did not discuss that the power of LOS path may be lower than NLOS path in some actual indoor positioning environment, for example, the signal can penetrate the obstacle and some energy will be absorbed in LOS path. Zhou et al. proposed an indoor localization method of using the adjacent angle power difference [14]. They use the OMP algorithm to obtain the initial estimation of the direction and adjust it through the power difference of the adjacent angle. And then, the previous location is used as the starting point of the next, so the estimation accuracy and stability of continuous moving target are effectively improved. He and So proposed LTE indoor localization system which combines positioning algorithms based on the Time Difference Of Arrival (TDOA) and fingerprinting [15], they determine a coarse target location through the TDOA algorithm and use it to identify the subarea of monitoring area which includes the best location estimate. Then, the fingerprinting algorithm based on deep neural net will perform the accurate location point by using the corresponding subarea.In general, positioning technology mainly uses the propagation time, propagation direction, and power attenuation of electromagnetic wave or acoustic wave in space. From the existing research progress, the time-based positioning requires high-precision clock synchronization or large bandwidth, and at least three base stations (BS) are required. The positioning based on signal strength is vulnerable to the environment and needs to update the priori information frequently. The positioning technology based on angle information requires the equipment to have an array antenna, or at least two BS are required. For indoor multipath effect, there are three main processing methods. First way is trying to avoid the influence of multipath signals during positioning [13, 16], the second way is to make full use of multipath signals [4, 17, 18], and the third way is to use multipath signals indirectly according to the characteristics of the environment, like some algorithms which is based on fingerprint information and machine learning [14, 15]. In the future indoor positioning technology, a major trend is to combine these principles to achieve high-precision positioning.In this paper, we propose a 3-dimensional (3D) indoor positioning method based on multipath information, which makes full use of OFDM technology and MIMO array antenna in wireless communication networks. The main contributions of this paper are as follows.(1)
TDOA and AOA estimation algorithms for indoor multipath signals, the algorithm can avoid the computational complexity of joint estimation(2)
making full use of the multipath signal of indoor complex environment. Combining the TDOA and AOA information with the indoor map to reconstruct the indoor multipath channel model(3)
realizing the indoor positioning in 3D spatial based on single BS, which has lower equipment cost and better application prospect than existing methodsThe main works of each section are as follows. We discuss the existing research of indoor positioning and the advantages of our research in the first section. In Section2, we introduce the indoor space model and signal model of this paper. In Section 3, we describe TDOA and AOA algorithms for multipath coherent signals. In Section 4, we describe the matching methods of AOA and TDOA, and how to obtain the target location. We simulate the proposed method in Section 5 and summarize the full paper in the last section.


## 2. System Model of 3D Indoor Positioning

### 2.1. Indoor Environment Model

The 3D indoor environment model of this paper is a cube space, in which the length, width, and height are 8m, 6m, and 4.5m, respectively; here, the m means meter, as shown in Figure1. Establish a 3D coordinate system with the top center as the origin. The x-axis is perpendicular to the front and rear planes, and the y-axis is perpendicular to the left and right planes. Take the positive direction of the x-axis as the north of indoor environment, the equations of each plane are shown in Table 1.Figure 1
3D indoor environment model, where BS is the location of the base station,UE is the location of the user equipment,L0,L1,L2,L3 are the channel paths, and R1,R2,R3 are the multipath reflection points.Table 1
Plane equation of indoor environmental model.
PlaneFrontRearLeftRightTopBottomEquationx=−3x=3y=−4y=4z=0z=−4.5SupposeUExs,ys,zs is the signal source location in the space, and the BSx0,y0,z0 is the reference antenna of array antenna,Lp is the path of the multipath channel, where p=0,1,⋯,P−1,Rpxp,yp,zp is the reflection point of the p-th multipath channel on the wall and the LOS path has no reflection point. Due to the signals after multiple reflections and long-distance propagation have severe fading, only the shorter P paths are considered. In Figure 1, we set the coordinate of UE is UE1.5,−2,−3,the coordinate of BS is BS0,0,0. Then, we can obtain the reflection points of the three NLOS paths which are R13,−1.333,−2,R21.125,−1.5,−4.5,R31,−4,−2.Since the array antenna located in the central of the top, we design it as a “+” shape, it can obtain the maximum planar antenna aperture. As shown in Figure2(a). The whole array has 31 antenna elements, which are divided into two identical Uniform Linear Arrays (ULA) and distributed on the x-axis and y-axis, respectively. Figure 2(b) is the ULA on x-axis. The 9th antenna of two ULA is a common array element and can be regarded as reference array element. As an example, we set the array element spacing of ULA is 0.03 m, which can be applied to any wireless network with frequencies below 5GHz. For higher frequency, the element spacing can be adjusted to meet the limit of less than half the signal wavelength.Figure 2
(a) The antenna array model. (b) The model of ULA onx-axis and spatial smoothing algorithm. (c) The model of Uniform Frequency Array and frequency domain smoothing algorithm.
(a)(b)(c)


### 2.2. Positioning Signal Model

One of the purposes of this paper is to use OFDM pilot subcarrier signals to realize TDOA estimation of multipath signals, so we do not care about the details of information transmission. Another advantage of using pilot subcarrier signals is that it does not occupy the information transmission rate. Consider an OFDM signal withK subcarriers, the transmission symbols of OFDM can be expressed as Equation (1).
(1)St=1TAUE∑k=0K−1Cktej2πfc+k∆ft,wheret∈0,T,and T is the data duration of OFDM symbols,AUE is the transmission gain of the UE, Ckt represents the data on the k-th subcarrier,fc represents the carrier frequency modulated by the communication system, and ∆f is the frequency interval of subcarriers.When the OFDM signal propagates in indoor environment, the impact response of multipath channel can be expressed as Equation (2).
(2)ht=∑p=0P−1aptδt−ζp,where P is the number of channels,apt is the amplitude fading of the p-th path, t is the transmission time of the signal, and ζp is the delay of multipath channel.The multipath signals received by a single antenna can be expressed as Equation (3).
(3)yt=1TAUEABS∑p=0P−1apt∑k=0K−1Cktej2πfc+k∆ft−ζp+nt,where ABS is the gain of the receiving array antenna, nt is the Gaussian white noise, since the change of channel is very slow compared with the sampling time, 1/T,AUE,ABS,apt can be regarded as constants, and we assume these components are 1. The phase change caused by reflection is very small compared with propagation delay, which has been ignored.Some time-based methods assume there is high-precision time synchronization between the transmitter and receiver [19]. The delays of these algorithms estimate are ζp as shown in Equation (3), but the high cost of atomic clock and the low sampling frequency of communication system make these methods not suitable for practical application. Theoretically, the signal of LOS path will first arrive at the receiver due to the shortest distance. Set L0-th path is the LOS channel and the signal from L0 arrive at the receiver at time tLOS,where tLOS=t−ζLOS,and t is the moment when the signal is emitted. The TDOA between NLOS channels and LOS channel is τp=ζp−ζLOS. Thus, Equation (3) can be changed to Equation 4.
(4)yt=∑p=0P−1∑k=0K−1Cktej2πfc+k∆ftLOS−τp+nt.


## 2.1. Indoor Environment Model

The 3D indoor environment model of this paper is a cube space, in which the length, width, and height are 8m, 6m, and 4.5m, respectively; here, the m means meter, as shown in Figure1. Establish a 3D coordinate system with the top center as the origin. The x-axis is perpendicular to the front and rear planes, and the y-axis is perpendicular to the left and right planes. Take the positive direction of the x-axis as the north of indoor environment, the equations of each plane are shown in Table 1.Figure 1
3D indoor environment model, where BS is the location of the base station,UE is the location of the user equipment,L0,L1,L2,L3 are the channel paths, and R1,R2,R3 are the multipath reflection points.Table 1
Plane equation of indoor environmental model.
PlaneFrontRearLeftRightTopBottomEquationx=−3x=3y=−4y=4z=0z=−4.5SupposeUExs,ys,zs is the signal source location in the space, and the BSx0,y0,z0 is the reference antenna of array antenna,Lp is the path of the multipath channel, where p=0,1,⋯,P−1,Rpxp,yp,zp is the reflection point of the p-th multipath channel on the wall and the LOS path has no reflection point. Due to the signals after multiple reflections and long-distance propagation have severe fading, only the shorter P paths are considered. In Figure 1, we set the coordinate of UE is UE1.5,−2,−3,the coordinate of BS is BS0,0,0. Then, we can obtain the reflection points of the three NLOS paths which are R13,−1.333,−2,R21.125,−1.5,−4.5,R31,−4,−2.Since the array antenna located in the central of the top, we design it as a “+” shape, it can obtain the maximum planar antenna aperture. As shown in Figure2(a). The whole array has 31 antenna elements, which are divided into two identical Uniform Linear Arrays (ULA) and distributed on the x-axis and y-axis, respectively. Figure 2(b) is the ULA on x-axis. The 9th antenna of two ULA is a common array element and can be regarded as reference array element. As an example, we set the array element spacing of ULA is 0.03 m, which can be applied to any wireless network with frequencies below 5GHz. For higher frequency, the element spacing can be adjusted to meet the limit of less than half the signal wavelength.Figure 2
(a) The antenna array model. (b) The model of ULA onx-axis and spatial smoothing algorithm. (c) The model of Uniform Frequency Array and frequency domain smoothing algorithm.
(a)(b)(c)


## 2.2. Positioning Signal Model

One of the purposes of this paper is to use OFDM pilot subcarrier signals to realize TDOA estimation of multipath signals, so we do not care about the details of information transmission. Another advantage of using pilot subcarrier signals is that it does not occupy the information transmission rate. Consider an OFDM signal withK subcarriers, the transmission symbols of OFDM can be expressed as Equation (1).
(1)St=1TAUE∑k=0K−1Cktej2πfc+k∆ft,wheret∈0,T,and T is the data duration of OFDM symbols,AUE is the transmission gain of the UE, Ckt represents the data on the k-th subcarrier,fc represents the carrier frequency modulated by the communication system, and ∆f is the frequency interval of subcarriers.When the OFDM signal propagates in indoor environment, the impact response of multipath channel can be expressed as Equation (2).
(2)ht=∑p=0P−1aptδt−ζp,where P is the number of channels,apt is the amplitude fading of the p-th path, t is the transmission time of the signal, and ζp is the delay of multipath channel.The multipath signals received by a single antenna can be expressed as Equation (3).
(3)yt=1TAUEABS∑p=0P−1apt∑k=0K−1Cktej2πfc+k∆ft−ζp+nt,where ABS is the gain of the receiving array antenna, nt is the Gaussian white noise, since the change of channel is very slow compared with the sampling time, 1/T,AUE,ABS,apt can be regarded as constants, and we assume these components are 1. The phase change caused by reflection is very small compared with propagation delay, which has been ignored.Some time-based methods assume there is high-precision time synchronization between the transmitter and receiver [19]. The delays of these algorithms estimate are ζp as shown in Equation (3), but the high cost of atomic clock and the low sampling frequency of communication system make these methods not suitable for practical application. Theoretically, the signal of LOS path will first arrive at the receiver due to the shortest distance. Set L0-th path is the LOS channel and the signal from L0 arrive at the receiver at time tLOS,where tLOS=t−ζLOS,and t is the moment when the signal is emitted. The TDOA between NLOS channels and LOS channel is τp=ζp−ζLOS. Thus, Equation (3) can be changed to Equation 4.
(4)yt=∑p=0P−1∑k=0K−1Cktej2πfc+k∆ftLOS−τp+nt.


## 3. AOA and TDOA Calculation

### 3.1. AOA Algorithm with Multipath Channel

Most existing algorithms obtain the AOA of the signal in 3D space by jointly estimating the pitch angles and azimuth angles [20, 21], which makes the calculation very complicated. To simplify the algorithm, we do not estimate the angles jointly.The spatial smoothing MUltiple SIgnal Classification (MUSIC) algorithm is used to estimate the AOA of multipath signals in this paper. The 1-dimensional MUSIC algorithm has the advantages of superresolution and high precision compared with the traditional beamforming algorithms and has lower computational complexity compared with the compressed sensing algorithms. And the computational complexity of MUSIC algorithm will increase exponentially with the number of dimensions when performing joint estimation, which is one of the shortcomings of the MUSIC algorithm. But we avoid this disadvantage by estimating the azimuth angles on two coordinate axes separately, and then match the two angles to obtain the AOA in 3D space.Taking the ULA on thex-axis as an example to introduce the principle of spatial smoothing MUSIC algorithm. The structure of ULA is shown in Figure 2(b), there are Mx antennas uniformly distributed in a line with the same spacing d. We set the ms-th antenna as the reference antenna and the right direction is the positive direction of the x-axis. When the far-field signals St are incident on the array antenna, the signals received by mx-th antenna can be expressed as Equation (5).
(5)ymxt=∑p=0P−1∑k=0K−1Cktej2πfc+k∆ftLOS−τp−τmx+nt,where τmx=mx−msdsinθ/c is the delay between the mx-th antenna and reference antenna, and mx=1,2,⋯,Mx,c is the speed of light. θ is the angle between signal propagation direction and the normal of ULA.Only one subcarrier frequency is needed to estimate AOA. Thus, the signal matrix of thek-th subcarrier received by the array can be expressed as Equation (6).
(6)Yx=y1ty2t⋯yMxt′,where “′” represents the transpose of matrix.The steering vector of array antenna is shown in Equation (7).
(7)bxθp=e−j2πfc+k∆f1−msdsinθp/ce−j2πfc+k∆f2−msdsinθp/c⋮e−j2πfc+k∆fMx−msdsinθp/c.The multipath signals matrix is shown in Equation (8).
(8)S=Cktej2πfc+k∆ftLOSCktej2πfc+k∆ftLOS−τ1⋮Cktej2πfc+k∆ftLOS−τP−1.The mathematical model ofYx can be expressed as Equation (9).
(9)Yx=BxS+Nx,where Bx=bxθ0bxθ1⋯bxθP−1, and Nx is the noise part.It can be seen from Equation (8) that the NLOS path signals can be represented by the LOS path signal as CktLOSej2πfc+k∆ftLOSe−j2πfc+k∆fτp,and the phase difference between each NLOS and LOS path is fixed. That means they are coherent signals. We use spatial smoothing to deal with multipath signals. Decompose the array antenna into staggered U subarrays, as shown in Figure 2(b). The number of antennas of each subarray is mU=Mx−U+1. Taking the first subarray as the reference subarray, the data of the u-th subarray can be expressed as Equation 10.
(10)Yxu=yxutyxu+1t⋯yxu+mU−1t′=BxDxu−1S+Nxu,where xu means the u-th subarray of x-axis array antenna, and u=1,2,⋯,U,Nxu is the noise part ofu-th subarray, the expression matrix Dxu−1 is shown in Equation (11).
(11)Dxu−1=e−j2πfc+k∆fdsinθ0/c0⋯00e−j2πfc+k∆fdsinθ1/c⋯0⋮⋮⋱⋮00⋯e−j2πfc+k∆fdsinθP−1/cu−1.The covariance matrix of the forward spatial smoothing MUSIC algorithmQxf is obtained as Equation (12).
(12)Qxf=1U∑u=1UBxDxu−1QsDxu−1HBxH+Nx,where Qs=ESSH,and the E⋯ represents the mean value,H represents the conjugate transpose.If taking the rightmost array as the reference array, the covariance matrix of the backward spatial smoothing MUSIC algorithmQxb as Equation (13) can be obtained by using a similar method.
(13)Qxb=1U∑u=1UBxJDxu−1QsDxu−1HHJBxH+Nx,whereJ is the mU×mU dimension transformation matrix as shown in Equation (14).
(14)J=00⋯1⋮⋮⋯⋮01⋱010⋯0mU×mU.The covariance matrix of forward and backward spatial smoothing algorithm is shown in Equation (15).
(15)Qxfb=12Qxf+Qxb.Setting the number of sampling snapshots in practical application isNS,the maximum likelihood estimation of the covariance matrix is Q^xfb=1/NS∑i=1NSQxifb. Eigenvalue decomposition of Q^xfb is shown in Equation 16.
(16)Q^xfb=VxsΣxsVxsH+VxnΣxnVxnH,where Σxs is the part of larger eigenvalues. Vxs is the eigenvectors corresponding to Σxs,VxsΣxsVxsH represents the signal subspace,Σxn is the part of smaller eigenvalues,Vxn is the eigenvectors corresponding to Σxn,VxnΣxnVxnH represents the noise subspace. Due to signal subspace which is the same as the subspace formed by the steering vectors of incident signals, the spectral formula of MUSIC algorithm is obtained as Equation 17.
(17)PxMUSIC=1bxθpHVxnVxnHbxθp.Since the noise,bxθpHVxn will not be equal to 0. By putting all possible values of θp into Equation (17), the right angles will make PxMUSIC get the maximum value.


### 3.2. TDOA Algorithm for Multipath Channel

We apply the spatial smoothing MUSIC algorithm to the frequency domain and realize the TDOA estimation based on the difference of subcarrier phases. The proposed algorithm can avoid the requirement of time synchronization and high sampling frequency. The phase of thek-th subcarrier from the p-th multipath to the array antenna is set as φk,p,where φk,0 is the phase of the LOS path. The phase difference between NLOS path and LOS path can be expressed as Equation (18).
(18)∆φk=φk,p−φk,0=2πfc+k∆fτp.Due toτp≫1/fc+k∆f,the period number of fc+k∆f in τp is fuzzy. But we can get the phase difference between the k-th and k+i-th subcarrier as Equation 19.
(19)Δφk,k+i=∆φk+i−∆φk=2πiΔfτp,where i∈0,K−1.Equation (19) avoids the fuzzy of period number caused by fc,and the iΔf can be considered the pilot subcarriers of OFDM signal. Pilot subcarriers are discretely inserted into the subcarriers of OFDM. And the pilot is a kind of signals known to both transmitter and receiver. As shown in Figure 3(a), the distance differences of multipath channels in our model are too short than the half wavelength of 7MHz, which makes the phase differences Δφk,k+i too small to be distinguished. Figures 3(b)–3(d) show that with iΔf increasing, the results of TDOA estimation become better. The simulation results in Figure 3 reflect that only when the frequency interval is properly selected, the TDOA can be estimated. For better TDOA resolution, only partial pilot subcarriers are extracted from the reference antenna to estimate TDOA in our model. Setting the interval ΔfG=iΔf=21MHz,and suppose there are GΔfG=12ΔfG in the OFDM signal. The g-th pilot subcarriers signal can be expressed as Equation 20.
(20)ygt=∑p=0P−1Cgtej2πfc+g∆fGtLOS−τp+nt,where Cgt represents the data of pilot signal, some constant terms of this formula are ignored for simplifying the calculation.Figure 3
Relationship between frequency interval and delay resolution, whereDelayns=13.31,22.91,19.22,21.66,SNR=20dB. (a) iΔf=7MHz. (b) iΔf=14MHz. (c) iΔf=21MHz. (d) iΔf=28MHz.
(a)(b)(c)(d)Setting the pilot subcarriers be uniformly arranged in the frequency domain as shown in Figure2(c), and the first frequency on the left side is the reference frequency, the right side is the direction of frequency increase. The pilot signals received by the reference antenna can be expressed as Equation (21).
(21)YG=y1ty2t⋯yGt′=LCG+NG,where NG is the noise matrix, CG is the pilot data received by the reference antenna, andL is the steering matrix of TDOA as shown in Equation (22).
(22)L=lτ0lτ1⋯lτP−1,where lτp can be expressed as Equation (23).
(23)lτp=e−j2πfc+1·∆fGtLOS−τpe−j2πfc+2·∆fGtLOS−τp⋮e−j2πfc+G·∆fGtLOS−τp.DividingG pilot subcarriers into W interleaved subarrays. Then, the MUSIC algorithm can be applied to the frequency domain using a similar way as AOA estimation. Taking the first subarray as the reference subarray, the signals of the w-th subarray can be expressed as Equation (24).
(24)YGw=ywtyw+1t⋯yw+gs−1t′=LDGw−1CG+NGw,where w=1,2,⋯,W, and gs=G−W+1 is the number of pilot subcarriers in each subarray,NGw represents the noise, the matrix DGw−1 is shown in Equation (25).
(25)DGw−1=e−j2π∆fGτ00⋯00e−j2π∆fGτ1⋯0⋮⋮⋱⋮00⋯e−j2π∆fGτP−1w−1.


## 3.1. AOA Algorithm with Multipath Channel

Most existing algorithms obtain the AOA of the signal in 3D space by jointly estimating the pitch angles and azimuth angles [20, 21], which makes the calculation very complicated. To simplify the algorithm, we do not estimate the angles jointly.The spatial smoothing MUltiple SIgnal Classification (MUSIC) algorithm is used to estimate the AOA of multipath signals in this paper. The 1-dimensional MUSIC algorithm has the advantages of superresolution and high precision compared with the traditional beamforming algorithms and has lower computational complexity compared with the compressed sensing algorithms. And the computational complexity of MUSIC algorithm will increase exponentially with the number of dimensions when performing joint estimation, which is one of the shortcomings of the MUSIC algorithm. But we avoid this disadvantage by estimating the azimuth angles on two coordinate axes separately, and then match the two angles to obtain the AOA in 3D space.Taking the ULA on thex-axis as an example to introduce the principle of spatial smoothing MUSIC algorithm. The structure of ULA is shown in Figure 2(b), there are Mx antennas uniformly distributed in a line with the same spacing d. We set the ms-th antenna as the reference antenna and the right direction is the positive direction of the x-axis. When the far-field signals St are incident on the array antenna, the signals received by mx-th antenna can be expressed as Equation (5).
(5)ymxt=∑p=0P−1∑k=0K−1Cktej2πfc+k∆ftLOS−τp−τmx+nt,where τmx=mx−msdsinθ/c is the delay between the mx-th antenna and reference antenna, and mx=1,2,⋯,Mx,c is the speed of light. θ is the angle between signal propagation direction and the normal of ULA.Only one subcarrier frequency is needed to estimate AOA. Thus, the signal matrix of thek-th subcarrier received by the array can be expressed as Equation (6).
(6)Yx=y1ty2t⋯yMxt′,where “′” represents the transpose of matrix.The steering vector of array antenna is shown in Equation (7).
(7)bxθp=e−j2πfc+k∆f1−msdsinθp/ce−j2πfc+k∆f2−msdsinθp/c⋮e−j2πfc+k∆fMx−msdsinθp/c.The multipath signals matrix is shown in Equation (8).
(8)S=Cktej2πfc+k∆ftLOSCktej2πfc+k∆ftLOS−τ1⋮Cktej2πfc+k∆ftLOS−τP−1.The mathematical model ofYx can be expressed as Equation (9).
(9)Yx=BxS+Nx,where Bx=bxθ0bxθ1⋯bxθP−1, and Nx is the noise part.It can be seen from Equation (8) that the NLOS path signals can be represented by the LOS path signal as CktLOSej2πfc+k∆ftLOSe−j2πfc+k∆fτp,and the phase difference between each NLOS and LOS path is fixed. That means they are coherent signals. We use spatial smoothing to deal with multipath signals. Decompose the array antenna into staggered U subarrays, as shown in Figure 2(b). The number of antennas of each subarray is mU=Mx−U+1. Taking the first subarray as the reference subarray, the data of the u-th subarray can be expressed as Equation 10.
(10)Yxu=yxutyxu+1t⋯yxu+mU−1t′=BxDxu−1S+Nxu,where xu means the u-th subarray of x-axis array antenna, and u=1,2,⋯,U,Nxu is the noise part ofu-th subarray, the expression matrix Dxu−1 is shown in Equation (11).
(11)Dxu−1=e−j2πfc+k∆fdsinθ0/c0⋯00e−j2πfc+k∆fdsinθ1/c⋯0⋮⋮⋱⋮00⋯e−j2πfc+k∆fdsinθP−1/cu−1.The covariance matrix of the forward spatial smoothing MUSIC algorithmQxf is obtained as Equation (12).
(12)Qxf=1U∑u=1UBxDxu−1QsDxu−1HBxH+Nx,where Qs=ESSH,and the E⋯ represents the mean value,H represents the conjugate transpose.If taking the rightmost array as the reference array, the covariance matrix of the backward spatial smoothing MUSIC algorithmQxb as Equation (13) can be obtained by using a similar method.
(13)Qxb=1U∑u=1UBxJDxu−1QsDxu−1HHJBxH+Nx,whereJ is the mU×mU dimension transformation matrix as shown in Equation (14).
(14)J=00⋯1⋮⋮⋯⋮01⋱010⋯0mU×mU.The covariance matrix of forward and backward spatial smoothing algorithm is shown in Equation (15).
(15)Qxfb=12Qxf+Qxb.Setting the number of sampling snapshots in practical application isNS,the maximum likelihood estimation of the covariance matrix is Q^xfb=1/NS∑i=1NSQxifb. Eigenvalue decomposition of Q^xfb is shown in Equation 16.
(16)Q^xfb=VxsΣxsVxsH+VxnΣxnVxnH,where Σxs is the part of larger eigenvalues. Vxs is the eigenvectors corresponding to Σxs,VxsΣxsVxsH represents the signal subspace,Σxn is the part of smaller eigenvalues,Vxn is the eigenvectors corresponding to Σxn,VxnΣxnVxnH represents the noise subspace. Due to signal subspace which is the same as the subspace formed by the steering vectors of incident signals, the spectral formula of MUSIC algorithm is obtained as Equation 17.
(17)PxMUSIC=1bxθpHVxnVxnHbxθp.Since the noise,bxθpHVxn will not be equal to 0. By putting all possible values of θp into Equation (17), the right angles will make PxMUSIC get the maximum value.


## 3.2. TDOA Algorithm for Multipath Channel

We apply the spatial smoothing MUSIC algorithm to the frequency domain and realize the TDOA estimation based on the difference of subcarrier phases. The proposed algorithm can avoid the requirement of time synchronization and high sampling frequency. The phase of thek-th subcarrier from the p-th multipath to the array antenna is set as φk,p,where φk,0 is the phase of the LOS path. The phase difference between NLOS path and LOS path can be expressed as Equation (18).
(18)∆φk=φk,p−φk,0=2πfc+k∆fτp.Due toτp≫1/fc+k∆f,the period number of fc+k∆f in τp is fuzzy. But we can get the phase difference between the k-th and k+i-th subcarrier as Equation 19.
(19)Δφk,k+i=∆φk+i−∆φk=2πiΔfτp,where i∈0,K−1.Equation (19) avoids the fuzzy of period number caused by fc,and the iΔf can be considered the pilot subcarriers of OFDM signal. Pilot subcarriers are discretely inserted into the subcarriers of OFDM. And the pilot is a kind of signals known to both transmitter and receiver. As shown in Figure 3(a), the distance differences of multipath channels in our model are too short than the half wavelength of 7MHz, which makes the phase differences Δφk,k+i too small to be distinguished. Figures 3(b)–3(d) show that with iΔf increasing, the results of TDOA estimation become better. The simulation results in Figure 3 reflect that only when the frequency interval is properly selected, the TDOA can be estimated. For better TDOA resolution, only partial pilot subcarriers are extracted from the reference antenna to estimate TDOA in our model. Setting the interval ΔfG=iΔf=21MHz,and suppose there are GΔfG=12ΔfG in the OFDM signal. The g-th pilot subcarriers signal can be expressed as Equation 20.
(20)ygt=∑p=0P−1Cgtej2πfc+g∆fGtLOS−τp+nt,where Cgt represents the data of pilot signal, some constant terms of this formula are ignored for simplifying the calculation.Figure 3
Relationship between frequency interval and delay resolution, whereDelayns=13.31,22.91,19.22,21.66,SNR=20dB. (a) iΔf=7MHz. (b) iΔf=14MHz. (c) iΔf=21MHz. (d) iΔf=28MHz.
(a)(b)(c)(d)Setting the pilot subcarriers be uniformly arranged in the frequency domain as shown in Figure2(c), and the first frequency on the left side is the reference frequency, the right side is the direction of frequency increase. The pilot signals received by the reference antenna can be expressed as Equation (21).
(21)YG=y1ty2t⋯yGt′=LCG+NG,where NG is the noise matrix, CG is the pilot data received by the reference antenna, andL is the steering matrix of TDOA as shown in Equation (22).
(22)L=lτ0lτ1⋯lτP−1,where lτp can be expressed as Equation (23).
(23)lτp=e−j2πfc+1·∆fGtLOS−τpe−j2πfc+2·∆fGtLOS−τp⋮e−j2πfc+G·∆fGtLOS−τp.DividingG pilot subcarriers into W interleaved subarrays. Then, the MUSIC algorithm can be applied to the frequency domain using a similar way as AOA estimation. Taking the first subarray as the reference subarray, the signals of the w-th subarray can be expressed as Equation (24).
(24)YGw=ywtyw+1t⋯yw+gs−1t′=LDGw−1CG+NGw,where w=1,2,⋯,W, and gs=G−W+1 is the number of pilot subcarriers in each subarray,NGw represents the noise, the matrix DGw−1 is shown in Equation (25).
(25)DGw−1=e−j2π∆fGτ00⋯00e−j2π∆fGτ1⋯0⋮⋮⋱⋮00⋯e−j2π∆fGτP−1w−1.


## 4. Positioning Algorithms in Indoor Multipath Environment

### 4.1. Matching of TDOA and AOA

Although the method for jointly estimating time, azimuth and pitch angles can directly obtain the correspondence between the TDOA and AOA in 3D space, the computational complexity will increase exponentially with the number of estimated parameters. The TDOA and AOA are individually estimated by the proposed method in the previous section, but it needs to be matched before positioning. According to Equation (5), except for the reference antenna, the pilot signals on them-th m≠ms array element also contain the information of tLOS,τp,τm, where the tLOS can be regarded as 0. τp has been estimated by the phase differences of pilot signals on reference array element. τm contains the angle information of the signals and the position information of the array element in the array antenna. Therefore, using the frequency domain smoothing MUSIC algorithm to process the pilot signals on the m-th array element can obtain the delay information of τp and τm. Based on this principle, we realize the matching of TDOA and AOA. The spectral formula of the m-th array element can be expressed as Equation 26.
(26)PMUSIC=1lτp,τmHVGnVGnHlτp,τm,where VGn is the noise feature vector, the expression of lτp,τm is shown in Equation (27).
(27)lmτp,τm=e−j2πfc+1·∆fGtLOS−τp−m−msdsinθpce−j2πfc+2·∆fGtLOS−τp−m−msdsinθpc⋮e−j2πfc+G·∆fGtLOS−τp−m−msdsinθpc,where m=1,2,⋯,ms−1,ms+1,⋯,Mx.Combining all the estimated AOA and TDOA in pairs and bringing them into Equation 27. When there is a large solution to Equation (26), it means that AOA and TDOA come from the same multipath signal. The multipath delay spread is larger than the time difference among array elements. To make the matching algorithm more effective, we use the farthest array elements. Similarly, using the same method to match the multipath TDOA with the AOA estimated by the array antenna on y-axis. If the AOA estimated by the array antenna on the x-axis corresponds to the same TDOA as the AOA estimated by the array antenna on the y-axis, it means that they come from the same signal.


### 4.2. Calculation of Target Position

Selecting the earliest arriving signal as the LOS path signal through TDOA, and the others are NLOS path signals. Then, constructing the reflection paths of NLOS channels combined with indoor map, the location is obtained through the LOS path and the reflection paths. For the 3D spatial coordinate system as shown in Figure1, the AOA estimated in previous content is the angles between the signal propagation directions and the normal of coordinate axes. Equation (28) can convert θp,μp to αp,βp,σp.
(28)αp=π2−θpβp=π2−μpσp=π−sin−1cos2αp+cos2βp,where αp,βp,σp represent the angle between the p-th multipath signal propagation direction and the positive direction of x-axis, y-axis, and z-axis, respectively, and αp,βp∈0,π,σp∈π/2,π in the model of Figure 1.The direction vector of thep-th multipath signal received by the array antenna can be expressed as Equation (29).
(29)n⟶p=cosαp,cosβp,cosσp.And the line equations wheren⟶p located can be expressed as Equation (30).
(30)SLp:xp=x0+γpcosαpyp=y0+γpcosβpzp=z0+γpcosσp,where x0,y0,z0 is the coordinate of one known point on the line SLp,γp is the scale factor between the point on the line and the point x0,y0,z0. For the multipath signals received by the array antenna,x0,y0,z0 is the coordinate of reference antenna.The NLOS paths are generated by the reflection of the walls, so the path lines will intersect the plane of the walls. Bring the plane equation parameters shown in Table1 into Equation (30) to obtain the coordinates of all intersection points. Since the coordinates of the reflection points must meet the conditions of indoor space, that is xp∈−3,3,yp∈−4,4,zp∈−4.5,0,we can exclude those intersections located outside the indoor space. Setting the coordinates of the reflection point of the p-th path in the indoor space is Rpxp,yp,zp,and the angles between the line where the path of emission signal is located and the positive direction of the 3 axes are αrp,βrp,σrp. According to the law of reflection for electromagnetic waves whose incident angle is equal to the reflection angle. If the plane is parallel to the x-axis, the relationship between αrp and αp isαrp=−αp,and if the plane is perpendicular to the x-axis, the relationship between αrp and αp isαrp=π−αp. The same principle can be applied to βrp and σrp. Bring Rpxp,yp,zp and αrp,βrp,σrp into Equation 30 to obtain the line equations of the NLOS paths before reflection, and we denote these equations of lines as RLrp,where rp=1,2,⋯,P−1.Theoretically, the intersection of lineSL0 and lines RLrp is the target location. A system of line equations for the P paths can be constructed as Equation (31).
(31)SL0:x,y,z=ox0,y0,z0+γ0n⟶0RLrp:x,y,z=Rpxp,yp,zp+γrpnr⟶rp,where nr⟶rp=cosαrp,cosβrp,cosσrp are the direction vectors of line equations RLrp,and rp=1,2,⋯,P−1.Due to the noise and the measurement errors, these lines may not intersect in 3D space. It means that Equation (31) may not have a common solution. In our method, the point in space with the closest distance to SL0 and all RLrp is regarded as the target location. The parameters in Equation 31 need to be solved can be defined as Equation (32).
(32)Ps=xUEyUEzUEγ0γ1⋯γrp⋯γP−1′,where xUE,yUE,zUE are the estimated location coordinates of the target.The direction vector parameters in Equation (31) can be defined as Equation (33).
(33)Pn=I−n⟶0′0⋯0I0−nr⟶1′⋯0⋮⋮⋮⋱⋮I00⋯−nr⟶P−1′,where I is a 3×3dimensional unit vector,0 is a 3×1 dimensional zero vector.The coordinate parameters of the reference antenna and reflection points are defined as Equation (34).
(34)PR=x0y0z0x1y1z1⋯xP−1yP−1zP−1′.According to equations (32)–(34), Equation (31) can be expressed as the matrix form: PnPs=PR. And the Ps can be solved through the method of least squares as Equation (35).
(35)Ps=Pn′Pn−1Pn′PR.Since the multipath signals are transmitted from one point, the linesSL0 and RLrp will not be parallel to each other in space, so Equation (36) must have a solution and the first three elements of Ps are regarded as the target position coordinates.


## 4.1. Matching of TDOA and AOA

Although the method for jointly estimating time, azimuth and pitch angles can directly obtain the correspondence between the TDOA and AOA in 3D space, the computational complexity will increase exponentially with the number of estimated parameters. The TDOA and AOA are individually estimated by the proposed method in the previous section, but it needs to be matched before positioning. According to Equation (5), except for the reference antenna, the pilot signals on them-th m≠ms array element also contain the information of tLOS,τp,τm, where the tLOS can be regarded as 0. τp has been estimated by the phase differences of pilot signals on reference array element. τm contains the angle information of the signals and the position information of the array element in the array antenna. Therefore, using the frequency domain smoothing MUSIC algorithm to process the pilot signals on the m-th array element can obtain the delay information of τp and τm. Based on this principle, we realize the matching of TDOA and AOA. The spectral formula of the m-th array element can be expressed as Equation 26.
(26)PMUSIC=1lτp,τmHVGnVGnHlτp,τm,where VGn is the noise feature vector, the expression of lτp,τm is shown in Equation (27).
(27)lmτp,τm=e−j2πfc+1·∆fGtLOS−τp−m−msdsinθpce−j2πfc+2·∆fGtLOS−τp−m−msdsinθpc⋮e−j2πfc+G·∆fGtLOS−τp−m−msdsinθpc,where m=1,2,⋯,ms−1,ms+1,⋯,Mx.Combining all the estimated AOA and TDOA in pairs and bringing them into Equation 27. When there is a large solution to Equation (26), it means that AOA and TDOA come from the same multipath signal. The multipath delay spread is larger than the time difference among array elements. To make the matching algorithm more effective, we use the farthest array elements. Similarly, using the same method to match the multipath TDOA with the AOA estimated by the array antenna on y-axis. If the AOA estimated by the array antenna on the x-axis corresponds to the same TDOA as the AOA estimated by the array antenna on the y-axis, it means that they come from the same signal.


## 4.2. Calculation of Target Position

Selecting the earliest arriving signal as the LOS path signal through TDOA, and the others are NLOS path signals. Then, constructing the reflection paths of NLOS channels combined with indoor map, the location is obtained through the LOS path and the reflection paths. For the 3D spatial coordinate system as shown in Figure1, the AOA estimated in previous content is the angles between the signal propagation directions and the normal of coordinate axes. Equation (28) can convert θp,μp to αp,βp,σp.
(28)αp=π2−θpβp=π2−μpσp=π−sin−1cos2αp+cos2βp,where αp,βp,σp represent the angle between the p-th multipath signal propagation direction and the positive direction of x-axis, y-axis, and z-axis, respectively, and αp,βp∈0,π,σp∈π/2,π in the model of Figure 1.The direction vector of thep-th multipath signal received by the array antenna can be expressed as Equation (29).
(29)n⟶p=cosαp,cosβp,cosσp.And the line equations wheren⟶p located can be expressed as Equation (30).
(30)SLp:xp=x0+γpcosαpyp=y0+γpcosβpzp=z0+γpcosσp,where x0,y0,z0 is the coordinate of one known point on the line SLp,γp is the scale factor between the point on the line and the point x0,y0,z0. For the multipath signals received by the array antenna,x0,y0,z0 is the coordinate of reference antenna.The NLOS paths are generated by the reflection of the walls, so the path lines will intersect the plane of the walls. Bring the plane equation parameters shown in Table1 into Equation (30) to obtain the coordinates of all intersection points. Since the coordinates of the reflection points must meet the conditions of indoor space, that is xp∈−3,3,yp∈−4,4,zp∈−4.5,0,we can exclude those intersections located outside the indoor space. Setting the coordinates of the reflection point of the p-th path in the indoor space is Rpxp,yp,zp,and the angles between the line where the path of emission signal is located and the positive direction of the 3 axes are αrp,βrp,σrp. According to the law of reflection for electromagnetic waves whose incident angle is equal to the reflection angle. If the plane is parallel to the x-axis, the relationship between αrp and αp isαrp=−αp,and if the plane is perpendicular to the x-axis, the relationship between αrp and αp isαrp=π−αp. The same principle can be applied to βrp and σrp. Bring Rpxp,yp,zp and αrp,βrp,σrp into Equation 30 to obtain the line equations of the NLOS paths before reflection, and we denote these equations of lines as RLrp,where rp=1,2,⋯,P−1.Theoretically, the intersection of lineSL0 and lines RLrp is the target location. A system of line equations for the P paths can be constructed as Equation (31).
(31)SL0:x,y,z=ox0,y0,z0+γ0n⟶0RLrp:x,y,z=Rpxp,yp,zp+γrpnr⟶rp,where nr⟶rp=cosαrp,cosβrp,cosσrp are the direction vectors of line equations RLrp,and rp=1,2,⋯,P−1.Due to the noise and the measurement errors, these lines may not intersect in 3D space. It means that Equation (31) may not have a common solution. In our method, the point in space with the closest distance to SL0 and all RLrp is regarded as the target location. The parameters in Equation 31 need to be solved can be defined as Equation (32).
(32)Ps=xUEyUEzUEγ0γ1⋯γrp⋯γP−1′,where xUE,yUE,zUE are the estimated location coordinates of the target.The direction vector parameters in Equation (31) can be defined as Equation (33).
(33)Pn=I−n⟶0′0⋯0I0−nr⟶1′⋯0⋮⋮⋮⋱⋮I00⋯−nr⟶P−1′,where I is a 3×3dimensional unit vector,0 is a 3×1 dimensional zero vector.The coordinate parameters of the reference antenna and reflection points are defined as Equation (34).
(34)PR=x0y0z0x1y1z1⋯xP−1yP−1zP−1′.According to equations (32)–(34), Equation (31) can be expressed as the matrix form: PnPs=PR. And the Ps can be solved through the method of least squares as Equation (35).
(35)Ps=Pn′Pn−1Pn′PR.Since the multipath signals are transmitted from one point, the linesSL0 and RLrp will not be parallel to each other in space, so Equation (36) must have a solution and the first three elements of Ps are regarded as the target position coordinates.


## 5. Simulation and Analysis

In this section, we will build a complete indoor positioning simulation based on the model in Figure1. Firstly, we analyze the advantages of using smoothing algorithms to process multipath coherent signals. Figures 4(a) and 4(b) show the AOA estimation results of classical MUSIC algorithm and spatial smoothing MUSIC algorithm, respectively. It can be seen that the spatial smoothing MUSIC algorithm can estimate the AOA but the classical MUSIC algorithm is not available. An array antenna with Mx elements can estimate up to 2Mx/3 multipath signals by using the spatial smoothing MUSIC algorithm. That means 16 antennas can estimate 10 multipath signals at the same time which can fully meet the needs of indoor environment. Using the same method can estimate angles μ0,μ1,⋯,μP−1 between the signal propagation directions and the normal of y-axis. The model has 4 multipath signals, but it can be seen from Figure 4(b) , the ULA on the x-axis can only estimate the AOA of 3 paths. That is because the AOA of the two multipath paths is too close to be distinguished by the array antenna. The same happens with the ULA located on the y-axis. But this does not affect the correctness of our algorithm, we will restore the missing angles through the matching algorithm.Figure 4
AOA estimation results of two algorithms. (a) AOA estimation results of classical MUSIC algorithm. (b) AOA estimation results of spatial smoothing MUSIC algorithm.
(a)(b)Figures5(a) and 5(b) show the TDOA estimation results of classical MUSIC algorithm and frequency domain smoothing MUSIC algorithm, respectively. It can be seen that the TDOA of coherent signals can be estimated by frequency domain smoothing MUSIC algorithm.Figure 5
TDOA estimation results of two algorithms. (a) The classical MUSIC algorithm. (b) The frequency domain smoothing MUSIC algorithm.
(a)(b)Since the estimated number of AOA is less than the actual number, the matching algorithm needs to use one of the AOA twice to recover the missing AOA information. The proposed matching algorithm makes up for this shortcoming by combining all angles and delays. For better resolution, the first array element which is farthest from the reference array element is selected for matching estimation. Figure6 shows the simulation results of the proposed matching algorithm, where the AOA of the x-axis and y-axis is matched with the TDOA, respectively. “∗” corresponds to the axis label at the bottom of the graph indicating the matching result between the TDOA and AOA of the x-axis. “+” corresponds to the axis label at the top of the graph indicating the matching result between the TDOA and AOA of the y-axis. Both the four larger values of “∗” and “+” above the dotted line in Figure 6 are chosen as the AOA and TDOA for the successful matching. It can be seen that each TDOA corresponds to AOA in the direction of x-axis and y-axis, respectively, where τ0,θ2,μ2 is the LOS path,τ1,θ3,μ3,τ2,θ1,μ3,τ3,θ1,μ1 are the 3 NLOS paths. The angles θ1=12.9° and μ3=−18.8° are used twice, respectively. In the actual environment, the corresponding angles of θ1 are 12.6°, 13.34°, and the corresponding angles of μ3 are -20.094° and -17.92°. They are very close and considered to be the same value by the array antenna, respectively.Figure 6
Simulation results of matching algorithms. The point above the dotted line means successfully matched.There are two types of AOA and one type of TDOA in our model that need to be estimated. Assuming that the angle range is [0°, 180°] and the resolution is 0.1°, the time delay range is [1, 30]ns and the resolution is 0.1ns. Then, traditional joint estimation MUSIC algorithm needs to calculate the spectral formula of MUSIC algorithm for1800×1800×300=9.72×106 times in total. We used the 3 times MUSIC algorithm and the twice matching algorithm to obtain these three parameters, respectively, which needs to calculate the spectral formula of MUSIC algorithm for 2×1800+300+2×12=3924 times in total. Compared with the existing joint estimation algorithm, the proposed algorithm greatly reduces the computational complexity.After obtaining the AOA and TDOA of the multipath signals in 3D space, we verified the target position calculation algorithm in Section4.2 through simulations. As shown in Figure 7(a),UE1.5,−2,−3 is the actual target location (represented by “+”). The thicker dotted lines represent the AOA of the multipath signals received by the array antenna. R1,R2, R3 represent the reflection points of NLOS paths, respectively. The solid lines are the path used for positioning. UEP1.5,−1.93,−3.05 is the estimated target coordinate (represented by “○”). It can be seen from Figure 7(b), although the positioning paths are far from the actual location of the target and do not intersect in 3D space, the closest point to all of these positioning paths does not deviate much from the actual location of UE.Figure 7
Simulation of positioning algorithm. (a) Location calculation of target. (b) The details of positioning.
(a)(b)Unlike most existing algorithms which treat multipath signals as a kind of interference or require multiple BS to coordinate positioning, the proposed method makes full use of multipath signals and only needs one BS to achieve 3D indoor positioning. In addition, our algorithms do not require precise clock synchronization of transceivers, and it is more accurate to judge the LOS channel by TDOA than by signal strength.In order to evaluate the influence of noise, we simulate the positioning error under a different Signal-to-Noise Ratio (SNR). The positioning error is defined as Equation (36).
(36)derror=1Tp∑i=1Tpxi−xUE2+yi−yUE2+zi−zUE2,where Tp is the number of times for positioning,xi,yi,zi is the target coordinates estimated by the proposed method, and xUE,yUE,zUE is the actual coordinates of the target.Figure8(a) shows the results of the evaluation about the influence of SNR. In the simulation experiments, the number of snapshots is set as 64. To avoid the influence of accidental factors, the positioning error at each SNR is the average value of 50 positioning errors, that is, TP=50. It can be seen that when the SNR is set to 24dB,the improvement of positioning errors will be very small. That is because when the SNR is improved, the errors caused by SNR will be very small, but the errors caused by resolution of the array antenna will become the main factor restricting the positioning accuracy. That is in the process of AOA estimation, the two angles with very small difference are regarded as the same AOA.Figure 8
Simulation of positioning error. (a) Influence of SNR on positioning errors, whereNs=64,Tp=50. (b) CDF curve of positioning errors, where Ns=64,Tp=100,SNR=20dB.
(a)(b)In order to evaluate the positioning errors of the proposed method, Figure8(b) shows the Cumulative Distribution Function (CDF) curve of errors, where the simulation conditions are set asNs=64,SNR=24dB,Tp=100. It can be seen that in 95% of the cases, the positioning accuracy is within submeter level, and within 0.4m in 60% of the cases. Note that the CDF curve is approximately stepped, the reason for this is the amplification effect of the reflection paths on the errors. For example, when the AOA estimated by the ULA at one of coordinate axes has errors, that will cause this multipath signal to have AOA errors on the other coordinate axes, too. And these errors will be increased again on the NLOS paths after reflection, and the longer the multipath paths, the more obvious the increase of these errors; finally, all these errors constitute the positioning errors.


## 6. Conclusions

The purpose of this research is to realize the 3D positioning with a single BS in complex indoor space. Signal propagation in indoor environment with serious multipath effects and those multipath signals are coherent, which make indoor positioning very difficult. Combining MIMO technology and OFDM technology in wireless communication system, we use the array antenna to estimate the AOA of the multipath signals and apply the spatial smoothing algorithm to the frequency domain to realize the TDOA estimation of multiple coherent signals. Then, we proposed a matching method of TODA and AOA, which reduced the computational complexity. After obtaining the TODA and AOA of the multipath signals, the LOS channel and NLOS channels are distinguished by TODA, and then, the AOA are combined with the indoor map to establish the reflection paths. The target location in 3D indoor space is obtained by combining the line equation where the reflection paths and the LOS path are located in 3D coordinate system. Finally, the proposed algorithm is verified by simulations. The results show the correctness of our method, and the localization accuracy is kept in the submeter level in 95% of the cases, and within 0.4m in 60% of the cases.Most of existing indoor positioning methods are based on 2D models, the algorithms require at least two or more BS for collaborative positioning, some methods try to avoid the effects of multipath signals. However, from the perspective of cost and application, single BS positioning is one of the most potential solutions for indoor positioning in the future. The method proposed in this paper not only utilizes multipath signals to realize the positioning of a single BS but also extends the indoor positioning algorithm to 3D space. That makes our positioning method have greater application prospects.Due to the resolution of ULA is not fine enough, the two AOA with very small difference are regarded as the same one, which will negatively affect the positioning results. This problem also occurs in other AOA estimation algorithms. How to avoid the influence of these errors remains to be further studied.



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*Source: 3144509-2022-04-14.xml*