Decimeter Positioning in an Urban Environment Through a Scalable Optical-Wireless Network

3.1 White Rabbit

The White Rabbit (WR) network timing protocol is a unique fiber-optic time and frequency transfer method that was originally developed as a beam-timing system and deterministic control network for the Large Hadron Collider at CERN (Serrano et al., 2009); (Moreira et al., 2009). WR is an evolution of the IEEE 1588 Precision Time Protocol and ITU Synchronous Ethernet (SyncE) protocol and is included as the High Accuracy profile of the 2019 version of the IEEE 1588 standard. To the best of our knowledge, WR is currently the only technology which combines signals for sub-nanosecond time and frequency transfer and digital communication (Gigabit Ethernet) within a single wavelength channel. WR is commercially available as open-source Ethernet equipment, including switches and nodes. Users can build fiber-optic Ethernet networks in which each WR device replicates the time and frequency of a single reference time scale realization typically based on atomic clocks. To this end, the reference time scale realization feeds its time and frequency signals into a WR device set up in grandmaster mode and will act as the time source for the entire WR network, as shown in Figure 2. WR connections are inherently hierarchical; WR master devices transfer the reference time and frequency to downstream WR slave devices (Serrano et al., 2009). WR devices often act both as a slave and a master, and therefore have multiple optical ports. In this case, one port is used to enslave the WR device to an upstream WR master device, while other ports can be configured as master ports to which multiple downstream WR slave devices are connected. Complex network topology with thousands of nodes are possible and fail-safe redundant connections have been demonstrated (Lipinski, 2016).

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FIGURE 2

Simple example of a White-Rabbit (WR) network lay-out. WR-compatible Ethernet switches facilitate the development of complex network topology combined with regular Gigabit Ethernet networking, along with the distribution of time and frequency signals produced by a reference (atomic) clock. Both WR switches and low-cost WR nodes can be used to synchronize PNT transmitters. Yellow and green lines represent fiber-optic connections and interfaces; electrical connections and interfaces are shown in blue.

When estimating the one-way delay from a round-trip delay measurement, the path is assumed to be symmetric in both the upstream and downstream directions. In practice, large (nanosecond-level) delay asymmetries occur. These can be identified and corrected as follows. First, asymmetrical delays between the transmitting and receiving paths inside WR devices and optical transceivers are calibrated once, prior to their installation into the network (Daniluk, 2015), (Peek & Jansweijer, 2018). WR employs bidirectional communication using two different wavelengths propagating over the same (single) fiber for precise round-trip delay measurements. The physical length of the fiber-optic path is therefore inherently symmetrical. However, chromatic dispersion in the optical fiber leads to a delay asymmetry which, if left uncompensated, could lead to timing errors at the nanosecond level. Thus, the WR calibration procedure also includes a calibration of the fiber delay asymmetry at the time of installation (Daniluk, 2015). Once operational, each pair of WR devices (master and slave) automatically performs quasi-continuous measurements of the round-trip delay over the fiber. The one-way propagation delay (and its variations) is then determined and compensated electronically with a control bandwidth of 30 Hz. Thus, the one-way delay is tracked continuously and timing noise caused by random path length variations of the optical fiber is suppressed within the control bandwidth.

With all delays measured and corrected, the time, phase and frequency derived from a low-cost local oscillator in the downstream WR slave device can be locked to those of the upstream WR master device. To this end, the quasi-periodic symbol changes in the 1.25 Gb/s bitstream are used as a transfer oscillator. This facilitates a phase-lock between two WR devices and allows the local oscillator of the WR slave device to replicate the time, phase, and frequency of the reference time scale realization with very little additional degradation of stability or accuracy. At each WR device, time and frequency are available both digitally (in the form of IEEE 1588 time stamps) and electrically from coaxial 1 pulse per second (PPS)/10 MHz output ports. The signals provided through these output ports are fully phase-coherent with the signals from the WR grandmaster and the reference time scale.

WR has been shown to be compatible with optical amplifiers and the existing long-distance fiber-optic telecommunication infrastructure carrying live data traffic (Dierikx et al., 2016); (Boven et al., 2019); (Dierikx et al., 2019). A residual time offset between WR devices of 0.4 ns was demonstrated over a distance of 169 km (Boven et al., 2019). This performance, together with the compatibility of WR with the existing fiber-optic infrastructure (see also Section 5) and the high scalability of its networks, makes WR a promising candidate for the creation of a ubiquitous time and frequency reference network. Specifically, WR can feed wideband terrestrial transmitters with accurate time signals so that a positioning accuracy of one decimeter can be achieved for a mobile user. Moreover, if the reference time is taken from a time scale realization designated by a national metrology institute or national time laboratory, as demonstrated in this paper, the network could simultaneously disseminate an approximation of the coordinated universal time (UTC) and the SI second independent from GPS, potentially with higher accuracy, to mobile users.

The residual time offset and time stability (TDEV) typically achieved using default WR gear (similar to the equipment used in this paper) are approximately a few 0.1 ns and 10 ps, respectively, over 200 s or longer (as discussed in the Subsection to follow). This satisfies the needs of ranging and positioning by means of time delay (pseudorange) measurements, which is another focus of this paper. Given appropriate hardware modifications, these figures may be reduced to below 0.1 ns and 1 ps, respectively (H. Li et al., 2015); (Rizzi et al., 2016); (Rizzi et al., 2018). This places lower bounds on the achievable accuracy and reproducibility of carrier-phase ranging to 3 cm and 0.3 mm, respectively. Further improvements can be expected once purely optical time and frequency transfer methods are implemented, which may be capable of delivering time with femtosecond-level stability and accuracy (Bergeron et al., 2019).

3.2 TNPS-Prototype Synchronization Architecture

A prototype TNPS has been deployed at The Green Village on the TU Delft campus (Figure 3). This site is only a few kilometers away from the Dutch National Metrology Institute, VSL, home of the Dutch National Time Scale, UTC(VSL). With support from the collaborative organization for ICT in Dutch education and research known as SURF, an optical fiber connection was arranged for the White Rabbit synchronization network between VSL and TU Delft. This network is set up as a loop that returns from TU Delft back to VSL, and thus enables monitoring of the synchronization stability. At the TU Delft campus, the loop includes two data centers identified as DT010A and DT001B. A WR link was established from data center DT010A to The Green Village using the existing TU Delft optical fiber infrastructure. A geographical representation of the setup is shown in Figure 3 and a schematic diagram of the synchronization infrastructure of the prototype system is shown in Figure 4.

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FIGURE 3

Geographical representation of the synchronization network, with time and frequency reference signals from UTC(VSL). An aerial photograph that features the southeastern part of the town of Delft and the TU Delft campus was obtained through Beeldmateriaal Nederland under a CC BY 4.0 license (Beeldmateriaal Nederland, 2021).

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FIGURE 4

Schematic diagram of the WR optical synchronization infrastructure. The infrastructure starts at the WR Grandmaster (GM) at VSL and leads to the WR switch (Boundary Clock – BC1) at one of the data centers. It then arrives at the WR timing nodes (TNs) at the experiment site of The Green Village. The loop runs from WR Grandmaster (GM), through the two WR switches (Boundary Clocks BC1 and BC2) and back to WR Slave (SL) again at VSL. The optical wavelengths for each connection are as indicated (in nm).

Each link in the WR network consists of a single mode fiber. Two wavelengths are used for transfer of WR synchronization packets between the master and slave device with one wavelength in each direction. In our implementation, WR traffic is separated from other traffic in the network with coarse wavelength division multiplexing (CWDM) filters, shown by the trapezoid symbols in gray in Figure 4 that are applied in the links that connect the sites at different geographic locations. The internal delays of these CWDM filters were calibrated before installation. The wavelengths between the sites were chosen to be close together (typically separated by only 20 nm) to minimize delay asymmetries introduced by chromatic dispersion over longer distances.

At The Green Village, the WR timing nodes are connected in a a pattern known as a daisy chain. In this pattern, each node is synchronized to the previous one in a master-slave pair. This daisy chain arrangement, unlike the lay-out of the WR network shown in Figure 2, was chosen based on equipment that was available at the time of the experiment. In this part of the network, there is no other traffic in the fibers and thus no WDM filters were used. Here, although the wavelength separation is 180 nm, the effect of delay asymmetry remains small because of the short optical fiber lengths of only 50 m between the WR timing nodes.

At VSL, the WR Grandmaster (WR-GM) is directly referenced to the national timescale realization UTC(VSL). Because of cable and internal hardware delays, the time broadcast by the WR-GM is delayed with respect to UTC(VSL) by 97 ns with an uncertainty of 1 ns root mean square error (RMSE). In this section, we quantify uncertainty in terms of the RMSE, taking into account both random spread (precision) and systematic effects (biases), if any.

The WR slave (WR-SL) at VSL is delayed with respect to UTC(VSL) by 91 ns with an uncertainty of 1 ns (RMSE). This means that the offset introduced by the WR link “WR-GM(VSL) - WR-BC1(DT010A) - WR-BC2(DT001B) - WR-SL(VSL)” is about 6 ns. (Note that both WR-GM and WR-SL are WR-ZEN TP, and both WR-BC1 and WR-BC2 are WRS-3/18, all by Seven Solutions). Based on calibration of the WDM filters, these asymmetries introduced an offset of 3 ns. The estimated offset introduced by fiber delay asymmetry caused by chromatic dispersion was 1 ns. The remaining offset of 2 ns was most probably caused by internal delay asymmetries in the WR devices in this loop. While these were not calibrated prior to the installation of the equipment, a fixed unknown time offset from the reference time scale is not critical for the operation of the TNPS prototype.

The link between switch WR-BC1(DT010A) and timing node TN1 (at The Green Village), is approximately 1 km. With the wavelengths selected close together, the offset introduced by fiber delay asymmetry caused by chromatic dispersion was estimated to be less than 1 ns. Because the WDM filters and internal delays of the WR devices in this link were not calibrated, this link introduces an unknown offset.

For the purpose of positioning with the TNPS prototype, the most critical part is the synchronization of the WR timing nodes located at The Green Village with respect to one another (rather than to the reference timescale). These TNs, by OPNT, are the basis for synchronized transmission of the ranging signals. Therefore, in the VSL laboratory, the internal delays of these WR timing nodes were calibrated for each pair of interconnected nodes by relative calibration (Daniluk, 2015).

In this method, by first order approximation, it is assumed that the sum of internal transmitter (Tx) and receiver (Rx) delays in the master and slave devices are equal. From the round-trip time measurement, the delay of the short fiber used in the calibration is eliminated. This result divided by 2 provides an estimate of the sum of internal Tx and Rx delays in each device. Furthermore, also as a first approximation, it is assumed that the internal Tx delay is the same as the internal Rx delay. Thus, the initial calibration parameters for the Tx and Rx delay in the WR timing nodes are set to the measured round-trip time minus the delay of the fiber, divided by 4.

With these initial settings, the time difference between the 1 PPS output from the WR master and the 1 PPS output from the WR slave can be measured with a time interval counter. Based on the reading of the time interval counter, the Tx and Rx calibration parameters can be iteratively fine-tuned until the 1 PPS difference becomes less than 0.1 ns. This is typically achieved after two iterations. Ideally, for high-precision applications of TNPS, even better synchronization would be desired. However, with the current generation of WR equipment the stability, repeatability and 1 PPS pulse shape do not lead to better performance.

3.3 TNPS-Prototype Synchronization Results

After relative calibration of the WR timing nodes in the VSL laboratory, five of the timing nodes were mounted in weather-proof boxes together with the transmitters and installed on site in lamp-posts (see Section 4.2). The sixth timing node was incidentally set up with the mobile receiver to synchronize it for the purpose of validation of an additional over-the-air time and frequency transfer experiment.

During the experiments with the prototype system, an on-site verification of the synchronization of the WR timing nodes was performed. This verification included two steps: (1) measurement of the absolute time offset between WR-TN1 and UTC(VSL) and (2) measurement of the synchronization between WR-TN1 and the other timing nodes locally at The Green Village. Finally, the time and frequency stability of the WR optical synchronization infrastructure was analyzed.

3.3.3 Stability analysis

The stability of time and frequency of the WR optical synchronization network was evaluated for the loop “VSL - DT010A - DT001B - VSL” and for the daisy-chain of timing nodes at The Green Village.

Figure 5 shows the time and frequency stability expressed as time deviation (TDEV) and modified Allan deviation (MDEV), respectively (Riley, 2018), of the loop “VSL - DT010A - DT001B - VSL” measured over approximately 100 days. The TDEV is typically below 10 ps. For averaging times below 200 s, TDEV is higher and is dominated by the noise from the time interval counter.

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FIGURE 5

Time stability (TDEV - left) and frequency stability (MDEV - right) of the loop “VSL - DT010A - DT001B - VSL” with 1-sigma error bars. The data presented in these graphs were taken from Koelemeij et al. (2022).

The noise introduced by the daisy-chain of timing nodes is shown in Figure 6 and includes TDEV of the link between WR-TN1 and WR-TN2 and of the entire chain from WR-TN1 to WR-TN6. The noise contribution from the daisy-chain of timing nodes is approximately a factor of 2 with respect to a single WR link between two timing nodes.

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FIGURE 6

Time stability (TDEV) of the link “WR-TN1 - WR-TN2” (in orange) and the link “WR-TN1 - WR-TN2 - WR-TN3 - WR-TN4 - WR-TN5 - WR-TN6” (in blue) with 1-sigma error bars. The data presented in this graph were taken from Koelemeij et al. (2022).

3.4 Discussion and Outlook

The transmitters in the TNPS prototype deployed at The Green Village were synchronized with a daisy-chain of WR timing nodes. The first timing node in this chain was connected to a WR network that was synchronized with the Dutch National Timescale UTC(VSL) within approximately 100 ns, also verified here. The accuracy of the synchronization with the reference timescale might be further improved by calibration of all components in the WR network between VSL and The Green Village. In this case, synchronization within 0.5 ns can be achieved. However, for the purpose of positioning with this prototype system, the accuracy of the synchronization with the reference timescale is not critical. The synchronization of the 1 PPS outputs of the timing nodes in the daisy-chain, calibrated in the VSL Time Laboratory to a level within 0.1 ns and verified on site at The Green Village, was found to be within 0.5 ns.

In order to set-up a large network, according to the lay-out shown in Figure 2, one would deploy a cascade with WR timing nodes connected to the final level of WR switches. Cascades of up to five standard 18-port WR switches (Rizzi et al., 2018) would suffice for covering an area as large as the Netherlands, with WR timing nodes geographically separated by 500 m.

4.1 Radio Channel and Ranging Signal Bandwidth

Typically, radio ranging is performed based on an estimation of the time duration between transmission of the ranging signal by the transmitter and arrival of this signal at the receiver. Estimation of this time delay is particularly challenging in a practical dense multipath radio channel, i.e., those with many reflections, because the receiver may mistakenly use a reflected signal for measurement instead of the desired direct-path or line-of-sight (LoS) signal. In addition, the receiver may not be able to distinguish the LoS signal from close-in reflections with a slightly longer delay. This will result in an offset (bias) in the estimated time delay of the signal.

In the following sections, we consider a baseband signal model with a ranging signal s(t). The transmitted bandpass signal at frequency f_c is then given by x(t) = Re {s(t)e^j2πf_ct}, where Re{·} indicates the real part of a complex variable. For an L-path multipath channel, the channel impulse response (CIR) can be modeled as shown in Equation (1):

1

where α_l and τ_l are, respectively, the real-valued path gain and time delay of the l^th path between transmitter and receiver. The received baseband signal is then given by Equation (2):

2

where * indicates convolution, a random variable is denoted by an underscore, e.g. n, θ₀ is the phase offset between transmitter and receiver at t = 0, f_Δ is the frequency difference between the receiver front end and the received signal, caused by a frequency offset between transmitter and receiver and/or by a Doppler shift (both assumed constant here for convenience of the discussion), and n(t) is zero-mean Gaussian distributed noise with variance . For ranging, the variable of interest is the time delay τ₁ of the LoS path, with its estimate indicated as . (In this paper, we do not consider non-line-of-sight [NLoS] scenarios.) In Section 4.3.1, the time delay will be estimated via a model of the CIR. This response can be obtained as the transmitted ranging signal s(t) is a priori known to the receiver. The model will be formulated in the frequency domain in terms of the channel frequency response.

Without considering any reflections in the time delay estimation (i.e. assuming a single path channel), the ranging precision can be bounded by the (in practice rather optimistic) Cramer-Rao lower bound (CRLB), which defines a lower bound on the variance of the unbiased estimation error . The CRLB for a single path LoS channel is given by Kay (1993) as shown in Equation (3):

3

and is determined by the signal-to-noise ratio (SNR) and the mean square or Gabor bandwidth F² of the signal, defined as shown in Equation (4):

4

where S(f) is the Fourier transform of s(t). Equation (4), shows that the Gabor bandwidth is related to the ranging signal bandwidth. From the time delay estimate , the estimate of the distance ρ between transmitter and receiver is found as , where c is the speed of light, and the variance of this range estimate thus becomes .

To reduce the range estimation error variance, a signal with a larger Gabor bandwidth can be used and/or the SNR can be increased. As a rule of thumb, doubling the RMSE bandwidth F will reduce the range estimator standard deviation by a factor of 2. Figure 7 shows the difference between the ideal CIR, as measured with an infinite bandwidth signal, and the CIR when measured with a signal of limited bandwidth (as shown by the dashed line in brown).

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FIGURE 7

CIR measured with an infinite bandwidth signal (δ-pulses in red and blue) and with a signal of limited bandwidth (dashed line in brown). The red δ -pulse represents the LoS path (i = 1), α_i and τ_i represent the amplitude and delay of the i-th multipath component, respectively.

The time resolution determines the minimum time difference in which the multipath components can still be separated. Multipath components that arrive with delays that are less than the time resolution from the LoS path will likely introduce a bias in the estimated time delay for the LoS path. This time resolution, in case of matched filtering, is determined by the width of the correlation peak, which is inversely proportional to the signal bandwidth. Therefore, the precision of time delay estimation and the separability of the multipath reflections are inversely proportional to the signal bandwidth. This makes the use of wideband ranging signals essential in achieving a high ranging and positioning precision and accuracy.

With respect to the application of autonomous driving as shown in Figure 1, the automotive radio channel can be characterized as a time variant multipath channel due to reflective vehicles moving in a reflective environment that includes buildings and infrastructure and vehicles moving with respect to one another. This results in a rapidly changing CIR in which the LoS path may be blocked intermittently. In addition, the different multipath components will exhibit various Doppler shifts. When a short duration ranging signal is used, the channel can be assumed quasi-static (i.e., during the measurements for a position fix). For a vehicle moving with a speed of 150 km/h, the displacement during a ranging signal burst of 160 μs, as used here, is only 6.7 mm, as shown in Figure 9. Given this displacement, the relative phase differences between multipath components and delays will change slightly in relation to the direction of arrival and direction of movement; however, the path amplitudes will remain relatively constant.

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FIGURE 9

Ranging signal transmission in time division multiplex (TDM) mode. A single Tx - Rx configuration in which the time-frequency reference is provided to the timing node through the optical telecommunication network (in yellow). The timing node delivers 1 PPS and 10 MHz electrical signals (red lines) to the X310 based radio transmitters. The transmitters stream the ranging signals periodically every T = 1 ms in a low duty cycle scheme. The ranging burst contains the TDM ranging signals of multiple transmitters. The receiver samples the incoming ranging signals and forwards the data to the host-PC for offline processing and time delay estimation.

4.2 TNPS-Prototype Architecture

The setup of the TNPS prototype includes six transmitters (Tx-1 to Tx-6) and one receiver (Rx), which are all based on an Ettus X310 universal software radio peripheral (USRP). Each USRP has two radio frequency (RF) channels that can be configured either as transmitter or receiver channels. A transmitter consists of a USRP connected to a timing node for time-frequency synchronization. One of the USRPs operated as two independent transmitters (Tx-2 and −3). The receiver unit consists of a USRP which can operate in asynchronous mode; it can also be connected to a timing node for synchronization to the same time-frequency reference as the transmitters. In this way, the performance of over-the-air and hard-wired synchronization with a timing node can be compared. The transmitters periodically stream the ranging signals; the receiver samples the received signals, stores the samples, and forwards them to a host-personal computer (host-PC) for offline processing.

After development and initial testing in the lab, the prototype system was deployed at The Green Village. The Green Village, located on the TU Delft campus, is an outdoor living laboratory for sustainable innovation. The main goal of this facility is to accelerate development and implementation of innovations by bringing together engineers, businesses, and scientists at a place where new technologies can be developed, tested, and demonstrated (The Green Village, 2021). This site offers a realistic environment for testing the TNPS in an area with buildings and surrounding infrastructure (see Figure 8). The site offers access to the TU Delft ICT facilities with (dark) single mode fibers on site (see Section 3.2). Tests have been carried out on a 40 m stretch of a 6 m wide local paved road on which the mobile receiver can move freely, as shown in Figure 8. The transmitters are located along this road, with antennas mounted primarily on lamp-posts at a height of approximately 4 m. The lay-out of the prototype-deployment was primarily opportunity-driven (e.g., using existing lamp posts to mount transmitter antennas). Tests have been carried out with the receiver attached to a trolley and also to a car to facilitate kinematic experiments at low speed only (walking pace).

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FIGURE 8

Prototype setup at The Green Village. Two of the transmitter antennas are visible in the lamp posts to the left (in red circles). The receiver antenna Rx (in yellow circle) is fixed to the trolley. The land-surveying stations at the right are used to provide the ground-truth trajectory of the receiver.

4.2.2 X310 USRP main features and configuration

The local sample clock of the USRP of 200 MHz was syntonized to the 10 MHz frequency reference. Signals were sampled in quadrature with a sample format of 16 bits-I (in-phase) and 16 bits-Q (quadrature-phase). The maximum effective radiofrequency (RF) bandwidth was limited to 160 MHz. Each X310 USRP has two RF channels with a central frequency that can be tuned from 10 MHz to 6 GHz. In the TNPS prototype, the USRPs are operated at a carrier frequency of 3.96 GHz based on an experimental license obtained from the Dutch Telecom Agency. The output power of a transmitter is P_tx = +14.5 dBm during burst transmission. The receiver gain was set to 20 dB since higher gain values impaired the received signals. At the transmitters and the receiver, identical omni-directional, ceiling mountable, wideband antennas from Taoglas are used with a gain of 3 dBi. The transmitters and receiver were connected to the antennas by cables of 5 m and 1 m, with measured cable losses of 5.6 dB and 0.9 dB, respectively. The received LoS signal power at a distance d (in meter) at the receiver-USRP input can be expressed as Equation (5):

5

The noise level within the 160 MHz measurement bandwidth and referenced to the receiver-USRP input was measured at −80.5 dBm. The SNR, defined here as the ratio of the received signal power and received noise power before correlation, varied from = −1 dB to 34 dB over the 40 m track because of variations in both distance and multipath fading. With the receiver located at the Reference Point (Ref. Pnt.) shown in Figure 16, the SNR values of the measured bursts from Tx-1 to Tx-4 and Tx-6 (with Tx-5 deactivated) as shown in Figure 10 were 15.2, 21.7, 22.4, 17.3, and 28.7 dB, respectively. Using Equation (5), the coverage distance is d = 320 m for SNR = 0 dB based on the received LoS path power. This would allow the geometry of The Green Village set-up as described earlier to be scaled by a factor of 10 to fit a highway scenario.

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FIGURE 16

The geometry of the six transmitters Tx-1 to Tx-6 with a 3.5 minute trolley run with a positioning update frequency of 1000 Hz. The reference point (Ref. Pnt.) indicates the measurement location of the results in Figure 10, and position solution both with (right) and without correction (left).

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FIGURE 10

Example of a received TDM ranging signal burst from five active transmitters. Tx-5 is deactivated) and the receiver is located at the Reference Point in Figure 16. The SNR values for the signals received from Tx-1 to Tx-4 and Tx-6, were 15.2, 21.7, 22.4, 17.3, and 28.7 dB, respectively.

4.2.3 Ranging signal structure

For ranging, a bursts transmission scheme was used, in which Tx-1 to Tx-6 periodically transmitted the ranging signal every 1 ms employing a time division multiplexing (TDM) scheme, in which the respective signals were synchronized to one another (Figure 9). The duration of a TDM ranging burst can be as long as 160 μs which will result in a transmission duty cycle of 0.16. For six transmitters, the ranging burst is divided into six time slots of 24.6 μs each separated by a guard time of 2.5 μs between successive slots. In the prototype, quadrature phase shift keying (QPSK) modulated PRN sequences and orthogonal frequency division multiplex (OFDM) signals are used as ranging signals. Figure 10 illustrates the received signal bursts from the transmitters using a PRN-sequence with Tx-5 deactivated. The embedding of the ranging signal burst in an OFDM signal is shown in Figure 11. The receiver can be connected to a timing node to synchronize the receiving window with the transmitting window. Alternatively, when not connected to a timing node, the receiver can operate in an asynchronous stand-alone mode in which the ranging bursts are detected in real-time by using a matched-filter technique in the USRP field-programmable gate array (FPGA). If the receiver clock is stable up to 0.1 ppm (10⁻⁷ s/s); then in a 160 μs time span to allow for the arrival of signals from all Txs, the Rx-clock offset may change by 16 ps, and hence generate a 4.8 mm offset.

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FIGURE 11

A burst-like OFDM signal packet is transmitted by each transmitter Tx-1 to Tx-6, for ranging in a TDM scheme with a period of T = 1 ms. Each signal packet contains three training symbols. The first two symbols are for synchronization and channel estimation and the third is for transmitter identification.

4.3 Signal Processing and Positioning

4.3.2 Time delay estimation

Based on the sampled channel frequency response H and assuming a circular-symmetric complex normal distribution of the noise, one can jointly estimate the time delay for both the LoS path and the reflections using the maximum likelihood (ML) principle (Dun et al., 2021a); (Progri et al., 2005). The cost function is calculated as shown in Equation (8):

8

with projectors in Equation (9):

9

and (·)^H denoting a Hermitian operator.

Rather than estimating L paths, one can simply consider a single path in the time delay estimation model. The cost function, with Equation (7), then turns into Equation (10):

10

This reduces the computational complexity compared to using the cost function described in Equation (8). This can lead to a better ranging precision (Dun et al., 2021a). Because only a single path is considered in the model for time delay estimation shown in Equation (10), the resulting estimator will be biased in a multipath channel. Using a wideband ranging signal, however, increases the time delay resolution and the bias will be limited. Figure 12 shows the envelope of the multipath error (i.e., the time delay bias using Equation (10) in a two-path channel, hence LoS and a single reflection) as a function of the relative distance c(τ₂ – τ₁), with the assumption of infinite signal bandwidth, i.e., no filtering applied, and the relative gain (α₂ /α₁) set to 0.6, similar to analyses described in Braasch (2017). Three types of ranging signals are considered in Figure 12, including two adjacent OFDM signals of 64 sub-carriers (each spanning a band of 10 MHz), in total spanning a 20 MHz band, 16 adjacent OFDM signals spanning a 160 MHz band, and the GPS L5 signal with a chip rate of 10.23 Mcps using a 0.5 chip-spaced correlator. Figure 12 demonstrates that a large signal bandwidth for ranging improves the separation of multipath reflections and effectively mitigates the multipath error. Using 160 MHz of bandwidth for ranging, the resulting maximum multipath error is about 0.5 m. This is much smaller than the multipath error of approximately 4.0 m that one would obtain using a bandwidth of 20 MHz.

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FIGURE 12

Multipath error envelopes for three unfiltered, infinite-bandwidth signals. Shown are the signals from two adjacent OFDM bands (20 MHz), a signal from 16 adjacent OFDM bands (160 MHz), and GPS L5 signal (10.23 Mcps) in a single reflected path scenario with the relative path gain α₂ / α₁ set at 0.6.

4.4 Ground-Truth Trajectory

The TNPS-receiver (Rx) and its antenna were attached to a simple trolley as shown in Figure 13 to facilitate both static and slow kinematic measurements. To determine the ground-truth position of the receiver in both modes, two 360-degrees prisms were also attached to the trolley; each was tracked by a robotized land-surveying total station. The total station measures vertical and horizontal angles and slant distance. In this short range set-up (maximum 40 m), the distance measurement determines the resulting positioning precision of the prism, which is typically at the several millimeter level. During the experiments, the receiver on the trolley moved on a flat level road. Therefore, horizontal two-dimensional ground-truth positions are determined. The heights of the receiver antenna and the two prisms were determined a priori, and they are assumed to remain constant during the experiment.

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FIGURE 13

Set up of the receiver (Rx) antenna and determination of its ground-truth trajectory. The receiver antenna and two 360-degrees prisms were attached to a trolley and the two prisms were tracked separately by two total stations.

The total station provides measurements with an update rate between 1–10 Hz, depending on the speed of the receiver. The TNPS provides receiver position solutions with an update rate of 1 kHz, which is much higher than the measurements obtained from the total stations. Hence, to construct the ground-truth of the receiver trajectory, the positions determined by the total stations need to be interpolated. A piece-wise linear least-squares interpolation was done for both coordinates with a moving window of five positions. In this case, the receiver is assumed to be moving along a straight line with constant velocity in this time window. Eventually, the positions of the prisms are interpolated with an update rate of 1 kHz based on the estimated linear model. The two total stations are not synchronized. The time lag between the two can be retrieved by shifting one of the time series trajectories until the motion patterns of the two trajectories are optimally matched.

The coordinates determined by the total stations are expressed in a local reference system with the origin at “total station (1)” shown in Figure 13. These are then rotated into a local east-north-up (ENU) reference system, after determining the azimuth with two GPS receivers on the baseline formed by the two total stations. Next, with the distances between the prisms and the Rx antenna on the trolley that were determined a priori, the ground-truth trajectory of the Rx antenna can be reconstructed. Finally, to compute the position error in the experiment, the trajectory determined by the TNPS and the ground-truth should also be optimally matched, as the receiver is not synchronized with the total stations.

To evaluate the quality of the reconstructed ground-truth trajectories, one can retrieve the horizontal distance between the two prisms based on their aligned trajectories and compare it to the (fixed) horizontal distance between the two prisms, as determined before the experiment. This is shown in Figure 14 for one of the runs. The RMSE of the resulting difference is 1.8 cm. Hence, one can assume that the error in the reconstructed ground-truth receiver position is at the cm-level and that most of the uncertainty is likely introduced by the interpolation. This is because the motion of the trolley was neither on a straight line nor at a constant velocity all the time. Additional error may be introduced by small changes in height, because the road was not perfectly flat and level.

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FIGURE 14

Difference in distance between two prisms based on reconstructed trajectories and a priori measured distance. This is presented as a function of measurement time (at the left) and in a histogram (at the right). The position update rate was 1000 Hz.

The total stations have also been used to determine the positions of all transmitter antennas. They are in the same local ENU system as was used for the receiver ground-truth trajectory. The phase center offset of the antennas, which was 4.5 cm in height with respect to the antenna ground plane as determined a priori in the lab, has been taken into account.

4.5 TNPS-Prototype Results

In this section, we provide results obtained with the TNPS prototype deployed at The Green Village in mid-September 2020. Several experiments were carried out in both static and kinematic modes, the latter with a trolley and a car.

The position results are by default based on time delay measurements as outlined in Section 4.3. Time delay estimation was done on a per packet basis and the position-time solutions were obtained correspondingly at a rate of 1000 Hz using the model shown in Equation (14). These position-time solutions are known as snapshot solutions and are obtained instantaneously, hence no time averaging or Kalman filtering have been applied. Obtained position solutions are compared to the ground truth trajectory discussed in Section 4.4. The resulting position errors are presented.

We have restricted positioning with the prototype-system to two dimensions. The receiver antenna is placed at a 1.5 to 2 m height and the transmitter antennas are all at a 3.5 to 5.5 m height. Thus, there is minimal diversity in height and the elevation angles are all fairly small. The receiver antennas on the trolley and on the car were at 1.573 m at 2.006 m in height, respectively. These heights are assumed constant and have been substituted in Equation (13) to allow for two-dimensional positioning using the prototype system.

Results of a kinematic run with the trolley are presented when the receiver was also connected to an active WR timing node. Time delay estimation and positioning were performed without this information, hence using the model in Equation (14) and estimating a receiver clock offset every epoch. Finally, we also present results of a kinematic run with a car with the receiver running in a fully asynchronous mode.

Based on the model shown in Equation (14) for estimating a two-dimensional position and a receiver clock offset, and assuming the variance matrix shown in Equation (15) to be an identity matrix (based on a standard deviation of σ_ρ = 3 cm), the precision of the two-dimensional position is shown as in Figure 15 for the area of the experiment. This quantity is known as the horizontal dilution of precision (HDOP) (Teunissen & Montenbruck, 2017), although we included here the measurement variance rather than presenting just the geometric factor. Clearly, the positioning geometry is good in the area enclosed by the transmitters. Also the track of the receiver in the kinematic experiment is within this area.

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FIGURE 15

Precision of horizontal positioning. This is shown by means of the horizontal dilution of precision (HDOP) scaled with the unit measurement standard deviation (σ_ρ = 3 cm), hence in m. The values at the edges of the graph have intentionally been clipped and set to 0.1 m,in order to include the details shown in the coverage area.

Figure 16 documents the horizontal positioning results of one of the runs with the trolley. In this case, the trolley moved forward on the local road, made a U-turn, went back, and eventually terminated at a point close to where it started. In total, the trolley moved by about 30 m. It occasionally made some stops on the way, and the run lasted 3.5 minutes. The TNPS-position solutions are shown in blue based on time delay measurements. The corresponding ground-truth is shown as a red dashed line. The graph at left shows the results directly from the measurements. For the graph at right, compensation was introduced to adjust for systematic (constant) biases in the measured pseudoranges of transmitters Tx-5 and Tx-6. After compensation the track is spot-on, and deviations are only at the 1–2 decimeter level.

Figure 17 presents the estimated receiver clock offset δ_r(t) as indicated in Equation (14). In this trolley run, the receiver was connected to an active WR timing node. Ideally, this should result in a zero clock offset. The estimated clock offset appears constant and the systematic offset of about 17 μs is caused by transmitter and receiver front end delays τ_h. The empirical standard deviation is only 4 cm, i.e., 0.1 ns in terms of time delay. These results show the high-precision wireless time-transfer potential of this system.

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FIGURE 17

Estimated receiver clock offset with wireless time transfer where the receiver is operated in synchronous mode, but was processed as if in asynchronous mode. Shown is cδ_r(t), Equation (11), with the offset cτ_h included. The empirical mean, for the most part caused by hardware delays τ_h, is 5143.41 m and the empirical standard deviation is 0.04 m.

Figure 18 shows the horizontal position error as a function of time, at left for a run with the trolley (the same run as with Figure 16), and at right for a run with a car, in which the receiver was not connected to a WR timing node, and hence running in a true asynchronous mode. We note that this latter run lasted 71 seconds. Table 2 presents the statistics of both runs in terms of mean position error, empirical standard deviation, and RMSE in meters, expressed in local east and north, for a run with the trolley and the car. The RMS position error comes close to a one decimeter in both cases.

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FIGURE 18

Positioning error of the receiver (Rx) trajectory for the trolley (synchronous, at left) and the car (asynchronous, at right). In both cases the asynchronous positioning model in Equation (14) was used with position coordinates and clock offset.

Building on the suggestion given in Section 4.3 the data of the run shown in Figure 16, with empirical statistical values given in Table 2, have been reprocessed. Only a few bands were used and effectively a frequency-sparse signal was employed (Dun et al., 2021a), rather than the full 160 MHz band. Muting 9 of 16 bands and using only the 7 that remained of 10 MHz each, including the bands at the edges (here, bands 1, 2, 5, 6, 12, 13 and 16) resulted in very similar time-delay based positioning results when the main-lobe of the cost function was correctly identified. We observed no noticeable degradation of performance nor any issues with bias or spread.

Figure 19 documents an example of a CIR obtained in the experiment, specifically of the signal received from transmitter Tx-2. After packet synchronization of the received signal and fast Fourier transformation (FFT), the channel frequency response as indicated in Equation (6) was obtained using a known training symbol. For the blue line, labeled 1D MLE, we consider only a single path, L =1, and show the cost function from Equation (10), as a function of time delay τ (expressed in meters). This is equivalent to using a single arm cross-correlator receiver. The red stems show the channel response using the joint estimation of time delay and gain with a model as shown in Equation (6) (Dun et al., 2020); (Berger et al., 2010). An entire range (dictionary) of delays was put forward with a resolution of 0.1 ns, although at the same time a penalty is introduced in the form of a minimum norm on vector x as the solution to Equation (6). This channel response estimate is labeled l₁-norm, and the solution is forced to be sparse. It is clear that multiple signal reflections were received, including several that were comparatively strong and close to the LoS path. This was likely caused by the objects directly adjacent to the road, including brick-wall houses, as shown in Figure 8. According to Figure 12, a fairly strong reflection can induce a multipath error in time delay range of only up to 0.5 m when using the full 160 MHz of bandwidth. This was confirmed empirically by the positioning errors that incidentally reached 0.3–0.4 m at most as shown in Figure 18, based on six transmitters, and noting an HDOP of approximately 1 along the trajectory covered in Figure 15.

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FIGURE 19

Estimated channel impulse response by using 1D maximum likelihood estimator (MLE) as per Equation (10) and sparsity-promoting l₁-norm with a redundant basis generated by delays at a resolution of 0.1 ns.

As a final experimental verification of the effectiveness of a large bandwidth for ranging to mitigate the impact of multipath, the data of the run performed with the trolley in Figure 18 at left were reprocessed using only 20 MHz of bandwidth (2 adjacent OFDM bands of 10 MHz each) rather than the full 160 MHz, as shown in Figure 20. The range of the vertical axis is now larger by a factor of 10 (compared to Figure 18). As shown by the blue curve in Figure 12, multipath errors can then be as high as several meters. With 20 MHz of bandwidth, the RMSE of the position error in east and north are 82.06 cm and 68.21 cm, respectively, and can be compared with values shown in Table 2.

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FIGURE 20

Positioning error of the receiver (Rx) trajectory for the trolley, based on time delay measurements, using only 20 MHz of bandwidth instead of the full 160 MHz. These findings can be compared to those in Figure 18 (at left). The position error RMSE in east and north are 82.06 and 68.21 cm, respectively.

Figure 21 also demonstrates the potential of the system to deliver high-precision positioning results by means of measurements of the carrier phase of the burst signal. The carrier phase is determined by phase rotation of the estimated (complex) gain of the first path, shown as x₁ in Equation (6). Carrier phase measurements are inherently ambiguous, and, because of initial phase offsets in both transmitter and receiver, the ambiguity does not initially emerge as an integer. The results shown in Figure 21 pertain to float solutions, based on unwrapped carrier phase measurements, with the remaining initial carrier phase cycle ambiguities estimated as constant real-values. As we present here a carrier-phase-only solution, at least two epochs of measurements are needed accompanied by a change of geometry. The first epoch is fixed and the second epoch varies across the duration of the 71 second run. Only two epochs of measurements are used for each solution shown. The solution is shown as a function of the second epoch, and only starts after five seconds to provide time for the necessary change in geometry. In this run, the receiver was installed on a car and operated in asynchronous mode. The carrier phase measurement model and additional results are presented in Dun et al. (2021b).

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FIGURE 21

Positioning error for two-epoch, carrier-phase-only float position solution estimation with empirical means of 2.5 cm (east) and 2.9 cm (north), empirical standard deviations of 3.4 cm (east) 3.6 cm (north), and empirical RMSEs of 4.2 cm (east), 4.6 cm (north).

The results shown in Figure 16 (at left) indicate a systematic bias in the estimated positions. Investigations in the laboratory revealed that the transmission times of the RF signals, particularly those from Tx-5 and Tx-6, show a delay of about 1.5 ns with a standard deviation of 180 ps compared to the other transmitters. Additional measurements revealed that the 1 PPS of the WR timing nodes were well aligned with a maximum deviation of 100 ps, but that the 10 MHz signals of the timing nodes were not well aligned to the 1 PPS signals and had different offsets of up to 600 ps (although repeatable with a standard deviation < 40 ps). These phase offsets of the 10 MHz should have been calibrated. By contrast, all USRP were connected to the 1 PPS and 10 MHz signals of a National Instruments CDA-2990 clock distributor, with deviations of 60 ps and 20 ps, respectively. The offsets in the measured RF transmission delays were on the order of 100 ps with a standard deviation of 25 ps. Based on these observations, the 10 MHz offsets are suspected to be the cause of the systematic offsets of about 1.5 ns in the measured ranges, most notably present in the pseudoranges of Tx-5 and Tx-6. These offsets are shown in Figure 22 (at left). The RF transmission delays have been determined a posteriori and were used to correct the timing offsets. The corresponding positioning results are shown in Figure 16 (at right).

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FIGURE 22

Difference between the measured time delay and the ground-truth propagation time delay without correction (left) and with a posteriori correction (right) of the transmitter timing offsets. In this run, the receiver was static during the first 45 seconds.

Figure 22 shows the residual range error for each transmitter without (at left) and with a-posteriori correction (at right). The receiver was synchronized, so the clock offset δ_r(t) as shown in Equation (11) should be a constant. Hardware delays τ_h are considered together with the receiver clock offset provided that they originate from the receiver, or, if they originate from the transmitter, they are identical across all transmitters. With given transmitter positions and using the known ground-truth position of the receiver, the distance d calculated in Equation (11) is known. As shown in Figure 22 in blue, red, yellow and purple, the residual pseudorange error for Tx-1 to Tx-4, which are fairly constant and at the same level, represent the clock offset. The residual error of the pseudorange from Tx-5 and Tx-6 (in green and light-blue) compared with Tx-1 to Tx-4, show a clear offset of about 1.5 and 1.9 ns, respectively. After correction, all residual errors are at the same constant level.