# Automated linear regression tools improve RSSI WSN localization in multipath indoor environment

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## Abstract

Received signal strength indication (RSSI)-based localization is emerging in wireless sensor networks (WSNs). Localization algorithms need to include the physical and hardware limitations of RSSI measurements in order to give more accurate results in dynamic real-life indoor environments. In this study, we use the Interdisciplinary Institute for Broadband Technology real-life test bed and present an automated method to optimize and calibrate the experimental data before offering them to a positioning engine. In a preprocessing localization step, we introduce a new method to provide bounds for the range, thereby further improving the accuracy of our simple and fast 2D localization algorithm based on corrected distance circles. A maximum likelihood algorithm with a mean square error cost function has a higher position error median than our algorithm. Our experiments further show that the complete proposed algorithm eliminates outliers and avoids any manual calibration procedure.

### Keywords

algorithm design and analysis correlation and regression analysis wireless sensor networks localization### Abbreviations

- EE
environment emulator

- IBBT
Interdisciplinary Institute for Broadband Technology

- LQI
link-quality indication

- RSSI
received signal strength indication

- WSN
wireless sensor network.

## 1 Introduction

Wireless sensor networks (WSNs) are expected to offer a cheap solution not only for monitoring and control applications, but also for more advanced applications like location-based services (e.g., tracking and tracing of persons and objects, indoor guiding of persons in complex buildings, offering location-based information, etc.). The main requirement for mass deployment of WSNs and corresponding services is the easy installation and configuration, which is realized by the introduction of self-organizing and auto-configuration mechanisms. For location-based services, the same requirements are valid: a cheap technology in combination with a simple deployment strategy avoiding complex and time-consuming manual configuration and calibration procedures. This article presents a new approach for developing a localization algorithm, and has validated the algorithm in a real-life test bed. The approach is based on the automatic selection of anchor nodes for received signal strength indication (RSSI)-based indoor localization, and hence avoids any manual calibration.

RSSI-based localization is two-sided [1]. The first method explores the physical (theoretical) relationship between the RSSI and the distance. Free space models and two ray ground models are the key words. The second (experimental) method uses fingerprinting: a RSSI database is filled with measurement records during an extensive calibration phase, and the location is estimated by fitting the measured RSSI to this database. An example of a mature application of this kind is MoteTrack [2]. In the presence of severe multipath fading with multiple reflections (more than 30 are reported in the literature [3]), the relationship between RSSI and distance is extremely hard to model, making both methods ineffective. Some authors conclude that the large amount of characterization will make the use of signal strength approaches with low power radios practically impossible [4].

This article is organized as follows: In Section 2, an overview of related work is given. In Section 3, the realistic wireless environment is described. Section 4 presents the discovery of multipath fading in a corridor and discusses the impact on power levels. In Section 5, preprocessing of the measurements is treated, followed by the positioning in Section 6. Section 7 compares the test results of the proposed algorithm with those of a more conventional maximum likelihood algorithm. Finally, in Section 8, conclusions are presented.

## 2 Related work

Possible range-based localization methods are angle of arrival [7], time of arrival [8], time difference of arrival [9], and RSSI. In this article, we only consider RSSI, because this can be implemented with inexpensive hardware. This means that the sensors calculate a distance from the physical RSSI measurements. Range-free localization methods do not use physical parameters but work with the content of a message and are also not treated here.

There are different technologies used for indoor localization: ultrasonic [10], infrared [11], Bluetooth [12], RFID [11, 12], WiFi [13, 14], UWB [15, 16, 17], and a combination of technologies [18, 19]. These technologies all have not only their respective advantages, but also shortcomings. Ultrasonic and infrared waves cannot propagate through walls. Bluetooth has not only a limited range, but also a restricted number of devices. Power consumptions of both RFID and WiFi are large compared with IEEE802.15.4 sensors. Furthermore, RSSI measurements on WiFi are tricky, since the absolute accuracy on the RSSI is not specified in the standard [20], making the RSSI manufacturer dependent. For IEEE802.15.4, however, there are very few chipmakers. Most devices use the CC2420 chip from Chipcon with a dynamic range of 100 dB, and the datasheet specifies the RSSI linearity (the relationship between RSSI and received input power) within the limits of ± 3 dB.

UWB is still a new technology following the IEEE802.15.4a standard. Only few devices are available [21] (at reasonable cost), and only very recently have theoretical performance bounds of wideband localization been described [22, 23, 24]. UWB empirical results on localization in harsh multipath environment are also brand new [25, 26].

Distance tolerances of these wireless indoor positioning techniques can be found in [27]. To our knowledge, an overview of empirical localization accuracies based on IEEE802.15.4 devices does not exist yet, but these studies are proliferating and some individual results can be found in [28, 29].

Tremendous effort is already put in localization algorithms [30, 31, 32, 33] in general. RSSI-based localization can be found with the Bayesian estimators [34, 35, 36] and maximum likelihood estimation [37]. These theoretical estimators have in common the ability to estimate the propagation parameters in their localization algorithm itself. We will not follow this approach and only feed good empirical input to our simulator. Preprocessing the data according to the known physical behavior will result in better models at the input and hence at the output of the positioning algorithm. The best anchor nodes are carefully selected with automated standard linear regression tools. Now the localization algorithm can be optimized on spatial constraints only.

The techniques described in [1, 38, 39] show some similarity with ours, but we push the limits one step further by an automated selection of anchors in a larger sensor network based on standard linear regression techniques.

The authors of [1] also suggest a new approach for range-based localization using RSSI. They also do not use fingerprinting and avoid the expensive training phase of MoteTrack-like algorithms and estimate the propagation parameters using the RSSI measurements. Our major improvement is that we use more than three anchors for the calibration and training phase at no additional time cost.

With the authors of [38], we share the experimental analysis of RSSI-based localization. Also, the linear fitting is common. Our experiments, however, are performed using IEEE802.15.4 devices and use a larger dataset in a larger realistic environment.

The calibration method described in [39] is the starting point of our study. In our article, a selection method of anchors is added and a localization algorithm is presented.

Our work completely exploits the observed correlation between the quality of anchors and the linearity of the calibrated path loss model as described in [34] and adds a new, fast, and simple algorithm using the basics not only of the underlying physics (propagation model and multipath fading), but also of linear regression (standard error, confidence, and expectation intervals).

## 3 Used equipment

^{2}. The dimensions of the third floor are 16.8 × 90.0 m

^{2}or 1512 m

^{2}. On the third floor, there are 56 nodes. All the nodes are mounted at the same height beneath a highly conducting ceiling. The central internal walls are constructed of plywood, while walls in the vicinity of the staircases and elevator are made of concrete. At the elevator, a lot of metal has been used. One side of the longest corridors is concrete and the other plywood. Only the third floor is discussed in this article. In Figure 2, this floor is shown with the positions of the 47 active nodes and its four narrow but long corridors. The anchors with numbers 10, 19, 32, 39, 40, 42, 44, 48, and 50 were closed down for maintenance during our experiment. Every node is connected to a corresponding environment emulator (EE). This component can measure real-time energy consumption, emulate battery depletion, and generate artificial sensor data, hence emulating real-world application scenarios. The EE is connected to an intermediate node or iNode. The iNodes are further connected to a central management server via ethernet and are responsible for the configuration and control of the sensor nodes. The iNodes are also connected with two IEEE802.11 (a/b/g) radios, and allow WiFi testing. In this article, we will only focus on the sensor nodes. Power is fed across ethernet through power over ethernet network switches. The control software supports both Tiny OS 1 and Tiny OS 2. The iNodes can be synchronized with a time server. A time accuracy of 2 μs is obtained and used for data logging. More details about our test bed can be found in [40].

On the iNode, we have a software tool called RadioPerf which is, in terms of functionalities, very similar to the iPerf tool. The tool includes a packet generator and Java-based configuration and visualization of measurements reported to the central data server. For this study, the following items are collected in a log file: the number of packets sent, the number of packets sent with an error, the minimum estimated noise floor, the average estimated noise floor, the maximum estimated noise floor, an identifier of the node generating the packets, the number of packets received, the number of packets lost, the minimum link-quality indication (LQI), the average LQI, the maximum LQI, the minimum RSSI, the average RSSI, the maximum RSSI, a time stamp, the node reporting these values, and a report sequence number. Our dataset is available at http://users.ugent.be/~inmoerma/LiReFloA.

## 4 Multipath fading

Every node broadcasts 240 packets. Every packet consists of 100 bytes, and the inter packet delay is 25 ms. Transmission is at channel 26 to avoid WiFi interference [41]. Upon swapping sending nodes, the test bed remains idle for 8 ms. The test is repeated for transmit power levels of 0, -1, -3, -5, -7, -10, and -25 dBm. Every second, available data were sent to our log file. Figure 1 was constructed with the average RSSI values of packets sent at full power (0 dBm). The distances between the nodes are known.

In order to explain the big differences in Figure 1, the longest corridor is selected, consisting of nodes 2, 5, 8, 12, 15, 20, and 21. These nodes are collinear, and a clear line of sight (LOS) exists.

- 1.
First walls have attenuation and what we notice is a stronger signal.

- 2.
Second the standard deviation on measurements due to time-fading is only a few dB [42], and what we see is a very large spread.

- 3.
Third the RSSI accuracy in the CC2420 datasheet is much better than the measured spread.

- 4.
Fourth fluctuations in transmit power are not an issue because Figure 3 only considers one transmitter (node2).

- 5.
Finally, the spread on receiver sensitivity of the CC2420 also does not exceed a few dB, and is therefore less than the observed spread.

The waveform with minima and maxima is also found in other practical experiments in corridors [43] and can only be justified with diffraction and the many constructive scatterings and reflections by the conductive metal ceiling and nearby walls in these corridors, acting as "street canyons" [44].

## 5. Preprocessing

Schematic description of the preprocessing

Procedure_Selection_of_Anchors_based_on_RSQ |
---|

Procedure_Selection_of_Anchors_based_on_Error_on_Distance |

Procedure Min-Max If there are larger than expected large circles for the used hardware: delete them |

Procedure Ignoring_Circles Delete the typical circles containing at least 5 others if there are no typical circles contained by at least 5 others If there are typical circles contained by at least five others delete them. |

Procedure Maximum_Likelihood Group the anchors based on distance Are there small distance circles contained by other small distance circles of other group members? |

If yes: new typical circle radius all members = geometric average of
new_Error_on_Distance = (Error_on_Distance of member) /sqrt (# remaining typical circles -1) adapt small and large circles according to new_Error_on_Distance continue If no: continue |

### 5.1 Selection of anchors based on the linearity of their calibrated path loss model

Figure 1 looks like a wide monotonously decreasing stripe clipped by a horizontal line at the bottom because of the sensitivity of the receiver. This suggests there still remains a linear relationship between RSSI and the logarithmic distance.

with pl(*d*) being the path loss in dB at a distance *d* (in m) from the receiver. Considering that the path loss is the (logarithmic) difference between sending power and receiving power and emitting at 0 dBm and reading the received power (RSSI) in dBm, we instantaneously get for (1), an intercept point of -40.2 and a slope of -20. For distances greater than 8 m, the standard proposes a steeper slope (-33) and thus proposes a model based on the two ray ground model. It is noted that this is invalid for frequencies other than 2.4 GHz. It is noted that, for WiFi, the RSSI is not calibrated in dBm [20], and an extra step is needed to include the relationship between RSSI and dBm.

In this article, we will calibrate the test bed by determining the intercept point and slope with the linear regression technique, as described in [39].

*r*-squared (RSQ) values for each sender (and all other nodes receiving) with a linear regression tool: RSSI as a function of the logarithm of the distance (in m). RSQ is the square of the Pearson correlation coefficient [46]. "Zero" indicates that there is no linear fit, whereas a 1.0 indicates that all points lie on a straight line. For the RSSI at full power, the RSQ is varying from 0.89 on node 43 to 0.37 on node 21 as shown in Figure 4. The higher the RSQ, not only the higher the quality of the anchor nodes, but also the smaller their quantity. We suggest choosing RSQ of 0.8 (corresponding to a Pearson correlation coefficient of -0.89), resulting in 10 anchor nodes. The top 10 best-fitting nodes are marked with a square in Figure 2. They are in the extremities of the building, not in the corridors. These nodes are located in positions less sensitive to multipath fading and are considered as more informative ones. This is also confirmed by another empirical study in an indoor environment [47]. In Figure 5, the intercept points and the slope factors are utilized to produce graphs of the 10 best fits with RSQ > 0.80. A mean of these 10 best fits is also included. This line is constructed using the average of the 10 intercept points and the average of the 10 slopes as a new intercept point and slope, respectively. Figure 6 gives the reported RSSI versus the logarithmic distance with the best node 43 sending and all others receiving. It is known from Section 3 that there are 47 active nodes. Hence, in this half-duplex communication system there are 46 potential receiving nodes. In this figure, the measurements from nine receivers are below the sensitivity level. Hence, there are 46 minus 9, i.e., 37 measurement points. The graph also includes a linear regression of the measured RSSI and the mean of the 10 best fits. The last two lines are almost coinciding. Measured points above the fit are due certainly to (constructive) multipath fading as explained in the previous section, whereas the measured points below the fit are due to (destructive) multipath fading or attenuation. Figure 7 gives the worst-performing node 21 (which is in the longest corridor). For this central node, the measurements from only one receiver are below the sensitivity level, yielding 46 minus 1, i.e., 45 measurement points. The fit and the mean of the ten best fits are no longer parallel. The slope of this node is not so steep, multipath fading boosts the tail, and the higher attenuation at low distance cuts the head. Furthermore, there is a greater difference between the measurements and the fit.

Adding more nodes with lower RSQ will pile up with flatter slopes. Calibrating the ten best sending nodes with their respective intercept points and slopes will improve the RSSI to distance conversion. These intercept points and the slopes calculated with known distances can now be utilized to calculate any distance with a measured RSSI as input.

Hashemi [48] reports that many researchers had added an error correction to (1) and (2) and minimized it in their investigation. In [1, 39, 49], this is, e.g., done for the attenuation of walls. It is known from Section 3 that many walls of our building are plywood. Passing through them does not affect the RSSI measurements considerably. Likewise, propagation paths from the sending nodes in the staircases will pass through the same highly attenuated concrete material for all the other nodes and therefore will mainly influence the intercept point. Since all anchors are individually calibrated, our model absorbs the latter effect. Although a model that includes walls can have a positive effect on the position accuracy, it will add complexity.

### 5.2. Selection of anchors with low standard error

With the ten best-correlated sending nodes, every node in the network hears at least 3 which should be sufficient for the 2D localization of a subject [50]. Some localization protocols work fine with anchor nodes in the corner [51]. Unfortunately, the building in this case (and many others) is not a square, and it is obvious that the absolute accuracy of far nodes will be lower. The challenge now is to find a few extra central nodes to increase this distance accuracy. Linear regression assumes a constant standard deviation for each point on the regression line [52]. There are two conditions for having a good RSQ. If the slope is flat and the points are spread around this line, then a low RSQ will result. On the other hand if the slope is not flat, but the points are spread too far around the line, then there will also be a bad correlation. The standard deviation is a measure indicating how close the points lie around the regression line and is, therefore, important for determining the ranging bounds. These bounds can be theoretically determined using the Cramer-Rao algorithm, like in [53].

In this article, we propose a new more empirical approach for determining and optimizing the above bound. Indeed, any standard linear regression tool can calculate the standard error on the vertical axis which is an estimate of the standard deviation. We suggest swapping the RSSI and the log(distance) axes and redoing the linear regression, resulting in the standard error on the logarithm of the distance. We define the error on distance parameter as two times the estimated standard deviation of the logarithmic distance frequency distribution. It is noted that the RSQ is invariant to the axes swap: the RSQ values do not change when the axes are swapped.

RSSI measurements that are too low (too much attenuation from the walls or too much destructive multipath fading) will result in distances at the right-hand side of the peak of the distribution and will thus correspond to estimated distances that are too high. RSSI measurements that are too high (certainly too much constructive multipath fading) will result in distances that fall at the left-hand side of the peak of the distribution and will thus correspond to estimated distances that are too low. The above mentioned property is widely used in the design of our positioning algorithm in Section 6.

Measuring a logarithmic distance of 1.5 ± 0.3 (31.6 m) will result in a distance of 10^{1.2} to 10^{1.8} or 15.8 to 63.1 m or 31.6 m (-50 to +100%). These error percentages are in a multipath environment an order of magnitude greater than the 10.1% encountered in less-complex environments [54], although equally well calibrated. It is also noted that the error percentages in plus and minus are asymmetric, because the distance on error is a logarithmic value, and thus the geometric mean needs to be considered.

We produce small and large distance circles with the error on distance and will use these from Section 5.4 onward. A typical circle corresponds to the calculated distance, obtained with the measured RSSI and the calculated slope and intercept for the corresponding anchor node. The radius of the small distance circle corresponds to the radius of the typical circle divided by 10^{error_on_distance}. The radius of the large distance circle corresponds to the radius of the typical circle multiplied by 10^{error_on_distance}.

*X*-coordinate and the average of the logarithm of the distance regression points as center

*Y*-coordinate. If an individual logarithmic distance-RSSI experiment is performed, then 95% of the measurements will be found between the expectation lines. The area between the confidence lines (or confidence band) shows where the regression line is expected for the given confidence level. Around the center point, the intervals are the smallest. Toward both ends of the regression lines, the intervals become larger. This can be attributed to the fact that the uncertainty on the slope increases as the regression point is no longer in the center. Using this property gives an insight in the quality of our regression model: Anchor nodes at the extremities of the building have their maximum accuracy at the geometric means of the minimum calculated distance (around 4 m) and the maximum calculated distance (around 83 m) or about 18 m. This is illustrated in Figure 12 with shaded rings for two anchors at the extremities of the building. At the geometric center of these rings, the accuracy of the regression model is maximal, since this corresponds to the central (

*X*,

*Y*) coordinates in Figure 11. Moving away from this central point in this figure results in wider confidence intervals resulting in a lighter gray shade as seen in Figure 12. The rectangle represents the building considered. Rings of central anchors will intersect with both the other rings and thus have a positive effect on the accuracy. These central anchors unfortunately cannot improve the accuracy of the model at the extremities of the building since there these are also working at the end of their regression lines.

### 5.3. Complexity of the calibration process and robustness against environmental changes

In the previous two sections, the best anchors are selected from the active nodes, and each anchor is calibrated with its own propagation constants (slope and intercept). This subsection discusses the complexity of such a calibration process and the robustness of the system against environmental changes.

After the experiment on the test bed the measurements (see Section 3) are imported in Matlab. For each sending-receiving pair, the RSSI is averaged and forms a RSSI-matrix, where element *a*_{ i, j } corresponds to the averaged RSSI reported from receiver *j* with sending node *i*.

With the (known) position of the nodes, a distance matrix is formed too. Since our nodes are fixed this matrix remains the same for all experiments. On the corresponding rows of both matrices our software now calculates the slope, intercept, RSQ, and error on distance. It selects the ten best-correlated nodes and completes the selection with the two first nodes in the sorted error on distances vector, which are not in the best-correlated list yet. At this stage, the anchors are selected and calibrated. The location information of the remaining nodes is discarded in subsequent procedures, and the anchors are used to localize these nodes. Using a single power level and sending 240 packets (with a 25 ms inter-packet delay) over a single channel, the whole calibration phase is completed in less than 10 min, including the measurements. Calibration can easily be redone, e.g., when a node becomes inactive. Sending less packets increases the effect of time fading, since there are less-averaged RSSI measurements. Averaging multiple-channel RSSI measurements will reduce the effect of different reflections.

For a fingerprinting-based algorithm, a training phase is not only typical, but also essential. During this time-consuming phase (usually several days), a large database is filled with RSSI measurements. During the final phase, RSSI measurement is compared with all the records of this database, and the record with the best match gives the estimated position. In this article, a different approach is used: the already deployed nodes in this study are utilized to quickly and automatically characterize the propagation constants (slope and intercept) of the medium. The knowledge of these two parameters is sufficient to localize (not previously stored) targets.

### 5.4. Min-max algorithm

Data available for positioning node 4

Anchor ID | RSSI (dBm) | Calculated distance (m) | Error on distance | Expected distance (m) | RSQ |
---|---|---|---|---|---|

3 | -55.9 | 11.3 | 0.23 | 6.6...19.4 | 0.87 |

6 | -63.9 | 12.3 | 0.28 | 6.5...23.2 | 0.84 |

9 | -59.3 | 10.1 | 0.31 | 5.0...20.4 | 0.82 |

13 | -62.6 | 12.3 | 0.27 | 6.7...22.7 | 0.85 |

30 | -78.0 | 35.6 | 0.32 | 16.9...75.2 | 0.71 |

31 | -89.8 | 108.1 | 0.32 | 51.5...227.2 | 0.71 |

33 | -90.4 | 122.2 | 0.32 | 58.2...256.7 | 0.81 |

47 | -89.6 | 105.7 | 0.31 | 51.2...216.8 | 0.87 |

Data available for positioning node 55

Anchor ID | RSSI (dBm) | Calculated distance (m) | Error on distance | Expected distance (m) | RSQ |
---|---|---|---|---|---|

13 | -89.3 | 66.9 | 0.27 | 36.3...123.2 | 0.85 |

30 | -68.9 | 20.2 | 0.32 | 9.6...42.7 | 0.71 |

31 | -74.0 | 38.8 | 0.32 | 18.5...81.5 | 0.71 |

33 | -44.0 | 4.6 | 0.32 | 2.2...9.7 | 0.81 |

35 | -66.3 | 21.4 | 0.33 | 10.2...45.2 | 0.80 |

38 | -61.9 | 13.7 | 0.31 | 6.8...27.8 | 0.81 |

43 | -62.4 | 9.7 | 0.26 | 5.3...17.8 | 0.89 |

47 | -58.3 | 10.7 | 0.31 | 5.2...22.0 | 0.87 |

56 | -56.3 | 7.7 | 0.33 | 3.6...16.6 | 0.82 |

Data available for positioning node 54

Anchor ID | RSSI (dBm) | Calculated distance (m) | Error on distance | Expected distance (m) | RSQ |
---|---|---|---|---|---|

3 | -84.1 | 51.3 | 0.23 | 29.9...87.8 | 0.87 |

6 | -85.1 | 43.0 | 0.28 | 22.8...81.4 | 0.84 |

9 | -88.9 | 66.1 | 0.31 | 32.7...133.6 | 0.82 |

13 | -77.1 | 31.0 | 0.27 | 16.9...57.1 | 0.85 |

30 | -70.0 | 21.7 | 0.32 | 10.3...45.8 | 0.71 |

31 | -46.0 | 6.3 | 0.32 | 3.0...13.3 | 0.71 |

33 | -73.4 | 36.8 | 0.32 | 17.5...77.3 | 0.81 |

35 | -60.0 | 13.9 | 0.33 | 6.6...29.2 | 0.80 |

38 | -54.4 | 8.1 | 0.31 | 4.0...16.4 | 0.81 |

43 | -70.0 | 16.4 | 0.26 | 8.9...30.0 | 0.89 |

47 | -73.6 | 32.8 | 0.31 | 16.0...67.3 | 0.87 |

56 | -73.6 | 26.7 | 0.33 | 12.4...57.7 | 0.82 |

^{-0,234}or 6.6 m and a maximum expected distance of 11.3 times 10

^{+0,234}or 19.4 m. The last column gives the RSQ from Section 5.1. Tables 5, 6, and 7 show the real distance between the anchors to node 4, 55, and 54, respectively.

Real distance between target node 4 and the 12 anchors

Anchor ID | Real distance (m) |
---|---|

3 | 13.0 |

6 | 12.2 |

9 | 12.6 |

13 | 15.4 |

30 | 43.1 |

31 | 43.4 |

33 | 72.0 |

35 | 72.9 |

38 | 68.2 |

43 | 67.3 |

47 | 72.2 |

56 | 75.8 |

Real distance between target node 55 and the 12 anchors

Anchor ID | Real distance (m) |
---|---|

3 | 82.5 |

6 | 77.8 |

9 | 73.9 |

13 | 68.4 |

30 | 33.7 |

31 | 33.9 |

33 | 4.8 |

35 | 12.9 |

38 | 15.4 |

43 | 10.9 |

47 | 6.8 |

56 | 5.0 |

Real distance between target node 54 and the 12 anchors

Anchor ID | Real distance (m) |
---|---|

3 | 52.7 |

6 | 47.9 |

9 | 43.9 |

13 | 38.4 |

30 | 8.8 |

31 | 5.6 |

33 | 25.5 |

35 | 24.7 |

38 | 20.0 |

43 | 19.6 |

47 | 24.5 |

56 | 28.1 |

Section 5.1 found ten good anchors, having a good linear regression between the RSSI reported by all receivers and the logarithmic distances between sender and receiver. Section 5.2 added two good central anchors having a low error on distance. Furthermore, it revealed where the anchors are the most accurate. In this section, we start polishing up the measurements.

When comparing Table 2 with Table 5, a large error between the calculated (122.2 m) and the exact distances (72.0 m) from anchor 33 is seen. Although not an outlier here (58.2 m is less than 72.0 m), this measurement can disturb any positioning algorithm. More extreme examples are found with other targets with calculated distance up to 134 m (1.5 times the size of the building) and maximum distance up to 275 m (three times the size). Multipath fading resulted in flatter slopes of the regression lines, and the extrapolation of these lines causes these large errors.

Similar to [55], a combination of a min-max algorithm with a maximum likelihood algorithm is needed to improve these situations. Contrary to [55], we suggest dropping all the measurements where the calculated distance is greater than a limit because the correction to that limit can lead to a large error. Based on Equation 2, a maximum distance for this hardware is calculated. Indeed, when both a sending power level of 0 dBm and a receiver sensitivity of -92 dBm (2) are considered, a maximum distance of 82.8 m is obtained. Thirty-five individual measurements are dropped in this way. For all the nodes, this additional step improves the distance error. If a distance calculation is higher than the limits for the hardware, then it probably comes from a connection with either too much attenuation or too much destructive multipath fading. In both cases, it comes from anchors operating at the end of their regression line. This algorithm when applied to node 4 (in Table 2) results in the elimination of the RSSI measurement of anchors 31, 33, and 47.

### 5.5. Elimination of circles

The elimination of circles discussed in this section is based on the accuracy of the model of Section 5.2. Anchor 54 is representative for a target in the center of the building, and anchor 55 is a good example of a target in an extremity of the building. These nodes perfectly demonstrate the accuracy of the model.

With two measurements (from two different anchors) of the same target, there are four possibilities: they are both too small, they are both (slightly or much) too large, the first one is (slightly or much) too small and the second is (slightly or much) too large or vice versa. On averaging the first two cases we get no improvement, since the closest distance to the target is corrected in the wrong direction. On averaging in the last two cases, only there is an improvement if there are no outliers. Deleting bad measurements and outliers is a major preprocessing step in the case of the heavily multipath faded environment, discussed in this article.

This number five is dependent on the total number of sending anchors (received by the target). We suggest using half of the total number of anchors minus 1. When this figure is set too low, good measurements may be eliminated. This will imply that the number of measurements is not large enough for the (good) positioning. When it is set too high, bad measurements may be allowed. This will have a negative impact on the next step of our preprocessing: the maximum likelihood algorithm.

It is noted that the remaining typical circles are more consistent. The typical circles that are too large (green) will be used in the maximum likelihood algorithm on the distance of the next section, where they can correct the distances of their group members.

### 5.6. Grouping anchors

When anchors are grouped, a maximum likelihood algorithm can be used. Usually this algorithm is found in the positioning algorithm itself, but we will use this technique in our preprocessing to accomplish a better distance estimation. In Section 7.2, we will compare the results of our complete localization algorithm with the more conventional maximum likelihood on the position.

Figure 2 shows three groups of anchors: nodes 3, 6, 9, and 13 in the first group; nodes 30 and 31 in the second; and the others in the third. The nodes in each group are neighboring anchors in one extremity of the building, the center and the extremity of the building at the other side, respectively. We suggest considering the anchors in one group as coinciding if one small circle is contained in the other(s). It is known from Section 5.2 that the radii of these small circles are available from the fifth column of Tables 2, 3, and 4 and calculated using the error on distance.

Under the assumptions that the probability distribution of the logarithmic distance is normally distributed for all the group members with the same variance, the likelihood function equals the product of these normal distribution functions of the *m* (see Table 1) remaining measured logarithmic distances in each group. Partial derivation of this likelihood function [52] proves the most likely logarithmic distance equals the average of the measured distance of the *m* group members. Hence, the most probable distance can be determined based on the (geometric) average of the *m* (see Table 1) remaining measured distances in each group. This geometric average is calculated and applied to all anchors in that group. (It is noted that the position of the anchors remains unchanged.) If a distance estimation of one anchor is too high, then it is compensated by the other group members. With distances too low also this is the case. By eliminating the outliers before this process, the chance of decreasing the accuracy of the good anchors is minimized. If the small circles do not contain others, then the target is probably nearby and the individual anchors are needed to increase the accuracy. In a more conventional study, the standard error for use in a maximum likelihood algorithm is determined using the Cramer-Rao Bound or calculated with a mean square error applied on the group members. In this article, we suggest using the standard error of the regression model and divide the error on distances by the square root of the number (*m* - 1) to obtain the corrected error on distance for the group members. This method provides larger large circles and smaller small circles. The advantage of doing so will be explained in the next section.

## 6 Positioning algorithm

The main path is leading through 6.1, 6.2, 6.3, 6.4, 6.5, and 6.7. In a very few cases, procedure 6.6 is encountered, and the use of procedure 6.8 is even rarer.

### 6.1. Calculate intersection points of the longitudinal borderlines of the building and the large circles

This is adopted for both the longitudinal borderlines of the building. An analogous procedure gives the leftmost right points of a trapeze. Without outliers on the constructive multipath fading, the large circles will not be too small, and the vast majority of the targets will continue to move on the main track toward 6.3.

### 6.2. Is the amount of constructive multipath fading acceptable?

When no outliers on the constructive multipath fading are present as shown in Figure 18, the trapeze has positive sides. Hence, this is a fast method to decide on the amount of constructive multipath fading. On the positive trapeze sides outcome, the algorithm continues with 6.3, and with negative sides 6.8 is the next step.

### 6.3. Calculate typical circles intersection points within the building

### 6.4. Is there more than one intersection point within the building?

Preprocessing steps resulted in more consistent distance circles. As discussed in Section 5.1, constructive multipath results in too small circles. When all the typical circles do not intersect at all within the building or when there is only one intersection point of all intersecting typical circles within the building, uncorrected constructive multipath fading is expected. This results in the selection of a different initial point. Very few targets follow the 6.6 procedure, and the vast majority of targets will continue with procedure 6.5.

### 6.5. Calculate initial position as the centroid of all typical circles' intersection points

A logical next step is the calculation of an initial position.

It is the result of a multilateration process, calculated as the centroid of all typical circles' intersection points with the typical circle that is intersected most. Figure 20 gives an example where the leftmost circle is intersected twice. The little square in the middle of the drawing represents the initial position. In a few cases a typical circle is intersected up to 14 times by other typical circles.

### 6.6. Calculate the initial position as the intersection of the diagonals of the trapeze

In Figure 21, the leftmost typical circle is smaller than the corresponding circle of the same anchor in Figure 20. It is likely that there is more constructive multipath fading for Figure 21. The typical circles do not intersect within the building, either because there are no intersections or because the intersections are not within the building. It is also noted that if the multipath fading for the two rightmost anchors decreases (and these circles become larger), then there will again be intersections within the building. Therefore the constructive multipath is expected when there are no intersections within the building. Furthermore, when there is at the most one intersection point of all the typical circles within the building, the initial point cannot be calculated with the multilateration step of Section 6.5. In this case the initial position is estimated as the intersection point of the diagonal of the (positive) sided trapeze, calculated from the large circles as shown in Figure 18. The procedure described in Section 6.7 will further increase accuracy.

### 6.7. Calculate the estimated position of the nearest intersection points of the longitudinal line and the typical circles

Around the initial point, accurate typical circles are expected, since the preprocessing resulted in consistent circles. It is known from Section 5 that all typical circles would intersect in the target if there were no multipath fading. Furthermore, Figure 11 shows that the 95% confidence intervals (in m) are the widest at high distances. An examination in the neighborhood of the initial point at distances larger than the limits of the confidence interval will include outliers. Therefore, we impose an absolute limit on the search region around this initial point. When set too high, non-detected outliers will spoil the good measurements, and when set too low, good measurements from far anchors will be eliminated. As a limit, we propose the same total confidence interval of 22 m for all the anchors. Models that are more complex can adapt this margin to the considered anchor. Figure 11 shows that for node 30 a 95% confidence interval of 22 m is reached at a distance of 50 m.

### 6.8. Calculate the position as the intersection of the diagonals of the trapeze

This procedure is executed when at least one of the concerned anchor-target paths exhibits too much constructive multipath fading. In our building the preprocessing step reduced the number of the affected targets from 5 to 3. Instead of finding out which circle is too small, we allow negative-sided trapezes. Considering that both circles are equally likely to be too small and that the negative sides are small, the position is estimated as the intersection of the diagonals of the trapeze. In Figure 19, this position is marked with a black square.

## 7. Results

In the first subsection of this section, the distance error is discussed, whereas in the next section, position errors are treated. The distance error only considers one of the 1942 sender/receiver pairs at a time. The error between the real and the calculated distance results in one point in the cumulative distribution plot (cdf). For the position error, all the distances from the different anchors are grouped, resulting in one of the 47 target positions at a time.

### 7.1. Distance error

### 7.2. Position error

*d*

_{ i, j }between the grid points

*j*and the anchors

*i*are calculated. Now, the anchor's RSSI measurements are used with conversion to the "calculated (tilded

*d*

_{ i, j }) distances" of Tables 2, 3, and 4. These points are compared resulting in the mean square error cost function (MMSE) (3) The most likely location is determined by minimizing this cost function [57, 58]:

^{2}or approximately 0.008 anchors per square meter) environment. As a common reference, the end of Section 5.2 is chosen: the best anchors are selected and calibrated. The effect of the presence of the rest of the preprocessing steps (5.4 through 5.6) is studied both on our algorithm and the more conventional maximum likelihood on the position. The figure shows that there is an improvement of the complete preprocessing on our complete algorithm, and the median of the error is reduced from 6.22 to 5.29 m. Furthermore, the worst result is for MMSE without preprocessing steps 5.4-5.6. Using our min-max algorithm, eliminating circles and grouping anchors have a positive effect on the position error of the MMSE algorithm. It reduces the median from 6.66 to 6.45 m. Our complete algorithm has a lower median than the MMSE maximum likelihood algorithm with all our preprocessing steps, and it has the lowest high percentiles.

## 8. Conclusion

We have presented a new approach for localization in a realistic indoor environment, where multipath fading is highly present. Our measurements show that the preprocessing steps decrease the median of the distance error from 8.95 to 4.03 m. Furthermore, the application of our complete algorithm eliminates the outliers and obtains a median of the position error of 5.26 m.

A maximum likelihood algorithm with a mean square error cost function has a position error median of 6.66 m. When our preprocessing is applied not only this median, but also the high percentiles of this algorithm are improved.

In our pre-existing sensor network with a large number of sensor nodes, the best available nodes are selected as anchors and calibrated. The whole process can be automated using standard linear regression tools. Time-consuming manual fitting and complex fingerprinting is avoided, thus, making it possible to do real-time localization in future dynamic wireless indoor environments.

## Notes

### Acknowledgements

The authors wish to thank the IBBT and all the designers of the W-ilab.t test bed in general and, Bart Jooris in particular for his support with the experiments.

## Supplementary material

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