Quasi-Coherent Phase-Based Localization and Tracking of Incoherently Transmitting Radio Beacons

The direct measurement of distance-dependent information between wireless units represents a challenge for wireless locating systems, because it requires the exact time synchronization of separate wireless units. To avoid these synchronization efforts, many wireless locating systems only evaluate phase difference of arrival (PDOA) measurements. While simple PDOA localization techniques rely on multiangulation, advanced PDOA concepts like the holographic extended Kalman filter (HEKF) directly evaluate the measured phases without non-linear preprocessing. However, these differential phase measurement approaches are less sensitive than systems that can measure absolute phase variations, which allow the tracking of much smaller position changes than the signal’s carrier wavelength. This paper proposes to extend the HEKF by the evaluation of absolute phases in an incoherent measurement setup, which consists of a continuous wave (CW) beacon and several receivers. The developed quasi-coherent holographic extended Kalman filter (QCHEKF) uses the overdetermined PDOA measurements to estimate the phase–frequency relation between each beacon–receiver pair. Then, the established phase–frequency relations allow the evaluation of absolute phase measurements and, thus, the accurate localization and tracking of a simple, unsynchronized, narrowband CW beacon, even under severe multipath conditions. This novel concept is experimentally validated via 3D localization results in a challenging indoor scenario using a 24 GHz CW measurement setup. Here, the QCHEKF improves the achieved localization accuracy in comparison to the HEKF by 35 % from 0.78 cm to 0.51 cm, while the maximum deviation from the trajectory reduces by 68 % from 5 cm to 1.6 cm. Furthermore, the QCHEKF enables the exact tracking of fast changes in direction, which is usually a significant challenge for standard wireless target tracking systems.


FIGURE 1.
Measurement setup with two receiver arrays and one beacon, which moves the distance d and emits spherical waves. distance information by evaluating the time-dependent phase relation for different frequencies within a limited bandwidth [1]. Unfortunately, available bandwidth is often limited due to governmental restrictions and implementation issues [11]. Hence, a tremendous effort is necessary to enable highly accurate ultra wide band (UWB) positioning [12]- [16], thereby providing localization results with errors of less than 1 cm.
Instead of evaluating the phases for different frequencies, phase difference of arrival (PDOA) systems evaluate the phase relation between spatially distributed antennas [11], as illustrated in Fig. 1. The most popular processing method is to estimate the angle of arrival (AOA) [17] for each beacon-receiver (RX) pair. Then, the AOA information at different RXs is combined to localize the beacon [18]- [20]. Since the phase-based AOA estimation results in undesired noise shaping, the accuracy of any successive signal processing is degraded [21] and, hence, a direct evaluation of phases is desirable. Therefore, holographic localization was proposed, which matches the received phases with hypothetical beacon positions using computational expensive brute force searches, as in [22]. To reduce the computational effort and further increase localization accuracy, the holographic extended Kalman filter (HEKF) was proposed in [23], which recursively estimates a beacon's position by evaluating the phase differences, measured between every antenna pair of each RX array.
PDOA systems inherently assume incoherent measurement principles, allowing localization with arbitrarily modulated signals based on relative phases [23]. However, evaluating absolute phases, which are related to TOA measurements between wireless units, provides a much higher sensitivity to position changes. Therefore, radio frequency identification (RFID) systems are often implemented with coherent beacons, as in [22]. Here, implementing coherent beacons depicts a challenging task, which can be solved by switched injection locking [24]. The relative phase information between measurements can then be used to improve localization accuracy, as in [25]. Most often, the timedependent phase relation is used to estimate velocities based on the Doppler frequency [26]. The Doppler information can then be well fused with AOA estimations [27], [28]. However, the Doppler frequency estimation implicitly assumes a constant velocity for the evaluated time frame, which is not sufficiently valid for an accelerating beacon in a highly accurate indoor positioning system. In this work, the phases will be evaluated directly.
To combine the advantages of incoherent radio frequency beacons, which are easy to implement, and the high sensitivity of coherent measurements, we propose a novel concept to localize an incoherent continuous wave (CW) beacon. In Fig. 1, the HEKF's phase difference evaluation between the different antennas of RX 1 enables absolute positioning of the CW beacon in the vertical direction, while RX 2 provides absolute position information in the horizontal direction. However, since the time-dependent phase relation of CW beacons is deterministic, except for phase noise and frequency instability [29], [30], beacon movements toward a receiving antenna can be effectively detected by its absolute phase measurements. In Fig. 1, the small position change d of the beacon in the horizontal direction can be accurately detected via the absolute phase measurements at RX 1, while the absolute phase measurements at RX 2 remain nearly constant.
The HEKF presented in [23] consists of a constant velocity system state model and the evaluation of phase differences in the state update. The QCHEKF extends the HEKF by the evaluation of absolute phase measurements to take advantage of their high measurement sensitivity. To involve the time-dependency of the absolute phase measurements in the incoherent CW measurement setup, the QCHEKF extends the HEKF's constant velocity model via a constant frequency model for each beacon-receiver pair. To estimate this phase-frequency relation between each beacon-RX pair, the QCHEKF inherently uses the absolute position information gathered from the phase differences. Afterwards, the constant frequency model provides absolute phase measurement estimations, which are used via the quasi-coherent phase evaluation to improve the localization in comparison to the pure phase difference-based HEKF. Due to the direct evaluation of phases, the QCHEKF completely omits hindering preprocessing steps, such as the AOA or the Doppler frequency estimation. This novel concept is validated by 3D indoor localization results using a 24 GHz measurement setup with strong multipath propagation. For the presented trajectory, the QCHEKF localization result yields a 2D root mean squared error of 0.51 cm and a 2D maximum error of 1.6 cm and, hence, is comparable to the best UWB localization results, despite the usage of a narrow band measurement setup. The QCHEKF is able to detect rapid changes of direction, which is a standard problem in target tracking [31].
The remainder of this paper is organized as follows. In Section II, the system model is presented, and the novel QCHEKF is introduced in Section III. In Section IV, the 133230 VOLUME 9, 2021 algorithm is validated via measurements. Finally, the article is finishes with a conclusion.

II. MODEL DESCRIPTION
In this section, the system model is described. First, the signal model chain from the beacon to one RX array, as illustrated in Fig. 2, is presented. Then, the measurement noise model is discussed. Finally, the system model describing the movement and oscillator behavior is presented.

A. SIGNAL MODEL
A moving beacon, located at time t at p where A B denotes the signal's unknown amplitude, f B denotes the approximate known transmit frequency, and φ B (t) denotes the transmit phase, which models both slowly varying frequency deviation and phase noise [29]. The signal is received by N R receive antenna arrays, with the n R th array comprising N A,n R antennas. The receive signal at the n A th antenna of the n R th array, located at p n R ,n A = [x n R ,n A , y n R ,n A , z n R ,n A ] T , is given by where a n R ,n A denotes the unknown signal attenuation caused by path loss, the receive and transmit antenna characteristics, and amplification factors in the processing chain, and denotes the signal's propagation time from the beacon to the antenna, with c 0 denoting the speed of light. At the n R th antenna array, the signal is first mixed using a CW signal with frequency f n R and time varying phase φ n R (t), which is also corrupted by phase noise, and then low-pass filtered, yielding the baseband signal where f n R = f B − f n R denotes the frequency difference and φ n R (t) = φ B (t) − φ n R (t) denotes the time varying unknown phase difference between the RX and the beacon, which is corrupted by their combined phase noise. Assuming the frequency difference f n R is small enough to satisfy f n R 1 2T S , with T S denoting the time between two sampling instances, the sampling theorem is satisfied and, therefore, the complete setup can be described solely at the discrete time instances t = kT S . From now on, for every variable, the time step information is marked in the index as variable k = variable(kT S ).
To separate the influence of the measurement system's frequency and phase relations from the influence of the beacon's movement on the measured phases, the beacon-RX phase difference φ n R ,k in (5) at time step k is defined as Expressing the slowly varying phase difference φ n R ,k via frequency variations enables the current phase difference to be described as a summation of all past phase shifts, yielding where φ n R ,0 denotes the starting phase difference and the overall frequency difference accounts for both the slowly drifting beacon-RX frequency difference and the phase noise of both the beacon and the RX [32]. Since the proposed algorithm will estimate the phase and frequency difference in a recursive manner, the beacon-RX phase difference (7) at time step k is formulated recursively as as in [33]. Inserting the beacon-RX phase difference φ n R ,k in (6) into (5) yields Evaluating the phase of (10) yields the measurement model h n R ,n A ,k at the n A th antenna of the n R th array as Stacking the measured phases yields the measurement vector as with the phases at one array

B. MEASUREMENT NOISE MODEL
The phases, measured at spatially distributed antennas in indoor scenarios, are corrupted by many different error sources. In addition to the phase noise, which was introduced in Section II-A, thermal noise, multipath propagation, and unknown antenna phase characteristics disturb the measured phases. In indoor scenarios, the multipath propagation represents the main error source for localization systems [34]. However, Kalman filters minimize a least-squares metric [35] and, hence, the proposed QCHEKF treats multipath propagation as additive white Gaussian noise (AWGN), which impairs the phase measurements as with w ϕ,k ∼ N ( 0, σ 2 ϕ I). In doing so, the AWGN model implies several assumptions on the measurement impairments. First, it assumes that the line-of-sight (LOS) is never blocked. Second, the AWGN model assumes that the impairments on the LOS signal are sufficiently small to neglect the non-linear distortion due to the phase extraction from the complex valued measurement data. Third, the AWGN model implicitly assumes the noise for different antennas to be uncorrelated, which is not true for multipath propagation. Therefore, it is advisable to decorrelate the LOS signal from the multipath propagation signal as much as possible by using widely spaced RX antennas [11], [36]. Further, the AWGN model implicitly assumes the noise for different time instances to be uncorrelated. Since the measurement errors caused by multipath propagation depend on the beacon's position, this is mainly valid for fast fading channels or a quickly moving beacon and, therefore, the proposed QCHEKF is most suited for tracking applications. Beside the uncorrelated noise assumption, (14) assumes normally distributed noise with constant power, which implies the complex valued measurements to be impaired by Gaussian distributed noise with constant signal-to-noise ratio [37]. This assumption is approximately valid, because both the multipath propagation receive power and the LOS signal power decrease with increasing beacon-RX distance.
Because of the phases' ambiguity, all phases are defined within (−π, π] in this paper and, therefore, the noise term w ϕ,k in (14) might cause a large difference between the measured ϕ meas,k and the hypothetical phases ϕ k . Hence, the measurement deviation between the measured and correct phases must again be mapped to (−π, π]. Evaluating this deviation yields which is valid as long as the noise standard deviation satisfies σ ϕ π. Despite the 2π-ambiguity, the noise shape in (14) is preserved within the deviation and, therefore, a Kalman filter based evaluation is well suited [38].

C. SYSTEM MODEL
Since the proposed algorithm mainly concerns the evaluation of phases, which are measured at spatially distributed antennas, arbitrary motion models can be incorporated. Most often, a constant velocity approach [39], such as denotes the beacon's velocity for each individual dimension, and w v d ,k ∼ N (0, σ 2 v ) denotes the velocity noise, is well suited for localization purposes. Furthermore, the phase and frequency relation of the complete beacon-RX processing chain has to be modeled. In communications, digital phase locked loops (PLL) are used to provide a stable phase relation between a transmitter and a RX [40]. However, these PLLs also correct phase deviations caused by beacon movements, so the estimation of the phase and frequency relation between each beacon-RX pair has to be incorporated into the localization algorithm. Generally, PLLs can be well modeled via a Kalman filter, as in [33]. For this purpose, similar to the constant velocity concept, a constant frequency approach is established for each beacon-RX pair as where w RX,n A ,k ∼ N (0, σ 2 n A ) denotes the frequency noise at the n A th RX array and w B,k ∼ N (0, σ 2 B ) denotes the beacon's frequency noise. In (17), the phase difference φ n R ,k−1 is created by the recursive summation of the past phase changes 2πT S f n R ,k−1 , as in (9). Note that the beacon's frequency noise term is the same for all RXs. Stacking the beacon's position d B,k and velocity v B,d,k for every dimension, and the phase and frequency differences between each beacon-RX pair, φ n R ,k and f n R ,k , yields the state vector x k and the complete system state model as where all phases are mapped to (−π, π], and F denotes the state transition matrix, which models the relation between x k−1 and x k according to (16) and (17). The system noise w k ∼ N ( 0, Q) in (18) includes all occurring system noise sources as w k = (w v x ,k w v y ,k w v z ,k w RX,1,k , · · · , w RX,N A ,k w B,k ) T , which are mapped in (18) via the noise shaping matrix G onto the system state according to (16) and (17). Here, the covariance matrix Q contains the noise variances on the main diagonal according to the noise vector w k . Using the state vector x k , the measurement function in (12) is denoted as h( x k ).

III. LOCALIZATION ALGORITHM
Because both the measurement deviation and the system noise are assumed to be normally distributed, an extended Kalman filter (EKF) based evaluation is well suited [38]. The EKF works in a recursive predict-update manner, which will be examined in the following section. Here, (·) k|k denotes an estimate at time step k and (·) k|k−1 denotes the estimate at time step k after the prediction step, meaning that the measurements at time step k have so far not been incorporated.

A. PREDICTION STEP
The prediction step performs a linear extrapolation of the current system state, see [41]. Despite the phase ambiguity, the prediction step remains unchanged in comparison to a standard EKF, which yields For consistency, all phases should again be mapped to (−π, π].

B. PHASE PREPROCESSING
Generally, the algorithm's concept is to evaluate both the phase differences between the antennas, which yield absolute information about the beacon's position, and the absolute phases, which provide highly accurate relative information about the beacon's position. To calculate suitable phase differences and choose one absolute phase, the phases at each array are preprocessed using a matrix T pre,n R as where the preprocessing matrix T pre,n R evaluates N A,n R − 1 phase differences between antenna pairs per array via the first N A,n R − 1 rows, and the absolute phase at one antenna per array via the last row. To assure that all antennas' phase information is incorporated, T pre,n R needs to have full rank. Hence, the evaluated phase differences should create a spanning tree between all antennas of the array [42] to assure a full rank of the first N A,n R − 1 rows of T pre,n R . Since the phase difference unambiguity range of an evaluated antenna pair increases with decreasing antenna distance, it is advisable to choose the evaluated phase differences in T pre,n R so that each receive antenna is connected to the spanning tree via the most closely spaced neighboring antenna. By stacking the preprocessing matrices T pre,n R of each array, the complete preprocessing matrix T pre evaluates the measured phases as ϕ pre meas,k = T pre ϕ meas,k . With the measurement deviation remains normally distributed for σ ϕ π. Hence, the measurement error of the preprocessed phase is normally distributed as T pre w ϕ,k ∼ N ( 0, R pre = σ 2 ϕ T pre T T pre ). The preprocessed residual r between the measured phases and the hypothetical phases is then given by C. UPDATE STEP Generally, the EKF update step minimizes a generalized least squares metric [21], [35], incorporating both the measurement and the predicted system state. By linearizing and minimizing the likelihood function starting from x k|k−1 , the update step is established as shown in the appendix, where H pre,k = T pre H k holds, with H k denoting the Jacobian of h( x k ) at x k|k−1 , and K k denoting the Kalman gain. Here, the difference to the common EKF is the mod 2π (·) mapping of the phase error. This is valid as long as the absolute differences between the measured ϕ meas,k and the hypothetical phases h( x k|k−1 ) do not exceed π, which is also a direct result of the update step derivation in the appendix.

D. INTERPRETATION
In [23], a HEKF was proposed that estimates the position of an incoherent beacon. To cope with the incoherent measurement process, the HEKF in [23] evaluated only phase differences. In this paper, the novel QCHEKF extends this concept by additionally evaluating absolute phases, because these are much more sensitive to position changes, and enable further improvement of localization accuracy in comparison to a pure phase difference evaluation. For this purpose, VOLUME 9, 2021 the absolute phase differences between the beacon and each RX φ n R ,k have to be estimated. Assuming the absolute phase behavior to be completely unknown, that is σ 2 n A → ∞ and/or σ 2 B → ∞, the same localization accuracy as in [23] is achieved. If the frequency differences f n R ,k remain sufficiently stable, the QCHEKF is able to estimate the absolute phase and frequency relation between each beacon-RX pair by comparing the phase-difference based position estimation with the absolute phase measurements. Thereafter, when the phase and frequency relation is estimated with sufficient accuracy, the QCHEKF is able to incorporate the absolute phase measurements in the localization process, further increasing the localization accuracy. For a reliable localization based on the evaluation of absolute phases, a high update rate is necessary to ensure that the absolute differences between the measured ϕ meas,k and the hypothetical phases h( x k|k−1 ) at the predicted position remain less than π. Otherwise, a position update towards an incorrect position will occur. Assuming the phase noise to be small, the main phase shifts occur because of position changes. Unfortunately, the exact behavior of the predicted position is difficult to analyze for every situation and, therefore, we suggest choosing the maximum position difference between successive measurement instances to be significantly less than half of the wavelength. Note that the absolute position estimation is still performed only based on the phase differences and, hence, the QCHEKF still provides accurate estimates for larger position deviations. Thus, in the event of an erroneous position estimation, which is related to the ambiguity of the evaluated absolute phases, the algorithm can slowly correct the error by adopting the beacon-RX phase difference φ n R ,k .

E. INITIALIZATION
Generally, Kalman filters exponentially converge toward the system state after the initialization [43]. In the beginning, the system state varies widely until the steady state is reached. This procedure is well suited to estimate the position and velocity for a pure phase difference evaluation, because phase differences provide a sufficiently ambiguous range [23]. However, as discussed in the previous subsection, the evaluation of absolute phase measurements requires the absolute differences between the measured ϕ meas,k and the hypothetical phases h( x k|k−1 ) at the predicted position to be less than π, which contradicts strong variations in the position estimate after initialization.
Hence, we propose starting the QCHEKF in a HEKF-based manner, meaning that only the phase differences are evaluated. For this purpose, coarse pre-estimates of the beacon's position p B,0 and frequency f B,0 are assumed. To exclude the absolute phases of the beacon-RX pairs from the localization process in the beginning, the frequency variation is assumed to be completely random in each sample via σ 2 n A → ∞ and/or σ 2 B → ∞. Then, the localization is conducted as for a pure phase difference-based HEKF, while the frequency and phase relations are estimated. To start the QCHEKF,  the variances σ 2 n A and/or σ 2 B are set to their desired values. Since the beacon's position and velocity are already roughly estimated, the QCHEKF then incorporates the relative phase and frequency relations, thereby increasing the localization accuracy.

IV. EXPERIMENTAL VERIFICATION
In this section, localization measurements within the 24 GHz industrial, scientific, and medical (ISM) band are presented. First, the measurement setup is described. Then, localization results are shown and analyzed.

A. MEASUREMENT SETUP
The measurements are conducted within the setup shown in Fig. 3. Because of the strong reflectors in the surrounding, the considered environment depicts a challenging multipath scenario. The measurement setup consists of N R = 3 identical receivers, each with N A,n R = 12 antennas, and one beacon, which is mounted on a robotic arm. The antenna geometry of each array with an aperture size of 25.4 cm × 9.1 cm and the phase differences, which are evaluated by T pre,n R , are shown in Fig. 4. To incorporate absolute phase measurements, the last row of T pre,n R evaluates the absolute phase measurement at antenna 1. The beacon's 24 GHz signal is created via a voltage-controlled oven-controlled 50 MHz crystal oscillator (VCOCXO) with the use of a PLL. According to the data sheet, the oscillator provides a 10 s Allan deviation of approximately 4 · 10 −11 , which corresponds to a frequency change of approximately 1 Hz after 10 s for the 24 GHz CW signal. Note that the behavior is degraded due to the acceleration force of the robotic arm [44]. At the RXs, the received signals are converted to a low intermediate frequency by mixing with slightly shifted 24 GHz signals, which are created by the identical VCOCXOs and PLLs as used for the beacon. To compensate for phase mismatch, coupling, and antenna position errors, the arrays have been calibrated in-situ with 50 measurements at known positions using the calibration concept proposed in [45]. Unfortunately, the robotic arm is able to provide highly accurate positioning, but not corresponding timing information when it performs continuous movements. Therefore, a direct comparison of the correct and the estimated beacon positions is not possible and only the distance between the estimated beacon positions to the overall trajectory can be evaluated here.

B. MEASUREMENT RESULTS
To assess the QCHEKF, the localization results are compared to the results of a pure phase difference HEKF. Since T pre,n R evaluates N A,n R −1 phase differences and one absolute phase via the last row, a pure phase difference HEKF is obtained by removing the last row of T pre,n R and omitting the constant frequency part in the system model. During the measurement, the robotic arm is moved continuously along a trajectory p rob (t), which consists of 21 straight sections at a constant height with a total trajectory length of 2.2 m, as shown in Fig. 5-7. While the trajectory parts with a linear motion are well suited for tracking systems, the changes of direction contradict the constant velocity model in (16) and are challenging for tracking systems. The complete measurement time was approximately 26 s, yielding measurements at 37,523 time instances, which corresponds to an update rate of 1.4 kHz.
Typically, the localization errors are calculated by directly comparing the estimated positions with reference positions. However, since an exact time synchronization of the robotic arm is not possible, each position estimate can only be assessed via its minimal deviation from the trajectory as Hence, a root mean square error can be defined as where K is the number of position estimations. The resulting RMSE can be interpreted as a 2D RMSE of a 3D localization process, because it most often depicts the 2D trajectory-beacon distance in the plane, which is orthogonal to the straight line currently driven by the robotic arm. Besides the RMSE, the maximal distance from the trajectory is used to assess the localization results.  To use the HEKF and QCHEKF, several parameters, summarized in Table 1, have to be chosen. Despite the beacon's position, which is known via the reference, the system states are unknown and initialized by zeros. The corresponding covariance matrices of the HEKF and the QCHEKF are initialized via a diagonal matrix with large entries to reflect a high uncertainty about the system state at the beginning. The used noise parameters were estimated by hand and are held constant for all subsequent evaluations. First, the values for σ ϕ = 5 • and σ v ≈ 5.1 · 10 −5 m s were chosen so that the HEKF's performance was optimized. Afterwards, σ n A = σ B ≈ 3.6 · 10 −4 Hz was determined so that the QCHEKF's performance was optimized. As discussed in Section III-E, the QCHEKF starts to involve the phase frequency relation with the tenth measurement instance by setting σ n A = σ B = 3.6 · 10 −4 Hz.
By evaluating the HEKF and QCHEKF using all 12 antennas at all receivers with an aperture of 25.4 cm × 9.1 cm, the localization results in Fig. 5 are obtained. The standard HEKF already provides very accurate localization results VOLUME 9, 2021 FIGURE 6. QCHEKF and HEKF localization result using antennas 1-8, yielding an array size of 6 cm × 9.1 cm. . This causes a maximal error of d max ≈ 5.0 cm. In comparison, the QCHEKF is able to trace the changes of direction, reducing the maximum error to d max ≈ 1.6 cm and improving the accuracy to RMSE ≈ 0.51 cm. The cumulative error function is shown in Fig. 8. It can be observed that the QCHEKF improves the localization accuracy whenever the  HEKF yields high errors due to stability issues induced by model errors.
The accuracy of the standard HEKF mainly depends on the array size and beacon distance, because these parameters directly influence the phase difference measurement sensitivity [11]. Hence, the HEKF's accuracy decreases for smaller arrays. By only evaluating the phase differences of antennas 1-8 in Fig. 4 at all arrays, the array size is reduced to 6 cm×9.1 cm, yielding the localization results in Fig. 6. Then, the HEKF's accuracy decreases to RMSE ≈ 1.3 cm and the maximum error increases to d max ≈ 8.7 cm. In comparison, the QCHEKF yields RMSE ≈ 0.9 cm and d max ≈ 3.4 cm. The cumulative error function in Fig. 8 again validates the QCHEKF's ability to prevent high localization errors.
By reducing the array's evaluated antennas to 1-3, which is the minimum for a 2D array, an array of size 2.5 cm × 2.5 cm is obtained. Now, the HEKF has difficulties reconstructing the driven trajectory, as shown in Fig. 7, yielding RMSE ≈ 3.0 cm and d max ≈ 7.6 cm. In comparison to the localization result with eight antennas per array, the maximum distance error is slightly reduced, because of the smaller error at p dev . Overall, many large errors occur, as can be observed from the cumulative error function in Fig. 8. The QCHEKF improves the localization accuracy to RMSE ≈ 2.2 cm and d max ≈ 5.1 cm, and effectively prevents the high localization errors in the cumulative error function in Fig. 8.
The RMSEs and maximal deviations of the different measurement evaluations are summarized in Table 2. Generally, the measurements have shown that the QCHEKF is able to improve the localization accuracy compared to the HEKF, particularly when the HEKF's localization performance deteriorates due to multipath propagation or a deficient motion model. Visible in the cumulative error function in Fig. 8, the potential improvement increases for smaller arrays, because the phase differences become less sensitive to position changes [11], whereas the sensitivity of absolute phases does not scale with the array size. Hence, even in the three-antenna case in Fig. 7, the QCHEKF provides very accurate information about relative movements. This can be seen by inspecting the localization behavior during the changes of direction, where the HEKF's phase difference evaluation results in large localization errors.

V. CONCLUSION
In this paper, the novel QCHEKF was presented, which incorporates the evaluation of absolute phase measurements into incoherent PDOA localization systems to enable highly accurate beacon tracking. Particularly relative motions can be tracked precisely, enabling, for example, the detection of a person's tremor or mechanical vibrations. In comparison to the UWB systems, the simultaneous tracking of many targets is easily implementable via slightly frequency-shifted beacons. The QCHEKF's localization accuracy depends on the stability of the frequency references used. Therefore, the relationship between the oscillator's frequency stability and the localization accuracy should be studied in future research. Localization accuracy could be improved further by sharing the signal of a common oscillator to all receivers, each with an individual PLL, and adopting the QCHEKF's system accordingly. In this paper, the QCHEKF relies on the evaluation of the permanently available LOS between the beacon and each receiver. To extend the QCHEKF for the varying conditions in complex indoor measurement setups, a detection of non-LOS scenarios, for example by an adaptive noise estimation [46], is desirable.

APPENDIX
In this appendix, (24) is minimized using one Newton step [47] beginning at the state prediction x k|k−1 . For this purpose, r( x k ) is Taylor expanded at x k|k−1 as r( x k ) = mod 2π T pre ϕ meas,k − T pre h( x k ) ≈ mod 2π T pre ϕ meas,k − T pre h( x k|k−1 ) + H k x k = r( x k|k−1 ) − H pre,k x k (29) which is valid as long as the absolute differences between the measured and the hypothetical phases do not exceed π, and where x k = x k − x k|k−1 holds. Inserting (29) in (24) yields The gradient of (30) is given by grad(J ( x k )) = −H T pre,k R −1 pre r k ( x k|k−1 ) − H pre,k x k + P − and as in [35]. Since the covariance matrix of a Gaussian distribution is equivalent to the inverse Hessian matrix of the likelihood function [41], the inverse Hessian of (30) is evaluated as P k|k = H T pre,k R −1 pre H pre,k + P −1 k|k−1 −1 = I − K k H pre,k P k|k−1 (35) as in [35], where again, (34) was used.