A 3D Ray Launching Time-Frequency Channel Modeling Approach for UWB Ranging Applications

Ultrawideband (UWB) has the ability to achieve decimetre level of ranging accuracy, hence, its wider usage nowadays in the field of positioning. In spite of the attractiveness of UWB, its performance is strongly dependent on the propagation channel. In this paper, an analysis of the the UWB channel for ranging applications using an inhouse developed 3D Ray launching (3D RL) algorithm is presented. A parametric study has been performed considering variations of cuboid size resolution of the simulation mesh, in order to analyze convergence impact on estimation accuracy, focusing on Radio frequency (RF) power levels as well as time domain characterization. The RF power results have been used to model the path-loss, small scale fading, and the power delay profile so as to obtain the statistics of the multipath channel as well as time of flight (TOF) estimation values. The results show that the 3D RL is a valuable tool to test UWB systems for ranging applications with a mean accuracy of up to 10 cm in multipath conditions considering complex scatterer distributions within the complete volume of the scenarios under test.


I. INTRODUCTION
The Federal Communications Commission (FCC) and the International Telecommunication Union Radiocommunication sector (ITU-R) define Ultrawideband (UWB) as an antenna transmission for which the emitted signal occupies a fractional bandwidth greater than 0.2 or a bandwidth more than 500 MHz. Thanks to its several attractive properties, UWB technology has seen tremendous advances in the field of wireless communications [1], imaging [2], radar [3], networking [4], and localization [5]. For instance, the large bandwidth allows for more reliable communications by providing a high resilience to frequency selective fading. Additionally, the spectral power spreading over a high bandwidth decreases the spectral density, consequently reducing interference to The associate editor coordinating the review of this manuscript and approving it for publication was Prakasam Periasamy . existing wireless systems. Finally, thanks to concept of pulse based UWB radio, the construction of communications systems with reduced complexity in the Radio frequency (RF) architecture is now possible.
Multiple solutions have been described in order to provide localization in indoor environments, based on technologies such as Wi-Fi/WLAN, Infrared, RFID, Cameras, among others [6]. However, the use of ultra-short pulses such as in UWB enables high time resolution, providing high accuracy. Therefore, UWB positioning has the ability to achieve decimeter-level location estimates using time of flight (TOF) techniques [6]. The reason is that the high bandwidth allows accurate TOF computations and it is robust against multipaths. Consequently, a considerable part of these systems is deployed in several environments such as hospitals, residential, industrial and others [6]. This is leading to an ever increasing adoption and deployment of UWB positioning VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ systems. Moreover, this necessity is likely to increase in the near future because mobile phone manufacturers such as Apple Inc are demonstrating attraction to this technology. For instance the new iPhone 11 embeds an UWB interface, and the company expects to release UWB beacons soon [7]. In spite of potential benefits of UWB, its performance is strongly dependent on the propagation channel. In fact, critical and real time applications such as positioning require lineof-sight (LOS) situations between the transmitter (Tx) and receiver (Rx) as well as a careful characterization of the channel. Therefore, to adequately deploy the necessary RF sources i.e., access points or anchors in wireless sensor networks, it is compulsory to adequately characterise the radio propagation losses as well as the time domain properties of the signal (i.e., delay spread and power delay profile). Literature shows that there are a wide range of works that characterise the channel using UWB measurements [8]- [10]. These works obtain statistical propagation models from the experimental results. However, measurement campaigns are usually time-consuming requiring dedicated material and human resources. As alternative, deterministic methods [11]- [13] based on numerical approaches which involve either solution of Maxwell's equations using full-wave simulation techniques or geometrical approximations such as Ray Launching (RL) [14] and Ray Tracing (RT) [15] can be employed. In principle, these techniques generate more precise results but require higher computational demands than statistical techniques. Another advantage in the use of deterministic methods is the possibility of using the available floor plan to generate a detailed information of the topology of the environment. This can be useful in providing more insights on the impact of several properties within the environment under analysis. In this sense, the use of deterministic based models, such as RT or RL enable precise characterization in terms of time domain properties as well as RF power level estimations, when compared with empirical/statistical based models. Time domain components can be evaluated at an individual component level as well as in ray beams. Indoor impact can be carefully determined, considering non-uniform topological scatterer distributions, as well as cluster configurations.
Despite the advantages of deterministic methods, to the author's knowledge, few references [36]- [41] briefly analyze the UWB channel using RL or RT techniques i.e., focus on a specific characteristic of the UWB indoor multipath channel such as power delay profile [36]- [38] and path-loss [42]. In [39]- [41], RT techniques are somewhat proposed for UWB characterization, in order to derive time domain characteristics, RF power estimations or excess delay characterization in indoor scenarios, considering indoor structural configuration (i.e., walls, hallways, simplified furnishing models). The results obtained provide higher accuracy, particularly when considering site specific conditions within the scenarios under analysis. However, complex clutter distributions or combined time-frequency analysis approaches have not been proposed. A possible reason for these few references is that back in the early 2000's when research on UWB was starting out, the use of deterministic techniques was more limited owing to high computational cost. Currently, UWB is being progressively adopted, thanks to multiple application domains, as previously stated. Additionally, computers exhibit higher computational capacity thanks to parallel-based computation, use of graphic processor acceleration or distributed computing architectures.
Research gaps which this work seeks to address include: a careful analysis the UWB channel using RL techniques. In the literature, we can find several works on UWB propagation models, applications and characterisation such as in Table 1. However, as far as the authors are concerned there is no work in the literature that provides a careful analysis of the UWB channel using 3D RL tool focusing on RF power levels as well as on time of flight (TOF) estimation.
In this paper, not only do we present a novel analysis of the UWB channel using an inhouse developed 3D RL, but also use the RL technique to obtain the accurate ranging results from the time domain properties of the signal. Accuracy of the results is shown by comparing simulation vs measurements in several spatial points of the scenario. Therefore, the specific contribution of this paper is three fold: 1) The UWB propagation channel is modelled in an indoor environment using an accurate 3D RL tool.
2) The convergence of the developed 3D RL tool is studied with respect to the size of the cuboid. 3) Using the time domain properties of the UWB signal, the 3D RL tool is used to obtain the accurate TOF ranging results which can be used in any positioning application.
The structure of the remainder of this paper is as follows. Section II presents the theory behind the analysis of radio signals using 3D RL tool. The inhouse developed 3D RL code is analytically described and validated using a set of measurements in section III. The statistical UWB models is presented in section IV. In section V, the estimation of position related parameters such as the range is presented. Finally, in the last section, conclusions are given.

II. RAY LAUNCHING ANALYSIS
Previously in the 1990s, RL and RT were together categorised as RT techniques. More recently, however, the distinctions have been made clearer. In the RL approach, the Tx sends thousands of rays in a solid angle and the algorithm follows the propagation of each ray until it either arrives at the Rx or becomes so weak to be significant. In RT, the rays are traced from the principal radio wave propagation regions that are estimated.
The majority of the works on RL or RT techniques in the scientific literature focus on narrow band channel measurements and modelling for office indoor scenario [43], public transportation buses [44], vehicular [45], aircraft [46], and metro environments [47]. These works utilise the conventional RL. However in systems where the signal occupies a wide bandwidth such as in UWB, a sub-band divided RL technique (i.e., simulating the channel at multiple center frequencies of the corresponding sub-bands) is recommended.

A. CONVENTIONAL RAY LAUNCHING
An inhouse implemented 3D-RL algorithm has been used for the UWB channel characterization. The developed algorithm is based on the conventional RL approach centred on the shooting and bouncing rays (SBR) technique, where a set of rays are launched from the Tx in a solid angle predefined as input parameter. The algorithm is based on Geometrical Optics (GO) and the Uniform Theory of Diffraction (UTD). GO can be seen as an asymptotic solution of Maxwell's equations and has proved to be precise in propagation prediction problems if the dimensions of the obstacles are large enough compared to the wavelength [48]. However, GO can lead to uncertainties in shadow regions where diffraction has higher impact, so it needs to be completed with the UTD to achieve accurate results in complex environments where a high quantity of obstacles is encountered [49], [50]. Thus, the combination of GO and UTD is an appropriate solution to model radio wave propagation provided the geometry and the scenario details, where an exact solution of Maxwell's equations could be unaffordable to compute.
The inhouse developed 3D-RL algorithm considers the morphology and topology of the scenario, as well as all the obstacles within it, by means of a realistic 3D recreation of the environment. Parameters such as number of reflections, radiation pattern of the transmitter and receiver antennae, angular resolution of launching rays and spatial resolution of cuboids size are considered as input parameters. The algorithm has been validated in the literature for propagation prediction at complex indoor environments, achieving a good trade-off between results accuracy and computational load [51]. In order to decrease computational load with a slightly decrease in precision, it can also be hybridized with other approaches, such as neural networks [52], collaborative filtering [53] or electromagnetic diffusion equation [43]. These hybrid approaches can be used depending of the complexity of the scenario and the expected results accuracy, which could serve for different radio planning engineering purposes.

B. SUB-BAND DIVIDED RAY LAUNCHING
Given the inherently large bandwidth of operation, the UWB propagation channel is frequency selective. Hence, channel charactersitics may vary considerably over the entire UWB bandwidth. Therefore, it is not enough to simulate UWB systems using the conventional RL technique applied to one discrete frequency. Overcoming this challenge requires use of a sub-band RL that has been already applied to works in [36] and [38]. The basic idea is simulating the UWB channel at multiple center frequencies of the corresponding sub-bands. We summarise sub-band divided RL into the following: 1) The entire bandwidth is split into several sub-bands where a constant frequency characteristic for materials can be assumed. The higher the number of sub bands the more accurate the algorithm, but at a higher computational cost. 2) At each center frequency, the conventional RL is deployed to extract the channel impulse response (CIR). 3) A complete frequency response over the entire UWB bandwidth is obtained by combining all the frequency responses from all the sub-bands. Each UWB radio channel can have a bandwidth of more than 500 MHz, depending on its centre frequency. In our work, we shall consider 4 GHz as the primary frequency of interest and a bandwidth of 500 MHz because: (i) several UWB commercial systems for positioning are available at this frequency, and (ii) the bandwidth of 500 MHz has been shown to be sufficient in obtaining accurate TOF results [54]. Using the Sub-band Divided RL technique, the frequency bandwidth of 500 MHz is thus divided into two sub-bands with 250 MHz each i.e., RL simulations and measurements are performed at 3.75 GHz, 4 GHz, and 4.25 GHz.

III. RAY LAUNCHING SIMULATION ALGORITHM
In order to calibrate the simulation scenario under test, a preliminary and simple experiment was performed to determine the frequency selectivity of the UWB channel at the considered frequencies. The Tx employed was a Hewlett Packward 8753D (30 KHz -6 GHz) signal generator, where as the Rx was a portable Agilent N9912 Field Fox spectrum analyzer, with 100 MHz bandwidth directly connected via a coaxial transmission line. Rx bandwidth has been selected in order to increase signal to noise ratio and hence detection capabilites. This is feasible owing to the fact that the 3D RL sub band simulations are based on a continuous wave approach.
The Tx power of the signal generator was set at maximum of 10 dBm. The measured RF power at 3.75, 4 GHz and 4.25 GHz were 7.5 dBm and 7 dBm, respectively, which indicating that the respective cable and connector losses accounted for 2.5 dB and 3 dB.
Simulations of a typical lab indoor scenario which is located at department of Electric, Electronic and Communication engineering of the Public University of Navarra in Spain was then performed. The dimensions of the scenario were 26 × 6 × 4 m 3 . The schematic view in Figure 1 shows all the interacting objects (IOs) such as walls, computers, monitors, chairs, desks, closets within the scenario and have been taken into account in the simulation process. In addition, material properties of the objects as defined by their dielectric constant and permittivity in Table 2, as well as propagation phenomena like reflection, diffraction or refraction have all been taken in account. The parameters for all of the materials included in the scenario, as provided in [55] are given in the 1 GHz -10 GHz frequency, valid for the UWB frequency range under consideration.
During the simulation, other parameters such as frequency of operation, the angular and spatial resolution, the maximum number of permitted rebounds ( the number of interactions between a propagated ray and the obstacles), the angular and spatial resolution, antenna type and gain, transmission power level have been considered according to Table 3. These parameters have been chosen to achieve adequate accuracy and optimization of the simulation results.   The 3D scenario in Figure 1 is designed as a structure made up of a matrix of fixed cuboids whose size has been varied according to Table 3. Essentially as a ray is sent from the Tx at the coordinates (x = 4.1 m, y = 1.47 m, z = 1.60 m) [see - Figure 2] through or to a given cuboid, its propagation parameters are stored in the corresponding matrix that represents the cuboid. Hence, it is possible to obtain the RF power from each cuboid for analysis as illustrated in Figure 3. Note that the RF power levels are obtained from the same spatial locations for both the measurements and simulations (obtained from 3D mesh of cuboids in which the scenario has been divided).

A. TEST SCENARIO VALIDATION
Validation of the RL simulation method was achieved by conducting measurements within the simulated scenario at the considered UWB frequencies. The Tx and Rx antennas were connected to the signal generator and portable spectrum analyzer, respectively through coaxial cables. Both Tx and Rx antennas were omnidirectional UWB Partron dielectric chip 0 dBi gain antenna manufactured by Abracon. Similar to the simulations, the Tx has been located at the coordinates (x = 4.1 m, y = 1.47 m, z = 1.60 m) as depicted in Figure 2. Measurements were performed along the dashed line also depicted in Figure 1 at a height of 1.6 m, with a spacing of 2 m between the points. At each point, the measurements were conducted for a duration of 60 s so that the maximum RF power level is obtained within the given time span and measurement bandwidth. The results of the test scenario validation are illustrated in Figure 3 and Figure 5. The error bars is the SD. The MAE is obtained from averaging the error obtained from RF power results in Figure 3.

B. RF POWER CONVERGENCE VERSUS THE CUBOID SIZE
One of the most relevant aspects of RL simulations is defining the appropriate 3-D cuboid mesh to obtain accurate results. As a result, convergence analysis was performed to find the optimal cuboid resolution needed to obtain accurate RF power estimations when modelling the UWB radio wave propagation channel in this environment. For every interval, eight different fixed cuboids sizes are considered and analysed i.e., the analysis considered both cubes and cuboids. In Figure 3 we show the mean and the SD of the RF power at 4 GHz obtained from the simulations versus the measured RF power. The estimated RF power becomes more accurate as the cuboid size intervals increase and converges at the interval 0.6 − 0.7 as seen in Figure 4. This implies that the maximum accuracy of the algorithm is reached when simulations are made within this cuboid size interval. A further assessment of the 0.6−0.7 interval shows that the cuboid size of 0.6/0.6/0.6 (0.6 m cuboids resolution in the x, y, z dimensions) in particular provides relatively accurate results. This is clearly evident in Figure 5 where a comparison between the measured and simulated Rx power is made. Looking at Figure 5, the first observation is that all results fit with the measured RF power trend curve, i.e., the power decaying with increasing Tx-Rx distance. The results also show that there exists a clear fit in the trends of the measured and simulated curves, which validates the algorithm. In fact in Table 4 the mean error and SD at the corresponding frequency of interest are shown. Most of the errors observed are mainly due to approximations made in simulation. Note that due to the relatively small differences in the RF power between the simulations and measurements, the analysis in respect to the RF power for the rest of this paper has been made with reference to the aforementioned cuboid size.

C. BI-DIMENSIONAL RF POWER DISTRIBUTION
The presented inhouse developed RL simulation tool provides 3D results, i.e., estimations for the complete volume of the analyzed environments are obtained. As an example, Figure 6 shows the bi-dimensional RF power distribution planes within the current scenario under analysis for different heights: from 0m to 0.6m, 1.2m to 1.8m (the transmitter location height) and 3m to 3.6m. Tx location is depicted by a red circle. The three height planes are shown for the chosen  It is worth noting that the obtained results show the typical rapid RF power variations of an indoor environment where the multipath propagation phenomenon has a big impact. At first glance, the differences between the heights can be clearly seen.
But on the other hand, the results between different operation frequencies seem quite similar. In order to gain insight in this, in Figure 7 RF power level difference planes are presented. Specifically, Figure 7 shows the difference between 3.75 GHz and 4.25 GHz results at 1.2-1.8m height (left) and  the difference between height 0-0.6m and 3-3.6m at 4 GHz (right). These results show that the RF power level can be as high as 20 dB (or more) in some points and zones of the scenario under analysis. This reinforces the idea of following the presented sub-band divided RL methodology when communication technologies such as UWB are employed, where the bandwidth is much wider than other common wireless communication systems (such as WiFi or ZigBee), exhibiting frequency dependent characteristics.

IV. STATISTICAL UWB CHANNEL MODELS
Once that 3D-RL simulation methodology has been validated, propagation characterization is described in this section. We will focus on the following primary characteristics of indoor multipath channel: i) path-loss, (ii) small scale fading, and (iii) power delay profile.

A. PATH-LOSS ANALYSIS
The path loss models can be used to estimate link budgets, capacity, cell sizes and shapes, etc. Generally, the path loss model in UWB systems is not only dependent on distance like narrow band systems, but also on the frequency and bandwidth [8].
Initially, the friis free space model (narrow band model), which includes effects of the Tx and Rx antenna can be used as good approximation for the path loss analysis. However, as it is stated in the final report of the IEEE 802. 15.4a UWB channel model [56], the specific antennas used for each case will have different specifications, and therefore, a model that only describes the wireless channel is presented in (1) where PL 0 is the path-loss at 1m distance, f is the bandwidth of the model (2)(3)(4)(5)(6)(7)(8)(9)(10), P(f ) r is the RF power level, P(f ) t is the Tx power level, PL(f ) is the path-loss defined in (2), f c is the central frequency, d is the distance, d 0 is the reference distance (1m), n is the pathloss exponent, k is the frequency dependence of the path-loss of the model, and η t and η r are efficiencies of the Tx and Rx antennas respectively. This model can be consulted in Section III of [56], in which UWB Model parameterization for 2-10 GHz is made under the Residential, Indoor office, Outdoor, Open outdoor and Industrial environments. Taking into account the morphological characteristics of the scenario under analysis in this work, the Indoor office environment model has been chosen in order to compare it with the 3D RL simulation results. The comparison is presented in Figure 8, for 3.75 GHz, 4 GHz and 4.25 GHz frequencies.
The results presented show that the 3D RL results follow the tendency marked by the IEEE 802.15.4a channel model for indoor office environments. It is worth noting that the obtained RF estimations start following the LoS line and by increasing the distance from the Tx the values change the tendency and follow the NLoS curve (at 15m approximately), which is due to the obstacles and elements that appear in the line of sight (e.g. a door, the glass wall, etc. [see - Fig. 1]).

B. SMALL SCALE FADING
Small scale fading is the fluctuation of power over very short periods of time caused by interference between multipath components. It is usually modelled as different distributions such as Gamma, Lognormal, Rician, Rayleigh, Nakagami, Weibull, or mixture between them [57]. Figure 9 presents the simulated cumulative distribution function (CDF) of the RF power for the respective frequency in the considered scenario along the linear distribution line in front of the Tx antenna. The comparison with the theoretical distributions which best fits the simulated data has been also presented. To measure the closeness of the data with the theoretical distributions, the Anderson-Darling (AD) goodness-of-fit (GOF) test with 5.0% for the significance level of the hypothesis (Ho) test has been used [58]. Table 5 presents the AD statistic results for the fitted distributions for the three considered frequencies (i.e., 3.75, 4 and 4.25 GHz), with the hypothesis result, p-value, AD statistic and squared coefficient value (SCV). The SCV is a measure of the data variability. Indeed, the SCV characterizes the amount of fading of the RF signal power [59]. It can be observed that the Lognormal and Weibull distributions are the best fits for this specific scenario for the three considered frequencies, where the test fails to reject the null hypothesis at the default 5% significance level.

C. POWER DELAY PROFILES
To gain insight into the small-scale variation in specific spatial locations of the scenario, the multipath variations has been analyzed with the power delay profile (PDP) for a specific spatial sample where high multipath is obtained. For that purpose, the CDF of a power delay profile at a point located at 5 meters distance in front of the Tx antenna has 97328 VOLUME 8, 2020 been computed and compared against Rayleigh distribution, which is commonly used to describe multipath statistics for narrowband channels at indoor environments [57].
The probability density function (PDF) of a Rayleigh distributed random variable is given by (3) where r is the magnitude of the RF power, and σ is the SD of the multipath propagation. The CDF of Rayleigh distribution is obtained via integration over the PDF and is presented in

V. ESTIMATION OF POSITION-RELATED PARAMETERS
Since a key application of UWB systems is positioning using TOF measurements, in this section, we provide an analysis of how 3D RL results can be tailored for ranging applications. Firstly, we perform extensive RL simulations to extract the TOF from the signals. Secondly, we carry out TOF measurements using commercial available UWB nodes to validate the simulations.

A. RAY LAUNCHING TOF SIMULATIONS
Estimation of TOF results is given by analyzing the ray path and considering propagated component, given by the cuboid simulation grid. The results are obtained by considering detection of ray components by surface detection of impinging rays, as a function of the direction of arrival of the rays launched.
In this way, depending on the relative locations of transceivers within the scenario, there is a deviation range in the detected components, exhibiting an average variation defined in (4).
where d cuboid is the cuboid edge dimension, given a uniform cubic lattice mesh, as depicted schematically in Figure 11. These variations are function of the cuboid location and the relative locations of Tx source and Rx observations point, as this will condition both the ray distribution (i.e., location of impinging ray/surface location) and the reference for subsequent distance estimation, which leads to larger variations the larger the cuboid edge size is in the mesh. The average variations for different cuboid sizes and different observation frequencies are depicted in Figure 12, for the set of simulation results obtained in Table 6. Looking at Figure 12, it can be seen that average variations increase as cuboid sizes increase, consistent with larger value of D as defined in (4) and hence, larger impact surface of the VOLUME 8, 2020   impinging rays within the cuboid under analysis. These variations are all below the corresponding average geometrical limits given by D as a function of the cuboid size and can subsequently error in TOF estimation can be reduced by adequate post-processing.

B. EXTRACTION OF TOF RESULTS
According to Molisch [9], the delay dispersion has a great impact on the performance of ranging devices. Therefore, in this paper, the TOF in Table 6 is extracted by detecting the first path of the signal (obtained from the PDP) for each of the locations along the straight path [see - Fig. 13]. In fact, Figure 13 shows 3 diffferent PDPs obtained by the 3D RL algorithm. This graph shows the amount of launched rays that reach 3 different locations within the scenario under analysis. The red dots correspond to a location close to the Tx antenna, the yellow triangles correspond to a point far from the Tx, and the blue squares to a midpoint distance. Each colored unit shows the information (RF power level and the arrival time) of each of the received rays at a specific location. As expected, the environment is very rich in terms of multipath propagation phenomenon, i.e., a lot of multipath  components are present in the whole volume of the scenario. In addition, 3 vertical lines have been depicted in the graph in order to show graphically how the ToF has been obtained from simulation results. The vertical lines show the time the first ray needed to reach the specific location/cuboid within the scenario. This time is the estimated TOF.
Following this idea, Figure 14 presents bi-dimensional planes showing the estimated TOF for 3 different heights. As expected, the results are similar since the distance differences are small due to the reduced size of the scenario.  Table 6.

C. TOF CONVERGENCE VERSUS THE CUBOID SIZE
Similar to the RF power 3D RL simulations, a convergence analysis was performed to find the optimal cuboid resolution needed to obtain accurate TOF estimations. In Figure 6, the MAE at different cuboid size intervals is shown. Similar to work in Figure 4, the TOF range becomes more accurate as the cuboid size intervals increase and converges at the interval 0.3 − 0.4, and then the error starts to increase afterwards. Therefore, it can be observed that 3D RL tool can be used to obtain decimetre level ranging accuracy as MAE of as low as 10 cm in Figure 15.

D. TOF MEASUREMENTS
Throughout this paper, we use two (one node configured as a Tx and the other as a Rx) TREK1000 development nodes manufactured by Decawave to measure the TOF of an UWB signal [54]. The nodes are fully compliant with the IEEE 802.15.4-2011 UWB standard, and are the best VOLUME 8, 2020 commercial products for TOF measurements [60]. For the purpose of this measurement campaign, nodes were configured to work with a 110 kb/s data rate and in the channel 2 (3990 MHz). The nodes operate in such a way that when a Tx sends a packet, the Rx hunts for the preamble or (once it has detected preamble) searches for start of frame delimiter (SFD) in the structure of the packet -which is the nominal point time-stamped by the IC. The 802.15.4 UWB standard requires that the time-stamp to be performed when the signal arrives at the antenna. Therefore, the Decawave Rx circuitry adds a correction factor determined by first path (leading edge) of the signal and also subtracts the Rx antenna delay to adjust the time stamp to the time at which the signal arrives at the antenna of the Rx.
A TOF measurement campaign was carried out in the Lab in Figure 1. The Tx was mounted on a mast at 1.6 m high at a fixed position. The Rx was also mounted on a tripod at the same height; however, it was moved from 2 to 20 m, with a 2-m step along a straight-path similar to dashed line in Figure 2. A laptop was connected to the Rx to store all measurements. At a rate of 3.57 Hz, TOF measurements were recorded over a period of 30 s, generating at-least 100 TOF estimations for each distance. The TOF results are multiplied by the speed of light to obtain the range and the results are in Figure 16.
The TOF ranging results are shown in Figure 16, in which the ideal ranges are compared with the experimental ones, as well as a histogram showing the number of occurrences for different range errors. As expected in Figure 16b, the range errors in LOS situations are modelled as a low-sigma Gaussian distribution. Looking at Figure 16, the mean error of up to 4 cm obtained during the measurements versus the 10 cm obtained during the simulations validates the 3D RL tool.

VI. CONCLUSIONS
We have presented a detailed analysis of the UWB channel for ranging applications using an inhouse developed 3D RL tool. RF power and TOF simulations have been performed in an indoor environment and validated using a set of measurements.
Using the sub-band divided RL technique, the UWB bandwidth of 500 MHz is divided into two sub-bands with 250 MHz each, therefore, RL simulations and measurements are made at 3.75 GHz, 4 GHz, and 4.25 GHz. A convergence analysis is performed which shows that adequate selection of the cuboid resolution is required to obtain accurate RF power and TOF results.
In addition, a detailed statistical UWB channel modelling is performed using the simulation results in terms of the pathloss, small scale fading, and PDP which are important when characterising an indoor radio channel. Therefore, similar to several narrowband systems, the 3D RL can be used as a valuable tool to test UWB systems for ranging applications with a mean accuracy of up to 10 cm in multipath conditions.