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Abstract
As a result of the influence of indoor multipath effects, visible light positioning (VLP) technology based on the received signal strength (RSS) has difficulty providing stable and highly accurate position estimation. In order to cope with this challenge, a trilateration-based positioning method using channel state information (CSI) is proposed. Two algorithms that can estimate the number of channel paths are studied, which are the prerequisites for the obtaining of an accurate ratio of received power of the line-of-sight (LOS) path to total received power. Unlike traditional trilateration positioning methods where total received power is used, the proposed method introduces CSI as a means of estimating the received power of the LOS path, which it uses for ranging. By reducing interference from the reflected paths in the received power, the proposed method has high-precision and high-reliability positioning capabilities. The positioning performance of the proposed method in an indoor environment with four LED bulbs and a single photodetector (PD) is evaluated through simulation. The simulation results demonstrate that the proposed method provides an average accuracy of 6.1 cm and a maximum positioning error of 17.7 cm in a cubic space with a size of 4 m×4 m×3 m. Compared to the existing least-squares (LS) method that uses total received power, the new method achieves approximately 83% improvement in the mean error of positioning and 81% in root mean square error (RMSE).
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1. Introduction
In the context of the Industrial Internet of Things, high-precision and high-robust indoor positioning is crucial for emerging applications that possess location-aware capabilities [1]. Typical application scenarios include tracking people and goods in smart factories, positioning and navigating automated guided vehicles (AGV), and underground parking lot vehicle guidance [2,3]. Due to the complex indoor environment, traditional indoor positioning methods that use indoor radio frequency (RF) wireless signals (including WiFi and Bluetooth [4,5]) are susceptible to multipath signal propagation, which results in poor position estimation.
As a promising technology that possesses high-precision indoor positioning capabilities, visible light positioning (VLP) technology based on visible light communication (VLC) has attracted significant attention as it inherits many unique advantages of VLC, which include strong anti-electromagnetic interference, wide license-free bandwidth, cost-efficiency, and high security [6,7]. The algorithms used in indoor positioning technologies include angle of arrival (AOA) [8], time of arrival (TOA) [9], time difference of arrival (TDOA) [10], and received signal strength (RSS) [11]. Among them, as it has simpler deployment, that is, it does not require precise synchronization between devices nor complex receivers, the RSS method has become a widely studied method.
However, the existing trilateration and fingerprinting positioning algorithms based on RSS information also have some limitations. Combined with machine learning (ML) algorithms, fingerprint positioning can achieve centimeter-level localization accuracy, including long short-term memory fully connected network (LSTM-FCN) [12], weighted K-nearest neighbors (WKNN) [13], and memory artificial neural network (M-ANN) [14]. However, high-cost offline data collection and map construction are required for these methods, and their performance is largely affected by the number of training samples and the spatial distribution of sampling points [15]. The trilateration positioning that uses an angle diversity transmitter (ADT) [16] and RSS-assisted perspective-three-point algorithm (R-P3P) [17] provides decimeter-level accuracy by only considering the line-of-sight (LOS) channel. However, in actual scenarios, the optical signal that is emitted by the transmitter reaches the receiver through the LOS path while also reaching it through multiple reflection paths that are generated on the surface of the reflector. This means that the total received power of the received signal consists of two parts: the useful direct path received power and the harmful reflected path received power. As only the power attenuation of the LOS path from the transmitter is considered in the distance model, the total received power cannot be used for the accurate determination of distance. Therefore, the practicality of these RSS-based VLP solutions in actual complex environments is compromised.
Considering the issues of the above RSS-based VLP schemes, an RSS-based visible light positioning method using channel state information (CSI) is investigated in this paper. With this method, the positioning accuracy is optimized from the perspective of estimating the received power of the LOS path, with no offline data collection. The CSI is introduced into VLP for estimating the normalized amplitude of the LOS path. The received power of the LOS path is then solved to calculate the distance instead of the total received power. The main innovations of this work have two points. Firstly, with the help of CSI, the received power of the LOS path is extracted from the total received power in order to reduce the error that is caused by the received power of the reflected path during the distance calculation process. Secondly, two algorithms are investigated to estimate the number of channel paths for obtaining accurate LOS path received power. The benefits of the proposed method include the fact that there is no need for offline data collection in advance, and that it can adapt to the changes of the environment while achieving stable and high-precision positioning. A typical indoor scenario with four LED bulbs as the transmitter and a single photodetector (PD) as the receiver is considered for the simulation environment. The classic least-squares (LS)-based positioning method using the total received power and the nonlinear LS estimation method proposed in [18] can be used as the benchmark method for performance comparison and analysis. Simulation results demonstrate that the positioning accuracy of the proposed method is better than the benchmark method in different situations including the entire room, the edge area, and the inner area. The overall distinct performance improvement proves the presented positioning method that uses the received power of the LOS path to be effective.
The remainder of this paper is arranged in the following way. Section 2 provides a description of the system model. The presented VLP scheme based on CSI is elaborated upon in Section 3.1. In Section 3.2, two proposed algorithms for channel path number estimation are described in detail. Section 4 presents and analyzes the simulation results of positioning performance. Finally, conclusions are provided.
2. System model
2.1 Optical wireless channel model
To analyze the accuracy of the positioning method, a typical indoor environment in a cubic space is assumed, the geometric model of which can be seen in Fig. 1. The dimensions of the room are 4 m×4 m×3 m, and four LED bulbs (transmitters) are deployed on the ceiling as anchor nodes for positioning. They also serve the function of sending positioning information, including unique identification, coordinate information, and transmitted optical power. The coordinates of LEDs (Tx1∼4) are (1 m, 1 m, 3 m), (3 m, 1 m, 3 m), (1 m, 3 m, 3 m), and (3 m, 3 m, 3 m) respectively. The entire room is divided into the inner area and edge area using a dotted line. The edge area is the area within 1 m of the wall, and the remaining area is considered to be the inner area. A photodiode on the ground acts as a receiver, and can obtain the received power and positioning information of different LEDs from different time slots in order to calculate its distance to the LEDs. As can be seen in Fig. 1, the signal that is sent by the LED is transmitted to the PD via free space, and the received signal includes the LOS signal and a large number of reflected signals that are generated by the reflective surface. Therefore, in this wireless optical channel modeling, a LOS path and the first-order reflection paths reflected by the wall are considered. The deterministic method is used for calculating the channel DC gain of the LOS path and the first-order reflection paths, as outlined in [18].
The total channel DC gain from LED to PD can be calculated by adding the contribution of LOS and the contribution of all reflections. Considering the delay difference between signals that pass through various paths to the PD, the discrete-time channel impulse response (CIR) can be constructed by the LOS path and multiple discrete non-line-of-sight (NLOS) paths. The discrete NLOS path gain can be calculated by adding the gains of all reflection paths where the delay is within one sampling interval. The discrete-time CIR is expressed as
where, L denotes the number of channel paths, and the l-th path gain is given by2.2 RSS-based positioning model
For the avoidance of mutual interference of signals from different LEDs, LEDs use time-division multiplexing access (TDMA) as a means of transmitting data. The TDMA scheduling scheme and signal frame structure are shown in Fig. 2. More specifically, time can be divided into multiple periods, each period including five time slots of identical duration. In time slot 0, all LEDs are in a silent state and are off, which can be used for measuring background light intensity. With each of the remaining four time slots (time slot 1∼4), only one LED is in the sending state to transmit positioning signal, whereas the other three LEDs are in the silent state, meaning that PD can receive signals from different LEDs and measure the corresponding received power.
As can be seen in Fig. 2, a positioning signal frame has three parts. The first part is a training symbol of length N, which is used for symbol timing synchronization. The second part is a data symbol that carries positioning information, including LED identification, coordinate information, and transmitted optical power. The final part contains multiple identical pilot symbols for channel estimation. Training symbol and data symbol have identical symbol length $N$ and CP length ${N_g}$. The length of the pilot symbol, ${N_p}$, is one-pth of the length of the data symbol, that is, $N = p{N_p}$, where p is an integer.
Assuming that the LED transmits the positioning signal with average transmitted optical power ${P_t}$, the received optical power from LED-TXi can be expressed as
For the positioning and navigation scenarios of the AGV in the smart factory, the LED is placed vertically downwards with an elevation angle of −90°, and PD is placed vertically on the ground with an elevation angle of +90°, which is reasonable. It can be deduced that $\cos (\phi ) = \cos (\psi ) = {H / {{d_i}}}$, where $\phi$ represents the irradiance angle of the LED, $\psi$ is the incidence angle of the PD, and $H$ denotes the vertical distance between LED and PD. The distance ${d_i}$ from LED-TXi to the PD is calculated using the following formula.
After the distance value from the LED to the PD is obtained, the relationship between the receiver coordinates $(x,y)$ and the transmitter coordinates $({x_i},{y_i})$ can be expressed using the following group of equations.
In order to minimize the squared Euclidean distance, the LS estimation method is used for estimating the receiver coordinates, which is calculated as
2.3 DCO-OFDM VLC system model
With the VLP system, the direct current biased optical OFDM (DCO-OFDM) is adapted to modulate the information that is sent by the LED, which includes positioning information, training symbols, and pilots. The detailed information of the DCO-OFDM system model can be seen in Fig. 3.
As the data symbol and pilot symbol are generated using the same process, with the exception of the different inverse fast Fourier transform (IFFT) sizes that are used, for ease of illustration, the pilot symbols are taken as an example for introducing the generation process. In this paper, the first ${{{N_p}} / 2} - 1$ elements of the Shapiro-Rudin sequence are modulated by BPSK for the generation of the pilot sequence, which is denoted as ${{\textbf X}^P} = [\begin{array}{{cccc}} {X_1^{}}&{X_2^{}}& \cdots &{X_{{N_P}/2 - 1}^{}} \end{array}]$. The definition of the Shapiro-Rudin sequence is as follows [19].
Based on satisfying the Hermitian symmetry that is required by the optical OFDM system, the frequency domain pilot sequence is modulated onto the subcarriers to form ${\textbf X} = [\begin{array}{{cccc}} 0&{X_1^{} \cdots X_{{N_P}/2 - 1}^{}}&0&{X_{{N_P}/2 - 1}^\ast \cdots X_1^\ast } \end{array}]$, where ${({\cdot} )^ \ast }$ indicates conjugation operation. Following IFFT and the addition of a cyclic prefix, the frequency domain symbols generate real-valued DCO-OFDM signals, denoted as $x(n)$.
To compensate for the non-linear distortion caused by the inherent nonlinear characteristics of the LED, the time domain DCO-OFDM signal can be pre-distorted and clipped to ensure that the LED works within the linear range [20]. The clipped signal ${x_{clip}}(n)$ is represented as
At the receiver, the light intensity is converted into an electrical signal by a photodetector (PD), and the analog-to-digital (A/D) conversion is then imposed for the generation of a baseband signal $r(n)$, which is expressed as
where $\gamma$ denotes the responsivity of PD, ${\otimes}$ represents convolution, and $h(n)$ indicates the CIR, as was described in section 2.1. $w(n)$ is the additive white Gaussian noise (AWGN) that is composed of shot noise and thermal noise [3], and the total electrical domain noise power $\sigma _{noise}^2$ is given by whereThe SNR in the electrical domain is defined as [3]
At the receiver, the role the symbol timing synchronization module plays is the detection of signal frames. This ensures the correct extraction of positioning information, a prerequisite for subsequent high-quality channel estimation. In our previous work [21], an efficient timing offset estimation method for DCO-OFDM was proposed and this is applied here for estimating the starting position of the OFDM symbol.
Following the removal of the CP and FFT, the a-th frequency-domain pilot symbol from LED-TXi is obtained, which is denoted as ${\hat{{\textbf X}}^{({i,a} )}} = [ {\hat{X}_0^{({i,a} )}}\quad{\hat{X}_1^{({i,a} )} \cdots \hat{X}_{{N_P}/2 - 1}^{({i,a} )}}\quad{\hat{X}_{{N_P}/2}^{({i,a} )}}\quad{{({\hat{X}_{{N_P}/2 - 1}^{({i,a} )}} )}^\ast } \cdots {{({\hat{X}_1^{({i,a} )}} )}^\ast }]$, where $a = 0,1, \cdots N1 - 1$, and N1 is the number of pilot symbols. The channel frequency response (CFR) estimated using the LS method is given by
3. Methodology
The distance solving method introduced in section 2.2 is derived from the channel gain model of the direct signal. Accurate ranging can be achieved if the received signal only contains the LOS signal. However, as there is a large number of reflected signals in addition to the direct signal in the real scenario, the received optical power equals the sum of the power of the LOS path and the power of the NLOS paths. The use of the total received power for solving the distance value will introduce errors, which can result in inaccurate position estimation.
3.1 VLP scheme based on CSI
For reducing the interference caused by the power of the NLOS path in the received power, a visible light positioning scheme based on CSI is proposed. The estimated CIR is utilized for estimating the proportion of the power of the LOS signal in the total received power, so as to deduce the power of the LOS path. The estimated LOS power is then used for calculation of the distance from the LED to the PD to achieve high-precision position estimation. The processing procedure of the CSI-based positioning scheme is shown in Fig. 4.
In order to improve the accuracy of CIR estimation, N1 estimated CIR vectors are used for averaging to reduce the influence of noise. The averaged CIR from LED-TXi is expressed as ${\hat{{\textbf {h}}}^i} = {{\sum\limits_{a = 0}^{N1 - 1} {{{\hat{{\textbf {h}}}}^{({i,a} )}}} } / {N1}}$. It is important to note that the length of the estimated CIR is greater than the number of actual channel paths. Without the consideration of noise, the CIR estimated using the pilot symbol consists of the real CIR followed by a string of zeros. After noise is added, the estimated CIR can be expressed as
Assuming the number of channel paths L to be known, the gains of the first L paths of CIR are normalized for estimating the proportion of the power of each path to the total received power. The normalized amplitude of the paths can be expressed as
Here, the amplitude of the first path is considered to be the percentage of the LOS path in the total received power, so the received power of the LOS path can be calculated as
The received power of the LOS path mentioned above is substituted into Eq. (4) to calculate the direct path distance from LED-TXi to PD, which is expressed as
Then, the receiver coordinates are obtained using the LS estimation method from section 2.2. Theoretically, the distance calculated by the received power of the LOS path is of greater accuracy than the total received power, which results in higher positioning accuracy. It should be noted that the calculation of the normalized amplitude of the path described above is based on the assumption that the number of channel paths is known, but the channel path number in the actual system is unknown and varies according to the location of the receiver. Therefore, estimation of the channel path number is an indispensable step in the positioning scheme that is proposed by this paper, and it is also the key for affecting positioning accuracy.
3.2 Channel path number estimation algorithm
From Eq. (17), it can be observed that the normalized amplitude of the LOS path has a direct correlation with the number of paths that are used. If too few paths are used, the proportion of the LOS path is too large, and the worst case degenerates into one, that is, the estimated received power of the LOS path is equal to the total received power. Conversely, if too many paths are used, the proportion of the LOS path is reduced, which affects the estimation accuracy of the received power of the LOS path. In this subsection, two algorithms for channel path number estimation are proposed with the aim of achieving high-precision positioning. The input of the two algorithms is N1 estimated CIR vectors.
1) Algorithm based on the statistical characteristics of CIR
By observing Eq. (16), it can be seen that the first L elements in the CIR vector are real path gains with noise superimposed, whereas the following N1-L elements only contain noise, where the mean of noise is zero and the variance is $\sigma _{}^2$.
Theoretically, for the first L elements, the mean of l-th element should be equal to the true path gain, which is expressed as $\textrm{E}[{{{\hat{h}}^i}(l)} ]= \textrm{E}[{{h^i}(l) + {w^i}(l)} ]= {h^i}(l)$. This is a value greater than zero, and the variance is equal to the variance of the noise, which is expressed as $\textrm{D}[{{{\hat{h}}^i}(l)} ]= \sigma _{}^2$. For the last N1-L elements, the mean of each element is zero. The ratio of the variance to the mean is defined as an indicator for measuring whether it is a true channel path, which is denoted as $\textrm{V}(l)$. The $\textrm{V}(l)$ of the first L elements is smaller than that of the last N1-L elements. From the above analysis, the following decision rules can be defined. For 0 < l < N1-1, the index of the maximum value of $\textrm{V}(l)$, $K$, is regarded as the first item in the CIR vector that does not contain the real channel path. K−1 is then regarded as the last real channel path, meaning that the number of channel paths is equal to K. Algorithm 1 outlines the steps for estimating the number of channel paths using the statistical properties of CIR.
2) Algorithm based on the correlation of elements in CIR
With this algorithm, the mean and clipping operations mentioned above must first be performed on the input N1 estimated CIR vectors as a means of obtaining the processed CIR vector that is composed of positive real numbers and zeros.
There is a greater likelihood that the element that is closest to the front of the estimated CIR vector is a true channel path, and the element that is closest to the back is more likely to be a non-channel path. More specifically, the first element of the CIR vector should be a real channel path item, and the final element should be a non-channel path item. Based on this, a characteristic parameter is defined, the ratio of the difference between different elements to the corresponding distance value, for evaluating the correlation between a non-zero element and other elements. If the i-th non-zero element is closer to the real channel path than the non-channel path, it can be classified to the real channel path, and vice versa. Through the iterative judgment of the classification of all non-zero elements, the maximum index of the elements classified into the real channel path can be obtained, which is the estimated number of channel paths. Algorithm 2 outlines the steps for estimating the number of channel paths using the correlation of elements in CIR.
4. Results and discussion
4.1 Simulation setup
In this section, an indoor scenario as shown in Fig. 1, is considered for evaluating the positioning accuracy of the proposed method. The main simulation parameters are summarized in Table 1, where the parameters of noise refer to [16], and the parameters of channel modeling refer to [21]. In view of the symmetry of the position of the LED lights in the simulation scene, we considered a quarter of the indoor area for simulation, where the x-axis is in the range of [0∼2 m] and the y-axis is in the range of [0∼2 m]. The ground with dimensions of 2 m×2 m is divided into a number of 1 cm×1 cm grids, and the receiver performs 100 positioning at each position to measure the positioning error in various positions.
Table 1. Simulation Parameters
Assuming the modulation bandwidth of the system following pre-equalization to be 125 MHz, and the received sample interval to be 4 ns. The length of training symbol $N$ and of the CP ${N_g}$ are set based on the criterion that the receiver is able to obtain accurate timing synchronization. The setting of the length of the pilot symbol ${N_p}$ must consider that all the components of the CIR are measurable. In this work, based on the simulation results of channel modeling, $N = 512$, ${N_g} = 16$, and ${N_p} = 32$ are set. The number of pilot symbols is set to 128. In the proposed method, the training sequence used for generating the training symbol and the pilot sequence used for generating the pilot symbol are modulated by BPSK, that is, the bit “0” is mapped to -1-j, and the bit “1” is mapped to 1 + j. In addition, at the transmitter, the training symbol, data symbol, and pilot symbols are assumed to have a consistent average electric power before clipping.
As the SNR of the LED furthest from the PD among the four LEDs is too low, errors in timing synchronization will be caused. Therefore, the CSI-based positioning algorithm measures the signals of the nearest three LEDs for calculating the position. By modeling the indoor environment, a rough estimate of the maximum number of paths ${L_{\max }}$ and the minimum number of paths ${L_{\min }}$ in the environment can be provided. In order to further enhance the accuracy of the estimation of the number of channel paths, the number of estimated channel paths is limited to the range of (${L_{\min }}$∼${L_{\max }}$). In the simulation, ${L_{\min }}$ is set to 4, and ${L_{\max }}$ is set to 8.
4.2 Performance evaluation
To show the influence the reflected path power has on the positioning accuracy, the positioning performance of different methods is simulated and analyzed in this subsection. In addition, the classic LS-based positioning method described in section 2.2 and the nonlinear LS estimation method in [18] serve as the benchmark method for performance comparison. In the nonlinear LS estimation method, the number of iterations is set to 5, and the 25 points surrounding the estimated position are selected for each iteration. These points are arranged in a 5×5 square centered on the estimated position, in which the spacing between their abscissa or ordinate is 1 cm. In the following simulation and analysis, the CSI-based positioning method using the channel path number estimation algorithms 1 and 2 is abbreviated as proposed methods 1 and 2. The LS positioning method that uses the total received power of four LEDs is abbreviated as benchmark method 1, and the method using the total received power of the nearest three LEDs is abbreviated as benchmark method 2. The nonlinear LS estimation method using four LEDs and the nearest three LEDs are referred to simply as benchmark method 3 and benchmark method 4, respectively. The mean error and root mean square error (RMSE) of position estimation are used for evaluating positioning accuracy, where mean error and RMSE are defined as
In Fig. 5, the two-dimensional (2D) distribution of the normalized amplitude of the LOS path from different LEDs can be seen, which reflects the strength of the multipath effects at various locations. This normalized amplitude shows the ratio of the received power of the LOS signal to the total received power. The larger the amplitude is, the less affected it is by the reflected signal, and the LOS signal is dominant. The smaller the amplitude is, the greater the proportion of the power of the reflection path is to the total power, meaning that the difference between the total received power and the received power of the LOS path is greater. Correspondingly, the greater the deviation between the distance calculated using the total received power and the true distance.
The 2D distribution of the average positioning error of different positioning methods (the benchmark methods 1–4 and the proposed methods 1/2) can be seen in Fig. 6, where the average positioning error is obtained by averaging 100 positioning errors for each position. In Figs. 6 and 7, the number of pilot symbols used by the proposed methods is 128.
Observation of the error distribution in Fig. 6 shows that almost all methods obtain the worst positioning accuracy in the corner of the room. This is due to the position being the farthest from all LEDs, which results in the most severe impact from the reflection path and the lowest SNR. When the position is approaching the center of the room, the LOS path dominates and the positioning error is significantly reduced as SNR increases. From the results in Fig. 5, it can be seen that the signal of LED-Tx4 is most affected by multipath in the corner that is farthest from it, which results in a larger error in calculated distance. Therefore, when the PD is near the corner of the room, the positioning accuracy of benchmark method 1 that uses the signals of four LEDs is lower than that of benchmark method 2 that uses the signals of the nearest three LEDs. This proves that increasing the number of LEDs is not always of benefit to positioning accuracy. When a distance value with a larger error is used to participate in the position calculation, this causes positioning accuracy to decrease. In addition, benchmark method 3/4 are slightly better localization accuracy than benchmark method 1/2. In contrast, the positioning accuracy of the proposed methods is significantly better than that of four benchmark methods, as using the estimated received power of the LOS path rather than the total received power for calculating the distance effectively reduces the ranging error, which results in excellent positioning accuracy.
The comparison results of the cumulative distribution function (CDF) curves of the average positioning errors of different methods under three different cases of the edge area, the inner area and the entire room can be seen in Fig. 7. It shows that whether in the inner area, the edge area, or the entire room, the CDF curves of the two methods are much higher than the curve of the benchmark methods in the entire distribution range of the positioning error, which indicates that the positioning performance is significantly improved.
To verify the effectiveness and generality of the CSI-based method proposed in this paper, the positioning accuracy of two different scenes are simulated and evaluated. Scene 1 is the simulation environment mentioned above in this paper. Scene 2 has 5 light sources whose coordinates are (0 m, 2 m, 3 m), (2 m, 0 m, 3 m), (2 m, 2 m, 3 m), (2 m, 4 m, 3 m), and (4 m, 2 m, 3 m). The positioning performance of different methods in different scenes can be seen in Table 2.
Table 2. Comparison of positioning performance of different methods
Considering the entire room in scene 1, compared to benchmark method 1, both proposed methods have the ability to achieve approximately 83% and 81% improvement in mean and RMS of the positioning error respectively. Similarly, both proposed methods can achieve 80% and 77% improvement compared to benchmark method 2. Under the condition of using the same number of LEDs, benchmark method 3/4 are slightly better than benchmark method 1/2, respectively, but they have huge gaps with the proposed methods.
For the edge area, the average positioning accuracy of both proposed methods is improved by about 36.1 cm, 27.9 cm, 33.4 cm, and 26.2 cm, compared to benchmark methods 1–4. In addition, the average localization accuracy of the proposed methods in the inner area is improved by about 13.8 cm, 13.4 cm, 12.4 cm, and 10.1 cm for benchmark methods 1–4. It can be seen that in the edge area that is seriously affected by the reflection path, the advantage of the proposed method compared to the benchmark method is greater than it is in the inner area, proving that this method can be used for effectively dealing with the challenge of complex indoor scenes with a large number of reflection paths. The proposed method in scene 2 also achieves excellent performance and a huge improvement compared to the benchmark methods. Generally, the significant improvement in positioning performance verifies that the CSI-based positioning method that is proposed by this paper can facilitate an effective reduction in the impact the reflected path has through the utilization of LOS path power, which improves positioning accuracy.
Below, the effect of the estimated number of channel paths on the localization accuracy is further investigated. The normalized distance estimation error is expressed as follows.
It can be seen from Eq. (32) that normalized distance estimation error is related to the number of channel paths used and the superimposed noise. The simulation results of the positioning performance of the new method in scene 1 and scene 2 when different numbers of channel paths are used can be seen in Fig. 8(a) and Fig. 8(b), respectively. The 99% error in Fig. 8 refers to the error value when the CDF is equal to 0.99.
Fig. 8. The simulation results of the positioning performance of the new method when different numbers of channel paths are used
Figure 8 shows that as the number of paths that are used increases, localization accuracy also increases. This is due to the fact that when the number is small, part of the real path information is discarded, leading to a larger ratio of estimated LOS paths than the actual one, thereby introducing errors into the distance calculation. When the number of paths that are used reaches a certain number, the normalized amplitude of the estimated LOS path is closer to the real situation, and the performance reaches a peak and has a tendency to be stable. As the number of paths that are used continues increasing, performance starts to degrade, as when the number of paths that are used is significantly greater than the actual number of paths, additional noise interference is introduced, which results in inaccuracies in the normalized amplitude of estimated LOS paths.
As can be seen from Fig. 8(a) and Fig. 8(b), the channel path number estimation methods proposed in this paper can achieve good performance in different simulation scenes, which verifies their effectiveness. However, it should also be noted that in some scenes, the performance of the proposed algorithms has some gaps compared to the above best case, showing that there is still room for further optimization of the estimation algorithm.
It should be noted that this work is purely simulation-based. Therefore, in the next step, we will build an experimental prototype to conduct experiments in real scenarios to further verify the localization performance of the proposed method.
The time complexity of the two proposed channel path number estimation algorithms can then be evaluated. For algorithm 1, it can be seen that the input to the calculation of ${E_l}$ in the Eq. (20) is N1 vectors of length ${N_P}$, so it has a time complexity of $\textrm{O}({N_P} \cdot N1)$. Similarly, the time complexity of calculation of ${D_l}$ is $\textrm{O}({N_P} \cdot N1)$. The second step has a time complexity of $\textrm{O}({N_P})$. It is known that the time complexity for searching for the maximum element in the array with length N is O(N), meaning that the time complexity for the third step is $\textrm{O}({N_P})$. Therefore, the time complexity of this algorithm is $\textrm{O}(2{N_P} \cdot N1 + 2{N_P})$. It can be seen that the time complexity of algorithm 1 has a direct correlation with the number and length of CIR vectors that are used.
For the first step of algorithm 2, the time complexity of searching for non-zero elements in the clipped CIR vector of length ${N_P}$ is $\textrm{O}({N_P})$. In the second step, the input to the calculation of $\textrm{dif}(l)$ in the Eq. (24) is $z\textrm{ - }1$ vectors of length ${N_P}$, meaning that its time complexity is $\textrm{O}({N_P} \cdot (z\textrm{ - }1))$. Similarly, for Eqs. (25)–(27), the time complexity is $\textrm{O}({N_P} \cdot (z\textrm{ - }1))$. The classification in Eq. (28) requires just one comparison per loop, and the time complexity of z-1 loops is $\textrm{O}(z\textrm{ - }1)$. In the third step, as the number of elements of ${{\textbf I}_r}$ is uncertain, the maximum time complexity that is considered here is $\textrm{O}(z\textrm{ - }1)$. Therefore, the time complexity of algorithm 2 is $\textrm{O}(4{N_P} \cdot (z - 1) + 2(z - 1) + {N_P})$. It is evidenced that the time complexity for algorithm 2 has a correlation with the length of the CIR vector and the number of its non-zero elements, regardless of the number of CIR vectors that are used. The complexity comparisons of both proposed algorithms can be seen in Table 3.
Table 3. Time complexity comparison
As the number of non-zero elements in the estimated CIR vector cannot exceed ${N_P}$, the following relationship is true.
Therefore, the upper bound of the time complexity of algorithm 2 is expressed as $\textrm{O}(4{N_P} \cdot {N_P} + 3{N_P})$. The time complexity of algorithm 1 is greater than that of algorithm 2 under the following conditions.
By simplifying the above equation, it can be obtained that
In this work, $N1 = 128$ and ${N_p} = 32$ are set, which satisfy the above condition. This means that in the simulation conditions that are set in this paper, algorithm 2 is better than algorithm 1 in terms of time complexity.
5. Conclusions
With the help of CSI, an alternative method of optimizing positioning accuracy is explored in this paper, which is different to the current popular machine learning optimization algorithms. This CSI-based positioning method uses the received power of the LOS path for calculating the distance rather than the total received power, which effectively reduces the interference of received power of the reflection path to ranging in order to obtain reliable distance measurements. The positioning performance of different methods is evaluated by computer simulation. The simulation results showed that the positioning accuracy of the proposed method is significantly better than that of the benchmark method in terms of both the edge and center areas of the room. Particularly in the edge area, the advantages of the proposed method are more obvious, which indicates that the proposed method is robust and can be adapted in the diversity of indoor environments. For the next step, an experimental prototype will be established as a means of verifying the performance of the proposed method.
Funding
Science and Technology Planning Project of Guangdong Province (2016A010101033); Key projects of basic and applied basic research in Jiangmen (2021030103250006686); Hong Kong and Macao Joint Research and Development Foundation of Wuyi University (2019WGALH21).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Dataset 1, Ref. [22].
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