4.1. Outdoor Experiments near the Surface of the Earth
Four series of outdoor experiments were carried out.
The measurement scheme for all outdoor scenarios was the same (
Figure 2).
The first series of measurements (outdoor, series 1, hereinafter O1) was carried out on a grass lawn. The nearest large objects (a building, a fence) were located at a distance of more than 7 m from the measurement line. Beyond the measurement site there were several trees. The emitter and the receiver were in line-of-sight (LOS) conditions on tripods of height h = 2 m. During the measurements, the position of the receiver was fixed, and the emitter moved in a straight line. The distance between the emitter and receiver was varied from 1 to dmax = 18 m in 1 m steps.
The second series of measurements (hereinafter O2) was carried out on the side of a forest road (the distance to the forest was ~2 m on one side and ~10 m on the other side). The signal emitter and the receiver were in LOS conditions on tripods of height h = 2 m. During the measurements, the position of the receiver was fixed, and the emitter moved in a straight line. The distance between the emitter and receiver varied from 1 to dmax = 13 m with steps of 1 m.
The third series of measurements (hereinafter referred to as O3) was carried out on a side of a countryside road (the distance to the nearest large object was more than 15 m). The emitter and the receiver were in LOS conditions on tripods of height h = 2 m. During the measurements, the position of the receiver was fixed, and the emitter moved in a straight line. The distance between the emitter and receiver varied from 1 to dmax = 31 m with steps of 1 m.
The fourth series of measurements (hereinafter O4) was carried out under the same conditions as the O3 series, i.e., on the side of the road, but the height of the tripods on which the emitter and receiver were raised was 1 m. During the measurements, the position of the receiver was fixed, and the emitter moved in a straight line. The distance between the emitter and receiver varied from 1 to dmax = 35 m with steps of 1 m.
In all the above scenarios, the aim of the measurements was to obtain the signal power estimate at the receiver input as a function of the distance between the receiver and the emitter.
According to the routine described in
Section 3.3, to determine the distance from the received power, one must know the attenuation index in specific conditions (here, in scenarios O1—O4). In the case of the power law of signal attenuation with distance (1), the attenuation index is the slope of the plot of the power difference (
Pd −
P0) (in dBm) versus the logarithm of the ratio of the measured
d and reference
d0 distances (here,
d0 = 1 m). Therefore, in order to evaluate the attenuation rate according to (1), the signal power and the distance were plotted in logarithmic scales (
Figure 3a) and the attenuation index
n was determined as the slope of the straight line approximating the dependence of the power difference on the logarithm of the distance ratio.
The power difference (
Pd −
P0) [dBm] as a function of the logarithm of the distance ratio (
d/
d0) is shown in
Figure 3a. In the range up to 10 m, this function is close to linear, then linearity is lost (the power-law dependence of signal power on distance is violated). In this regard, the question arises, what is considered index
n? Should we calculate it for the linear part or include the nonlinear part as well? The attenuation index depends on the set of measurements for the different distances at which the approximation is made. Thus, if we want to stay within the power law, we have to determine the range of distances over which it operates (within certain limits).
The answer to this question can be given as follows: we want to find the maximum range in which the distance error median, when using the power law of power attenuation, does not exceed a given value, e.g., 1 m, which is good accuracy for systems based on signal power measurement. In order to identify a linear section, the sets of distances from 1 m to di, di = 2, 3, … k, were successively considered, where k is the maximum distance for a particular experiment. For each of these distance sets [1, 2], [1, 2, 3], …, [1, …, k], the attenuation index n was calculated. Using the value of n, distance errors were determined for each distance. The limit distance dlim was taken to be the maximum distance at which the maximum median error did not exceed 1 m. The value of n obtained for the set of distances [1, …, dlim] was further considered as the attenuation index for this scenario and for these specific measurement conditions.
Using the measured signal strengths and the estimate of the attenuation index, we estimated the distances
d*. The estimated and actual distances are compared in
Figure 3b, where the diagonal corresponds to a perfect match, and deviation from the diagonal corresponds to the distance estimation error. This figure shows how power measurement errors lead to distance errors. Each of the curves corresponding to different scenarios also shows the standard deviation of the distance measurement error at a given point. As can be seen from the figure, the power spreads at each point are much smaller than the large-scale power variation with distance, most likely due to the presence of the second ray reflected from the earth’s surface [
16].
In
Figure 3c–e for scenarios O1–O4 the mean absolute error, standard deviation and relative error are plotted as functions of distance, respectively. The deviation of the power-distance dependence from the power law strongly affects the magnitude of the error, which increases with distance and can reach half the distance (
Figure 3c).
Figure 3 and
Figure 4 show us the dependence of the distance error on the actual distance.
Figure 4 shows boxplot for each of the points at which the distance was estimated. As can be seen, the median errors follow the trend (they are regular). The deviation from the median value increases with distance, which is explained by the presence of a systematic error of the distance measurement method [
15]. This systematic error is due to the limited number of ADC quantization levels in the receiver threshold device, which limits the resolution of the power measurement method. Since the threshold level of the log detector is discrete, the measurement error increases with increasing distance (and decreasing received signal power). Theoretical estimates of measurement error for this method are given in [
15].
In all four series, the dependence of the measured power on distance has a similar type of behavior, namely, up to a certain limiting distance
dlim the signal power obeys a power law, then the law is violated and large-scale fluctuations in signal power are observed. This behavior is explained by the presence of two rays (direct and reflected from the earth’s surface), which add up at the reception point and significantly affect the power of the total signal, and also deviate it from a power-law dependence. The theoretical two-ray model of UWB signal propagation is described in detail in [
16].
The area where the power-law dependence is preserved depends on the height of the emitter and receiver above the ground: the higher the suspension point, the larger this area, i.e., the larger the range of distances at which high accuracy is achieved.
Table 1 summarizes the experimental results for the four scenarios. The calculated values of the attenuation index
n, the value of the limiting distance
dlim at which the maximum median error does not exceed 1 m, the values of MAE, MedAE, STD, RMSE, and relative error averaged over the distance range [1,
dlim] are given.
To obtain an idea of the accuracy that can be achieved by focusing only on the linear section,
Table 2 shows the errors over the entire range of measured distances [1,
dmax] at the value of the attenuation index
n calculated on the linear section of the dependence of (
Pd −
P0) [dBm] on (
d/
d0).
In four series of outdoor experiments at distances up to 10–15 m, the average absolute error was 0.21 m, the average median error was 0.21 m, the average RMSE was 0.23 m, the average standard deviation was 0.07 m, and the average relative error was 2.6%.
The results of all four scenarios are united by the presence of a common range of distances from the emission point to 10–15 m, within which the median error is ~0.2 m (see
Table 1). Next, a region begins where the dependence of (
Pd −
P0) [dBm] on (
d/
d0) ceases to be linear (the power-law dependence of power on distance is lost) and the error increases. As a result, at distances of about 30–35 m, the median error increases to 3.4 m (see
Table 2).
4.2. Indoor Experiments in a Corridor
For all indoor scenarios, the scheme of measurements was the same (
Figure 5).
The experiment was carried out in a corridor 4.6 m high, 3.1 m wide, and 45.5 m long. Two receivers were located on a line at a distance of
d12 = 19 m from each other on tripods at a height of
h = 2 m above the floor. Between them, on the same line a signal emitter was placed under LOS conditions (
Figure 5). During the measurements, the position of the receiver was fixed, and the emitter was moved along the same line. The distance between the emitter and receiver varied from 1 to
dmax = 18 m in 1 m steps.
The introduction of receiver R2 is a feature of the experiment arrangement. In indoor experiments, receivers R1 and R2 were located in different parts of the room, so that the propagation conditions were different (different configurations of reflective surfaces), which allowed us to increase the set of scenarios and thereby enrich the experimental data. To reduce laborious operations (e.g., installing the emission source at different points of the room) and the time of the experiment, data from R1 and R2 were taken in one pass. The results for each receiver were analyzed independently.
The two series of measurements obtained with two different receivers are hereafter referred to as C1 and C2. Two other series of measurements (C3 and C4) were performed under the same conditions, but to reduce the spread of values in the receiver, we additionally processed (“sifted”) the measurement results and took the minimum value from every 50 consecutive measurements (sliding window).
Measurements in the corridor show qualitatively the same dependencies as in open areas. There is an initial section within which the change in power is described by a power law with a high degree of accuracy (
Figure 6a). Therefore, once we have determined the attenuation rate in this region, we can measure the distance inside with an average median error of ~30 cm (see
Table 3).
This is followed by a pattern of deviation from the power law, similar to that observed in open areas, caused by the presence of rays reflected from the surfaces of the floor, ceiling, and walls, which leads to the appearance of slowly changing trends.
The measured distances and the actual distances are compared in
Figure 6b. In
Figure 6c–e for different scenarios, the mean absolute error, standard deviation, and relative error are shown as functions of distance, respectively.
Figure 7 shows the distance estimation error (boxplot) versus actual distance for scenarios C1–C4.
In
Table 3, the experimental results for four corridor scenarios (C1–C4) for the distance range [1,
dlim] are presented. In
Table 4, the results on errors are given, calculated over the entire range of measured distances [1,
dmax] for the value of the attenuation index
n, calculated over the linear section of the dependence of (
Pd −
P0) [dBm] on the logarithm of (
d/
d0).
The average absolute error in four series of experiments C1–C4 at distances up to 6–7 m was 0.31 m, the average median error was 0.31 m, the average RMSE was 0.32 m, the average standard deviation was 0.04 m, and the average relative error was 8.3%.
At distances up to 18 m, the average absolute error in four series of experiments C1–C4 was 1.22 m, the average median error was 1.23 m, the average RMSE was 1.23 m, the average standard deviation was 0.1 m, and the average relative error was 12.2%.
Results for scenarios C1–C4 were obtained on the same day and under the same conditions. Series of measurements C1 and C3 were obtained with receiver R1 (
Figure 5), the position of which did not change in both series of experiments. In series C1, raw data were used, and in series C3, measurements were further processed in a sliding window. The results for C2 and C4 were obtained with receiver R2. In scenario C4, measurements were also further processed. Analysis of the graphs in
Figure 7 shows that if we ignore the outliers (blue dots), then we observe good agreement between the results obtained for the same receiver in different experiments. That is, the results for pairs C1 and C3, and C2 and C4, are practically repeated up to the difference in the error spread for each of the distances, which is associated with processing the measurement results in scenarios C2 and C4. This suggests that the measurement results are determined primarily by the geometry of the room, and the signal power measurements are quite stable and slightly susceptible to random fluctuations. For large variations in signal strength due to multipath propagation, observed at distances greater than 6 m, one can try to eliminate using fingerprinting-like approaches.
4.3. Experiments in a Corridor with a Large Number of Metal Structures
The experiments were carried out in the basement, in a corridor 2.6 m high, 2.9 m wide, and 46.3 m long. The corridor contained a large number of different metal structures: beams and purloins, cabinets, and boxes along the entire length of the corridor.
Two receivers were also used (
Figure 5), located at a distance of
d12 = 41 m. On the line between them, under LOS conditions, a signal emitter was placed. During the measurements, the position of the receivers was fixed, and the emitter moved between them in a straight line. The distance between the emitter and receiver varied from 1 to
dmax = 40 m in steps of 1 m.
Several series of measurements were carried out. In series B1 and B2, the height of the tripods on which the emitter and receivers were mounted was 2 m. In series B3 and B4, the height of the tripods was 1 m.
The power difference (
Pd −
P0) [dBm] as a function of the logarithm of the distance ratio (
d/
d0) is shown in
Figure 8a. Unlike the outdoor scenario, here, the power-law dependence of signal power on distance breaks down at shorter distances (about 6–7 m). The measured distance
d* and the actual distance
d are compared in
Figure 8b. In
Figure 8c–e for different scenarios, the mean absolute error, standard deviation, and relative error are plotted as functions of distance, respectively.
At a height of 2 m, the tripod stands almost at the level of the doorway, which is located at a distance of 29.4 m from the first receiver and at 8.4 m from the second receiver. This may explain the sharp degradation in quality for scenarios B1 and B2 at these distances.
Basement measurements provide a clear example of the dramatic effect of multiple reflections on the estimated signal strength. Whereas outdoors or in a corridor without metal structures the presence of a regular oscillating trend is clearly visible, associated with the second ray reflected from the earth’s surface, then in a place with numerous reflective surfaces unevenly surrounding the emitter and receiver, the regular structure of power oscillations is completely lost outside the area close to the emitter, where the direct path prevails, and where the power-law dependence of the received power on distance applies. This effect of numerous reflective surfaces once again emphasizes the difficulty of determining distance in such conditions.
The area where the average error is 0.56 m extends to 5–8 m (see
Table 5), and then we are dealing with unpredictable and irregular variations in signal power and the relative error reaching 100–200% (
Figure 8d). Nevertheless, even in such conditions, trends (albeit irregular) still exist, so a fingerprinting method could be applied here in the future.
Figure 9 shows distance estimation errors (boxplot) versus actual distance for scenarios B1–B4.
Table 5 presents the experimental results for four scenarios in the corridor with a large number of metal structures (B1–B4) for the distance range [1,
dlim].
Table 6 shows the errors calculated over the entire range of measured distances [1,
dmax] for the value of the attenuation index
n, calculated over the linear section of the dependence of (
Pd −
P0) [dBm] on the logarithm of (
d/
d0).
In four series of experiments in the corridor with a large number of metal structures, at distances up to 5–8 m the average absolute error was 0.56 m, the average median error was 0.56 m, RMSE was 0.56 m, the average standard deviation was 0.05 m, and the average relative error was 11.8%.
At distances up to 40 m in four series of experiments, the average absolute error was 6.04 m, the average median error was 6.04 m, the average RMSE was 6.06 m, the average standard deviation was 0.27 m, and the average relative error was 28.9%.
4.4. Experiments in the Conference Hall
The experiment was carried out in an empty conference hall 4.1 m high, 10.4 m wide, and 18.6 m long.
Two receivers were located at a distance
d12 = 8 m from each other on stands with a height of
h = 2 m (
Figure 5). Between them, on the line connecting the receivers, a signal emitter was placed under LOS conditions. During the measurements, the position of the receiver was fixed, and the emitter was moved along the same line. The distance between the emitter and receiver varied from 1 m to
dmax = 7 m in 1 m steps.
Several series of measurements were carried out.
Series H1 and H2, H3 and H4 were performed on different days. Series H5 and H6 were carried out under the same conditions as H3 and H4, but with a different antenna orientation: the antenna was rotated around the vertical axis by 90 degrees.
The dependence of the power difference (
Pd −
P0) [dBm] on the logarithm of the distance ratio (
d/
d0) is shown in
Figure 10a. In this case, the characteristic dimensions of the hall coincide with the power-law portion of the dependence of signal power on distance, which is confirmed by the measurement results. A wide, uniform space without large reflective surfaces gives an average measurement error of 0.15 m over distances up to 7 m (see
Table 7).
The measured distance
d* and the actual distance
d are compared in
Figure 10b. In
Figure 10c–e for different scenarios, the mean absolute error, standard deviation, and relative error are plotted as functions of distance, respectively.
Figure 11 shows distance estimation errors (boxplot) versus actual distance for scenarios H1–H6.
Experimental results for six scenarios in a conference hall (H1–H6) for the distances up to
dlim are given in
Table 7. Here, the value of
dlim coincides with
dmax.
In six series of experiments in the conference hall, at distances up to 7 m the average absolute error was 0.15 m, the average median error was 0.14 m, RMSE was 0.15 m, the average standard deviation was 0.04 m, and the average relative error was 3.4%.
4.5. Experiments in an Office Space
The experiments were carried out in a laboratory room (office) 4 m high, 6.1 m wide, and 6.4 m long.
Two receivers were located at a distance
d12 = 5 m from each other on stands with a height of
h = 1.65 m (
Figure 5). Between them, on the line connecting the receivers, a signal emitter was placed under LOS conditions. During the measurements, the position of the receiver was fixed, and the emitter was moved along the same line. The distance between the emitter and receiver varied from 0.5 m to
dmax = 4.5 m with 0.25 m steps.
Two series of measurements were carried out for two pairs of receivers located in different parts of the room (series L1–L4).
The dependence of the power difference (
Pd −
P0) [dBm] on the logarithm of the distance ratio (
d/
d0) is shown in
Figure 12a. The measured distance
d* and the actual distance
d are compared in
Figure 12b. In
Figure 12c–e for different scenarios, the mean absolute error, standard deviation, and relative error are plotted as functions of distance, respectively.
In office premises, the distances between the emitter and the receiver are small, so the dependence of the power difference (Pd − P0) [dBm] on the logarithm of the distance ratio (d/d0) is almost linear over the entire range of distances.
Figure 13 presents the distance estimation error (boxplot) for each of the actual distances for scenarios L1–L4.
Experimental results for four scenarios in an office premises (L1–L4) for the distances up to
dlim are given in
Table 8. Here, the value of
dlim coincides with
dmax.
In four series of experiments in the office environment, at distances up to 4.5 m the average absolute error was 0.16 m, the average median error was 0.15 m, RMSE was 0.16 m, the average standard deviation was 0.04 m, and the average relative error was 5.9%.