Open access peer-reviewed chapter

Averaging Indoor Localization System

By Eman Shawky Abd El-Fattah Amer

Submitted: November 28th 2020Reviewed: April 1st 2021Published: August 21st 2021

DOI: 10.5772/intechopen.97531

Downloaded: 79

Abstract

This chapter aims at improving the accuracy of estimation the localization by using the RSS method to estimate the positions and take into account the effects of both LOS propagation. The proposed system is depending on developing a mathematical model for the noisy VLC positioning system. For improving the results, adopting the KF is combined with the proposed system, which is considered an optimal estimator. The performance of the proposed technique is determined by evaluating the positioning errors in a typical room. Also this chapter develops the accuracy of the positioning system by using different ideas with average techniques. The discussion of the results for averaging technique is displayed.

Keywords

  • RSS
  • KF
  • VLC
  • localization

1. Introduction

This chapter aims at improving the accuracy of estimation the localization by using the Received Signal Strength (RSS) method to estimate the positions and take into account the effects of both line of sight (LOS) and non-line of sight (NLoS) propagations. The proposed system is depending on developing a mathematical model for the noisy Visible light communication (VLC) positioning system. For improving the results, adopting the Kalman filter (KF) is combined with the proposed system, which is considered an optimal estimator. The performance of the proposed technique is determined by evaluating the positioning errors in a typical room. Also, this chapter develops the accuracy of the positioning system by using different ideas with average techniques.

The remaining of this chapter is organized as follows: Section 2 discusses the optical channel in indoor systems. Section 3 is devoted to explaining the methodology of localization using RSS techniques. A mathematical derivation for performance evaluation is developed in the same section as well. In Section 4, The proposed KF algorithm is presented with explaining its algorithm for estimation correction. There is an average technique aiming to use the average method as shown in Section 5. Using both effects of LOS and the first reflection of NLoS propagation is done in the average proposed system. Adopting KF with averaging is shown in Section 6. The discussion of the results for averaging technique is displayed in Section 7. Section 8 shows the comparison between the results with some recent references. Finally, the concluding remarks are given in Section 9.

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2. Optical channel model

The characteristics of the channel modeling have been analyzed with the effects of the channel distortions in [1]. The power associated with the channel is separated into two factors, these being optical path loss (PL) and multipath dispersion. The PL is calculated from the knowledge of the receiver size, the transmitter beam divergence, and separation distance. However, a NLoS configuration (diffuse systems) mainly used in the indoor environment, uses reflections of the room surfaces and furniture. These reflections could be seen as unwanted signals or multi-path distortions which predict the PL more complex. The OW channel transfer function is defined by

H=HLoS+HNLoSE1

According to Figure 1, it describes HloSas the contribution due to the LoS, which is independent of the modulation frequency and it depends on the distance between transmitter and receiver. In a VLC system, the direct current (DC) gain of a VLC channel is expressed by.

HLOSi=m+12πdi2cosmϕiARcosψiTsψigψi,E2

Figure 1.

The channel model of VLC system.

The received power therefore becomes

Pr=HLOS.Pt,E3

where Ptis a transmitted power mis the Lambertian order, diis the distance between transmitter iand the receiver, ϕiis the irradiance angle, ψiis the incidence angle, Tsand gare the gains of the optical filter and concentrator at the receiver (assumed here as unity gain), and ARis the detector effective area. Where the channel DC gain HNLOSof the first reflection is shown as in the following equation related to Figure 1.

dHNLOSip=m+12πDip,12Dp,22cosmϕipcosαipdAp×ρcosβpcosψpTsψpgψpAR,E4

where Dip,1is the distance between transmitter iand reflection point p, Dp,2is the distance between reflection point pand receiver RX, ϕipand ψpare the NLoS irradiance and incidence angles with respect to point p, respectively, αipand βpare the incidence and irradiance angles at reflection point pon the wall, respectively, ρis the wall reflectivity, and dAprepresents the area of the reflection point on the wall.

The total NLoS channel gain for ith transmitter HNLOSiis given by collecting the reflections from the 4 walls [2]:

HNLOSi=j=14HNLOS,wallji,E5

where HNLOS,walljiis the collection of reflections from transmitter ito wall j, and can be obtained by integrating (4) over xzor yzbased on the wall location, such that

HNLOS,wallji=xzoryzm+12πDip,12Dp,22cosmϕipcosαipρ×cosβpcosψpTsψpgψpARdAp.E6

The determination of the parameters for the previous equation is as follows:

Dip,1=TX,ipTX,ipTDp,2=pRXpRXT,E7

where Tis the transpose operator for row vector a. By using triangle calculations, the angles ϕip,αip,ψp,and βpcan be found as follows:

cosϕip=zizDip,1,αip=π2ϕip,cosψp=zz0Dp,2,βp=π2ψp.E8

The simulation of the power distribution is done for two cases, first case is using four transmitters which are distributed in different positions (1.25, 1.25, 3) m, (1.25, 3.75,3) m, (3.75, 1.25, 3) m, (3.75, 3.75, 3) m in room with size (5,5,3) m, as shown in Figure 2. Second case is using only one transmitter that is located in the center of the ceiling. This power distribution is shown in Figure 3. The simulation is done with using some parameters where FOV=70oand assume the filter gain = 1, the number of LEDs by array 60x60. The concentrator gain = 1 where the active area of photo diode (PD) = 1cm2.

Figure 2.

The power distribution for 4 LEDs.

Figure 3.

The power distribution for one LED centered in the ceiling.

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3. RSS mathematical analysis

The proposed technique depends on estimating the receiver position using RSS method, then further improving the acquired estimation by adopting the Kalman filtering algorithm. In the initial estimation, RSS technique is used taking into account the effect of LoS. Specifically, A mathematical model is developed for the noisy VLC positioning system and estimate both the angular and horizontal-distance errors. Because of the dependence of horizontal-distance error on the irradiance angle error. The performance of the proposed technique is determined by evaluating the positioning errors in a typical room. Also, the results are compared to that of the traditional RSS system. Depending on Figure 1; the analysis assumes that ψi=ϕifor any i1,2,3,4. Assume that the light emitted from each LED is distinguishable at the receiver. Accordingly, The index iis dropped from the developed equations.

If the transmitter and receiver are aligned together, then ψ=ϕ=0, d=V, where dis a direct distance between transmitter and receiver and Vis the horizontal distance of the transmitter, using Tsψi=1, and gψi=1as shown in Figure 1. In this case, the power of LoS can be approximated as the following:

PR0=m+1AR2πV2PT.E9

In the general case (ϕ0) as view in (2), and by using Figure 1 where cos2ϕ=V2d2and by multiplying the equation by V2V2, then, the received power can be modeled as:

PR=cosm+3ϕPR0,E10

The last equation expresses the ideal system case, which means there is no noise affecting the system. From which:

ϕ=cos1PRPR0k,E11

where k=m+3. In the previous works, PRois known while here, PRois not known exactly with noise power Pn. Assuming constant noise power in the room then the noise add to both the receiver power and irradiance angle. Neglecting the effect of NLoS (as it is very small), include the noise effect to (10) as follows:

PR+Pn=cosm+3ϕ+ΔϕPR0+Pn,E12

where Pnand Δϕare the receiver power and irradiance angle noises, respectively. Substituting ϕfrom (11):

Δϕ=cos1PR+PnPR0+Pnkcos1PRPR0k.E13

From Figure 1, the horizontal distance without any noises is given by

dL=Vtanϕ.E14

In the case of a noisy channel, the horizontal-distance error ΔdLis estimated from:

dL+ΔdL=Vtanϕ+Δϕ.E15

The value of the horizontal-distance error ΔdLis used to determine the localization more accurate after calculating the position of the receiver by trilateration method as shown in the next section.

According to the RSS method, the positioning error is simply obtained from the distance errors. The positioning algorithm uses three maximum power levels to determine the location of the user [3]. Here, RSS algorithm is used to estimate x0y0, the position coordinates of the receiver. Let xiyi,i1,2,3, be the three coordinates of three transmitters. From ΔdLfind the positioning estimate:

x0xi2+y0yi2=d̂L,i2,E16

where

d̂L,i2=dL+ΔdL,E17

for any i1,2,3, where dL,iis the horizontal distance of the receiver from the transmitter i. This method can be clear as shown in Figure 4, where RSSI is received signal strength indicator.

Figure 4.

A trilateration method for RSS to calculate the position of a receiver by using three transmitters.

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4. Proposed KF algorithm

In this section, a KF algorithm is proposed to further improve the previous estimation (introduced in the last section) of the receiver position. Specifically, the estimation of the irradiance angle developed in the last section is further improved by adopting a KF algorithm.

The proposed system with KF is shown in Figure 5 where the PD collects the received power and inserts it into the proposed system. The process of the proposed system is analyzing the mathematical equations to calculate the irradiance angle ϕand the error of the irradiance angle Δϕ. Both of the two calculated values insert into the Kalman filtering process. The optimal value of the estimated irradiance angle is obtained after using the KF algorithm. The estimated irradiance angle inserted into the RSS process to calculate the estimated horizontal distance d̂. The trilateration method has been applied to calculate the estimated position of the receiver by using the estimated horizontal distance d̂. KF algorithm recursively estimates the state of variables in the system in two phases; prediction and measurement [4, 5].

Figure 5.

The block diagram of the proposed system with KF.

4.1 Predict step

We denote the state vector by x=xaxbT, where xarepresents a measured angle, xbis the error in the angle, and Tis the transpose operator. Based on the estimate at iteration k1, the state xk1k1. The next step kof the system dynamics xkk1is evaluated as:

xkk1=Fkxk1k1,E18

where Fkis the state transition matrix. The corresponding state covariance matrix is given by:

Pkk1=FkPk1k1FkT+Qk,E19

where Qkis the process noise covariance.

4.2 Measurement step

The updated state variable xkkand updated state covariance matrix Pkkare given by

xkk=xkk1+KkykPkk=IKkHkPkk1,E20

respectively, where Kkis the Kalman gain, ykis is the error vector, and Hkis is the observation model.

Kk=Pkk1HkTSk1yk=zkHkxkk1.E21

Here zdenotes the measurement vector, given by:

zk=Hkxk+RkE22

where Rkis the measurement noise matrix. Also Skis the innovation matrix, which relates the covariance of state variables to measurement vector:

Sk=HkPkk1HkT+Rk.E23

Finally, after getting the estimated angle then recalculate the positioning error using equations developed in Section 3.

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5. Proposed localization methodology using an averaging RSS technique

Second technique contains the averaging localization method and Kalman filtering with averaging schemes. For the averaging technique, the position of the receiver has been estimated by RSSI technique for multiple times (e.g., Nsamples) and the acquired estimations are averaged over all samples.

To enhance the results of improving the localization, The algorithm of Kalman filtering has been adopted for estimation the received power over Nsamples, followed by using RSS technique on the average received power. These techniques have been analyzed mathematically, with respect to the effects of both LoS and first-reflection from NLoS propagation.

Typical room is considered for evaluating the positioning performances for proposed techniques and the results of them are compared with the traditional RSS system.

For determining the receiver location, the trilateration method is used with the RSS from three LEDs transmitters having the maximum received levels [3]. Our techniques depend on the average of estimated receiver position over a certain number of measurements to decrease the localization error. This decreasing in error gets at the cost of exceeding the system mathematical complexity. Figure 6 shows a simple block diagram that demonstrates this approach.

Figure 6.

Block diagram of proposed averaging positioning scheme.

5.1 RSS technique

Using (2), the received LoS power from transmitter i1,2,3,4can be written as:

PR,i=m+12πdi2cosm+1ϕiARPT,i,E24

where PT,iis the transmitted power of ith LED. Here, assume that ψi=ϕi, which is determined from Figure 1 as:

cosϕi=Vdi,E25

where Vis the vertical distance between transmitter and receiver, assumed constant. Accordingly, the distance between transmitter iand receiver can be evaluated as:

di=m+1Vm+1ARPT,i2πPR,im+3.E26

If consider the effect of NLOS as well, the total power collected at the receiver is obtained by modifying (24) to:

PR,i=HLOSi+HNLOSiPT,i.E27

5.2 Linear LS method

To estimate the receiver location, the linear LS estimation is commonly used. Let xiyi,i1,2,3, be the horizontal coordinates of transmitter iand dL,ibe the horizontal distance of the receiver from transmitter i. The range equation can be written in the form:

x̂xi2+ŷyi2=dL,i2,i1,2,3,E28

where x̂ŷis the estimated horizontal location of receiver. The last system of equations can be written in matrix form as:

AX̂=B,E29

where

X̂=x̂ŷTA=x2x1y2y1x3x1y3y1B=b21b31T.E30

Here for any m23,

bm1=x̂x1xmx1+ŷy1ymy1.E31

The solution of (29) is:

X̂=ATA1ATB.E32

5.3 Complexity analysis

The complexity of proposed averaging RSS technique can be analyzed by counting the number of mathematical operations required to solve the LS method once and then multiplying the resulting by the number of samples. Specifically, the total number of floating-point operations is 39N+1flops, where Nis the number of samples. That is, the complexity increases linearly with the number of samples.

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6. Kalman filtering with averaging

KF estimates the states of a linear system from the noisy measurements then produces the estimation of unknown variables that aim to get more accurate than those which based on a single measurement value.

At this section, a KF algorithm is adapted to enhance the estimation performance of the receiver positioning system. In the first, KF estimates several samples of measured received powers. Then, the average of these estimated power values is evaluated. Using the output of KF which is the estimated average power, the position of the receiver can be calculated by using RSS technique. The block diagram of the proposed Kalman filtering with averaging technique is shown in Figure 7.

Figure 7.

Block diagram of proposed Kalman filtering technique.

KF algorithm is shown in the previous chapter in Section 4 recursively estimates the state of variables in the system in two phases; prediction and measurement [4, 5]. We denote the state vector by x. This state vector represents measured received power and number of samples which are used in the process. Based on the estimate at iteration k1, and have state xk1k1.

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7. Simulation and discussion for averaging technique

In this section, The simulation results are presented and compared them to that of traditional systems. Table 1 shows the main parameters used in the simulation.

ParameterValue
Room dimensions5×5×3m3
Number of transmitters4
Transmitted power30 W
Locations of LEDs1.25,1.25,3, 1.25,3.75,3, 3.75,1.25,3, 3.75,3.75,3m
FoV of photodetector70
The active area of the photodetector1 cm2

Table 1.

Main parameters in VLC.

In case of demonstrating the relation between the SNR and the average positioning error, Figure 8 shows that using five different positions of the receiver and the average positioning error in a meter. This figure shows that the proposed system outperform the traditional RSS method by nearly 1 cm at SNR=10 dB while adopting KF decreases the error by 11.5 cm that means improvement by 52.27%.

Figure 8.

The relation between average positioning error in meter with SNR for different positions.

Figures 9 and 10, plot the true path with three different methods. The simulation is done at a FoV of 70and an SNR of 20dB. These figures are simulated as a plan view to show the estimation position of the receiver in the room. The proposed RSS technique achieves a tiny improvement while the proposed RSS with KF is the closest to the true path.

Figure 9.

Positioning error for traditional proposed, and KF correction RSS techniques at a FOV of70and an SNR of20dB for a y path.

Figure 10.

Positioning error for traditional, proposed, and KF correction RSS techniques at FOV of70and an SNR of20dB for a x path.

The pedestrian is moving in random directions inside the room. In Figure 11, compare between three techniques: Traditional RSS, proposed RSS, and proposed RSS with Kalman filtering. The comparison is done at a FoV of 70and an SNR of 20dB. It is clear from the figure that adopting KF estimation further reduces the positioning error and provides an estimate that is very close to reality.

Figure 11.

Relation between positioning error and number of pedestrian steps at a FOV of70and an SNR of20dB.

In the second technique, simulation results for the proposed system are presented and compared with that of traditional systems. The main parameters used in the simulations for the VLC link are listed in Table 2.

ParameterValue
Room dimensions5×5×3m3
Number of transmitters4
Total transmitted power30 W
Locations of LEDs1.25,1.25,3, 1.25,3.75,3, 3.75,1.25,3, 3.75,3.75,3m
FOV of photodetector70
SNR20
Active area of photodetector1 cm2
Wall reflectivity ρ0.8
Number of samples50
Range of receiver in room(1–3.5) m over both x, yaxes

Table 2.

Simulation parameters.

7.1 Positioning error

In the simulation, the performance measure is determined by the positioning error:

εposition=x̂x02+ŷy02,E33

where x0y0is the receiver horizontal location and x̂ŷis its estimated location. Figure 12 shows the average error in receiver positioning for different number of samples. It is clear that the error can be reduced to less than 10%of its maximum value by averaging over 50 samples. This reduction comes at the cost of increasing the mathematical complexity of the system as shown in Figure 13. The complexity is calculated according to the number of operations, which increases as the number of samples increases.

Figure 12.

Comparison between average positioning errors versus number of samples in averaging RSS technique.

Figure 13.

Complexity of the averaging RSS method.

7.2 Averaging RSS and traditional RSS techniques

The RSS variations of the positioning error at every sample are shown in Figure 13 for receiver position x0y0=11, considering the effect of LoS only.

The positioning error using the proposed averaging RSS technique (with 100 samples) is plotted in same figure as well. The improvement using proposed technique is clear from the figure. The traditional RSS errors are more than 0.6 m (42.4%), where the error when employing the proposed averaging RSS is only 0.217 m (15.3%). That is, an improvement of about 27.1% is getting when adapting the proposed system. Both LoS and NLoS effects are studied for position of the receiver x0y0=11as well and the results are plotted in Figure 14. Traditional RSS errors are more than 0.7 m (49.5%), where the errors when using proposed averaging RSS are only 0.255 m (18%). The improvement of nearly 31.5% is achievable by using the proposed scheme.

Figure 14.

Positioning error for both traditional RSS and averaging RSS techniques at position11, considering the effects: (a) LOS only, (b) both LOS and NLOS.

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8. Kalman filtering, averaging RSS, and traditional RSS techniques

In this section, different comparisons are shown between the performance of three methods; The traditional RSS technique, proposed averaging RSS technique, and the proposed Kalman filtering with averaging. We use same values which given for the VLC link of Table 2.

8.1 LoS propagation

The effects of LoS only on two tracks’ estimations for both xand ypaths are presented in Figures 14 and 15 for both xand ypaths, respectively.

Figure 15.

Track estimations using traditional and proposed techniques for: (a) anxpath, (b) anypath.

From the figures, both tracks’ estimations are the nearest to the real one when employing the proposed techniques. Also, The results show that adopting KF estimation reduces the positioning error and improve the estimation. Table 3 for three techniques summarizes the error and improving percentages.

Localization methodAverage positioning errorPercentage improvement
Traditional RSS18 cm
Averaging RSS12 cm33.3%
Kalman filtering5 cm72.2%

Table 3.

Accuracy for different techniques.

8.2 Both LoS and NLoS propagations

The effect of both LOS and NLOS on Kalman filtering tracks’ estimations for both xand ypaths are presented in Figures 16 and 17 for both xand ypaths, respectively. It is clear that the track estimation gets slightly worse when including the effect of NLoS.

Figure 16.

Comparison between Kalman filtering track estimation for both LOS and NLOS propagations for anxpath.

Figure 17.

Comparison between Kalman filtering track estimation for both LOS and NLOS propagations for anypath.

8.3 Kalman filtering response

The response for a random position estimation for the KF is shown in Figure 18. The filter input is a measured value of received power, while the filter output is the corresponding estimated value at different number of samples. The filter response (estimated value) is near to the actual value where the samples are greater than 11.

Figure 18.

Response of Kalman filtering technique.

8.4 Position estimation accuracy comparison

As mentioned in the introduction, several techniques have been proposed for indoor localization based on VLC technology. In this section, a comparison is provided between the position estimation accuracy and that of previous works for same simulation parameters. The results of this comparison are summarized in Table 4.

ReferenceSys. parametersRef. acc.Present workPresent work
AVG sys. Acc.KF acc.
[6]LoS, FoV =80, PT=10W, AR=0.5cm2, 5,5,3m35 cm3.7 cm3.1 cm
[7]LoS/NLoS, FoV =10180, PT=1.9W, AR=0.81cm2,ρ=60%, 4,4,3.5m313.95 cm9.1 cm4.8 cm
[8]LoS, FoV =85, PT=1W, AR=0.81cm2, 5,4,3m310 cm6.17 cm1.75 cm
[9]LoS, FoV =80, PT=17W, AR=1cm2, 3.6,3.26,2.5m314.5 cm17.4 cm3.5 cm
[10]LoS, FoV =25, PT=17W, AR=1cm2, 6,6,3m35 cm11 cm2.3 cm

Table 4.

Position estimation accuracy comparison.

It is clear from the Table 3 that both proposed averaging RSS and Kalman filtering with averaging techniques achieve better accuracy than that proposed in [6, 7, 8]. Since the authors in [9, 10] have adopted Kalman filtering, they have better accuracy than the proposed averaging method. However, employing Kalman filtering with averaging gives a better accuracy.

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9. Concluding remarks

First, the proposed techniques have been analyzed mathematically, taking into account the effects of LoS propagation. The positioning estimation accuracy of proposed techniques has been evaluated in a typical room. The results reveal that an improvement of about 52% in the average positioning error is achievable using the proposed technique with KF, when compared to that of the traditional RSS.

Secondly, both averaging and Kalman filtering by averaging schemes are adapted to improve the positioning system. Specifically, in the averaging technique, the receiver position has been determined by using the average of the samples of RSS estimations. The position is determined by RSS estimation of a Kalman filtered averaged multiple received power samples in the second proposed system, Kalman filtering with averaging algorithm.

Simulation results reveal that an improvement of about 33.3% in estimation accuracy is achievable when using the averaging scheme as compared to that of traditional RSS scheme. This improvement increases to 72.2% when adopting proposed Kalman filtering with averaging scheme.

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Eman Shawky Abd El-Fattah Amer (August 21st 2021). Averaging Indoor Localization System, Adaptive Filtering - Recent Advances and Practical Implementation, Wenping Cao and Qian Zhang, IntechOpen, DOI: 10.5772/intechopen.97531. Available from:

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