Joint Delay and Phase Discriminator Based on ESPRIT for 5G NR Positioning

A joint delay and phase discriminator based on the ESPRIT algorithm is proposed for positioning with the fifth generation (5G) New Radio (NR) downlink cellular signals. Since ESPRIT requires knowledge of the number of paths of the radio channel to operate, the proposed discriminator is coupled and studied with two channel order estimators: minimum descriptive length (MDL) and efficient channel order determination (ECOD). To mitigate the delay and phase outliers appearing when the MDL overestimates the channel order, a path selection criterion based on the signal-to-noise ratio (SNR) of the path is proposed. The ESPRIT discriminator is evaluated in terms of the distribution of the estimated channel order, and the delay and phase errors in the absence and presence of multipath. It is observed that the path selection criterion makes the MDL suitable for positioning as no delay and phase outliers are produced. In the presence of multipath, the ESPRIT discriminator achieves lower mean and root-mean-square delay and phase errors than the commonly used early-minus-late power (EMLP) and 2-argument arctangent (ATAN2) discriminators.


I. INTRODUCTION
The fourth generation (4G) Long-term Evolution (LTE) and fifth generation (5G) New Radio (NR) cellular systems have evolved to become a relevant source of positioning thanks to their favorable signal characteristics and good coverage in urban environments [1]. Positioning with cellular signals is of particular interest to the users of the Global Navigation Satellite System (GNSS) who may experience reduced performance in challenging environments such as in urban canyons or indoors [2]. The integration of the cellular downlink delay measurements with GNSS allows to improve the availability, continuity, accuracy, and integrity positioning performances [3]- [7]. To further improve positioning performances, the signal phase measurements representing the carrier phase can also be obtained from the received signal and included in the navigation processing [8].
Downlink delay and phase measurements can be obtained by tracking the pilots of the transmitted cellular signals. LTE The associate editor coordinating the review of this manuscript and approving it for publication was Venkata Ratnam Devanaboyina . and 5G NR waveforms utilize the orthogonal frequency division multiplexing (OFDM) scheme and contain signals for various purposes including channel estimation, positioning, and time/frequency synchronization [9], [10]. Although the cellular base stations can transmit a dedicated positioning signal called the positioning reference signal (PRS), it has not been widely adopted in LTE, and the extent of its utilization in 5G NR is still uncertain. Instead of the PRS, other downlink signals can be exploited for positioning, some of which do not require the user subscription or specific network configuration. One such signal in LTE systems, called the cell-specific reference signal (CRS), has been shown to provide satisfactory performance for ranging purposes [11], [12]. In 5G NR systems, the CRS is no longer present and is replaced by other reference signals such as the channel state information reference signal (CSI-RS) [13]. For potential downlink positioning in the frequency range 1 (FR1), the CSI-RS configured as the tracking reference signal (TRS) is further considered. The TRS is nominally used for fine synchronization and channel analysis for the user equipment (UE). Although the CSI-RS is UE-dependent, the considered configuration has a high subcarrier density making it suitable for the evaluation of the delay estimation and positioning.
The receiver obtains the delay and phase measurements by continuously tracking the received signal. In terrestrial scenarios, the effects contributing to the delay and phase measurements include the geometrical distance between the transmitter and receiver, transmitter and receiver clocks, and various instrumental delays in the signal processing chain. The signal tracking is typically performed by a closed-loop architecture, where the tracking loops continuously update the delay and phase estimates. The tracking loops utilize so-called discriminators whose outputs they drive to zero by employing a feedback loop to the received signal. The discriminators may introduce additional biases and errors to the measurements due to the multipath and noise. The delays are estimated to the subsample accuracy as depending on the sampling rate, a single sample may represent a positioning error equal to several meters. The early-minus-late (EML) delay discriminator was proposed by Yang et al. [14] in the frequency domain for an arbitrary OFDM waveform and fixed correlator spacing of half a sample. The EML power (EMLP) delay discriminator was studied by Serant et al. [15] in the time domain for an arbitrary correlator spacing and terrestrial digital video broadcast OFDM signals. The work was expanded into the frequency domain by Chen et al. [16]. A phase discriminator based on the 2-argument arctangent (ATAN2) operator applied to the channel frequency response (CFR) estimated using the reference pilots was proposed by del Peral et al. [17] for LTE positioning and further expanded in the work of Shamaei and Kassas [18]. Positioning of an airborne drone receiver implementing delay and phase ranging using the EMLP and ATAN2 discriminators was demonstrated in real flight scenarios [8].
Although the EMLP and ATAN2 discriminators approach closely the performance of the maximum-likelihood estimator in the additive white Gaussian noise (AWGN) channel, they experience degraded performances in multipath channels that introduce additional biases to the delay and phase estimates. The multipath channels are typically experienced in urban environments, where achieving the maximum benefit of cellular positioning is desired due to the limited visibility of GNSS satellites. Delay estimators robust to multipath are based on super-resolution algorithms (SRAs). Various SRAs were studied for indoor positioning by Li and Pahlavan [19]. The ESPRIT algorithm was used as a delay estimator coupled with a Kalman filter for positioning with LTE signals [20], [21]. Due to its wide estimation range, the ESPRIT algorithm was also used for the signal acquisition in LTE positioning [8], [18]. A joint delay and phase estimation using ESPRIT was proposed by Dun et al. [22]. Using the ESPRIT algorithm as a joint delay and phase discriminator that can be implemented in a tracking loop has not yet been investigated.
The ESPRIT algorithm requires knowledge of the number of paths of the radio channel to operate. As the true number of paths is unknown, the estimation of the channel order in multipath environments is needed. The most recognized channel order estimator is the minimum descriptive length (MDL) method [23]. The MDL is based on information-theoretic criteria and has been widely adopted for positioning with SRA algorithms [18], [19], [21]. However, the MDL may overestimate the channel order that can cause ESPRIT to produce delay outliers in the form of false paths arriving before the true earliest path [18], [21], [24]. The overestimation may occur even for a high signal-to-noise ratio (SNR) [25]. To avoid the overestimation, different channel order methods may be used. Lavias et al. [26] proposed the efficient channel order determination (ECOD) method that is based on eigenvalue differencing and does not overestimate the channel in sufficiently high SNR. For positioning, Driusso et al. [20] addressed the effect of the MDL overestimation on ESPRIT by applying a threshold to the delay measurements. The utilization of the ECOD and the study of alternative ways to select the true earliest path during the overestimation remain unexplored topics in the context of cellular positioning.
In this paper, a joint delay and phase discriminator based on the ESPRIT algorithm is proposed for positioning with 5G NR cellular signals. The main contributions of the paper are summarized as follows: (i) A waveform suitable for the performance evaluation of delay and phase discriminators in 5G NR systems is proposed. (ii) A discriminator based on ESPRIT capable of joint delay and phase error estimation suitable for the signal tracking loops is proposed. (iii) The impact of the channel order overestimation of the MDL method on the ESPRIT algorithm is investigated. A path selection criterion based on the SNR of the path is proposed. (iv) The ESPRIT discriminator is coupled with the MDL and ECOD methods, and the resulting combinations are compared in terms of the distribution of the estimated channel order, and the delay and phase errors in the absence and presence of multipath. The remainder of the paper is organized as follows. Section II proposes the 5G NR downlink waveform for positioning evaluation. Section III defines the commonly used EMLP and ATAN2 discriminators. Section IV proposes the join delay and phase discriminator based on ESPRIT. The results of the performance evaluation of various delay and phase discriminators are presented in Section V. Conclusions are given in Section VI.
Notation: Matrices are denoted as uppercase boldface letters, such as X ∈ C M ×N . Column vectors are denoted as lowercase boldface letters, such as x ∈ C M ×1 . I P×P is a P × P eye matrix, 0 P×Q is a P × Q zero matrix, and 0 P×1 is a zero column vector of length P. The operators (·) T , (·) H , (·) −1 , and (·) † denote the transpose, the Hermitian transpose, the inverse, and the Moore-Penrose pseudoinverse of a matrix, respectively. | · |, (·), and (·) * denote the absolute value, the argument, and the conjugate of a complex VOLUME 9, 2021 number, respectively. The operator min (x) denotes the minimum value of a vector x. The operator sort (x) produces a vector whose values are sorted in ascending order.

II. CELLULAR DOWNLINK WAVEFORMS FOR POSITIONING EVALUATION
The 5G NR and LTE cellular downlink waveforms are based on the OFDM transmission scheme that is described by a time-frequency grid in which a symbol time index i and frequency subcarrier index n uniquely identify a single transmitted complex data symbol. A digital baseband signal x i [l] of the i-th OFDM symbol is expressed as for l = 0, 1, . . . , N fft + N cp − 1, where n is the OFDM subcarrier index, X i [n] is the complex data symbol transmitted on the n-th subcarrier, N fft is the total number of subcarriers available for modulation that are symmetrically arranged around the direct current (DC) component with the index n = 0, and N cp is the number of samples within the cyclic prefix (CP). The CP is a sequence of the last N cp samples of the symbol prepended to its beginning to provide a guard interval preventing the inter-symbol interference from the previous symbol whilst allowing for a circular convolution of the channel. The digital signal in (1) can be conveniently generated using the inverse fast Fourier transform (IFFT). The continuous signal is then generated using the digitalto-analog converter. The resulting subcarrier spacing is f = The 3GPP specification maps the N sc ≤ N fft transmitted resource elements to the N fft available subcarriers. The OFDM subcarrier index n on which the k-th resource element is transmitted can be obtained using the mapping function κ (k) as for k = 0, 1, . . . , N sc − 1, where ξ ∈ {0, 1} reflects whether the DC component is used for transmission or not. The unused subcarriers are left empty. The 5G NR and LTE waveforms may contain various signals that can be exploited for positioning. Although a dedicated positioning signal called the PRS exists in the 3GPP specifications, it is not considered for the positioning evaluation in this paper as the extent of its utilization in 5G NR is still uncertain.

A. 5G NR DOWNLINK EVALUATION WAVEFORM
The 5G NR OFDM transmission scheme is logically organized into 10 ms long radio frames. The radio frame is composed of 10 · 2 µ slots, where µ ∈ {0, 1, 2, 3, 4} is called the numerology parameter whose value is driven by the frequency band and the signal bandwidth [9]. In the normal CP mode, the slot is composed of N DL symb = 14 symbols. The smallest logical block of the OFDM grid is called the resource block (RB) and comprises N DL RB = 12 subcarriers and N DL symb symbols. The subcarriers are spaced f sc = 15·2 µ kHz apart. The DC component is used for data transmission and ξ = 0 in (2).
To assign values to different parameters and allow for the evaluation of positioning performances using reference pilots, a potential 5G NR downlink waveform that is not necessarily linked to any particular deployment or specification is proposed. The waveform is configured to the maximum allowable transmission bandwidth of B = 100 MHz in the FR1, time division duplexing (TDD) scheme, normal CP mode, numerology µ = 1, and cell ID 42 [27]. In this configuration, the radio frame is composed of 20 slots and the subcarrier spacing is f sc = 30 kHz. The length of the FFT/IFFT is N fft = 4096 resulting in the sampling frequency f s = N fft f sc = 122.88 Msps. After the guard band removal, the number of RBs used for transmission is N RB sc = 273. The TDD mode is configured to the slot format 32 that reserves the first ten symbols of the slot for the downlink and the last two symbols for the uplink whilst the remaining two slots are left empty [28]. The values of the relevant parameters are summarized in Table 1.
Although being UE-dependent, the CSI-RS is used for signal tracking and positioning in the proposed 5G NR evaluation waveform. The CSI-RS is configured as the TRS resulting in its periodic transmission every slot, frequency density of ρ = 3 pilots per RB, full occupation of the 273 available RBs for the UE, and no code division multiplexing [9]. The assumed configuration provides P = N RB sc ρ = 273 · 3 = 819 pilots spanning the full bandwidth and is considered relevant for the evaluation of positioning performances as it maximizes the presence of the pilots. The subcarrier spacing between adjacent pilots is The initial position of the first CSI-RS pilot is for convenience configured with the time offset l 0 = 0 and the subcarrier offset k 0 = 0. The distribution of the CSI-RS pilots assumed in the 5G NR waveform is shown in Fig. 1a. The downlink symbols are considered to be fully occupied by the physical downlink shared channel (PDSCH) that is simulated by a random data generator. No data transfer is present on the uplink channel.

B. LTE DOWNLINK EVALUATION WAVEFORM
For comparison with 5G NR, an LTE downlink evaluation waveform is also considered. The OFDM scheme of LTE systems can be interpreted as a subset of the 5G NR scheme configured with µ = 0 [10]. A notable difference is a smaller maximum transmission bandwidth of B = 20 MHz and frequency division duplexing (FDD) scheme that is used in the deployments on the European continent [29]. On the  contrary to 5G NR systems, the DC component is omitted from transmission in LTE and ξ = 1 in (2). The parameters of the proposed LTE waveform are summarized in Table 1.
To allow the UE to perform coherent demodulation and to estimate the quality of the downlink channel, the CRS pilots are transmitted in the downlink waveform. The configuration of the CRS is only driven by the cell ID and the antenna port. The distribution of the CRS pilots assumed in the LTE waveform is for the first antenna port of the cell ID 42 shown in Fig. 1b.

III. DELAY AND PHASE DISCRIMINATORS
Delay and phase discriminators allow for continuous signal tracking in the closed-loop receiver architecture as they estimate the delay and phase offsets between the received signal and the locally generated replica. The block diagram of the signal tracking stage of a typical OFDM positioning receiver implementing the delay and phase discriminators is shown in Fig. 2. In the tracking stage, the receiver updates the delay and phase estimates with the output of the tracking loops. To drive the outputs of the discriminators to zero and maintain the lock of the signal, the delay and phase estimates are fed back to the demodulation block in the next receiver iteration to remove the estimated offsets from the received signal. The delays are estimated to the subsample accuracy and removed from the input signal at the integer and fractional parts. The signal tracking stage is preceded by the signal acquisition stage that initializes the delay and phase estimates. The signal acquisition is for simplicity omitted from the block diagram shown in Fig. 2 as it does not involve the discriminators.
The relation between the received signal and the local replica at the receiver is expressed by the CFR that can be estimated from the pilots in the OFDM waveform aŝ where p is the pilot index, i is the symbol time index,Υ i,p is the received pilot symbol with the removed delay and phase estimates from the previous iteration of the tracking loops as shown in Fig. 2, and ϒ i,p is the pilot symbol replica at the receiver. The pilot index p = 0, 1, . . . , P − 1 is assigned to each of the P ≤ N sc pilots of a given reference signal. For equally spaced pilots with a constant transmission pattern, the OFDM index n p of the p-th pilot can be expressed as where P is the pilot spacing, k 0 is the subcarrier offset of the first pilot from the first subcarrier used for transmission, and κ (k) is the mapping function in (2). Considering the pilot spacing and number of the pilots are the same for all VOLUME 9, 2021 FIGURE 2. Block diagram of the signal tracking stage of a typical closed-loop OFDM positioning receiver. The receiver tracks the input signal using the tracking loops comprising of discriminators, loop filters, and numerically controlled oscillators (NCO). The tracking loops keep updating the code and phase estimates, which are then fed back to the demodulation block and removed from the received signal in the next receiver iteration to maintain the lock of the signal. The integer part of the delay estimate ( τ ) is used to shift the FFT window in the time domain and the fractional part (τ − τ ) is applied as a phase rotation of the subcarriers in the frequency domain. The traditionally used EMLP delay and ATAN2 phase discriminators are proposed to be replaced by the ESPRIT discriminator that provides both estimates.
symbols containing the CSI-RS pilots in the 5G NR downlink evaluation waveform from Section II-A, the symbol time index i is for simplicity omitted for the remainder of this section.

A. EARLY-MINUS-LATE POWER DELAY DISCRIMINATOR
The EMLP delay discriminator estimates the delay offset by correlating the received signal with the local replica at the receiver. The replica is delayed and advanced by ±δ samples and the resulting two correlators are referred to as the early and late branches. The time shift property of the discrete N fft n τ allows to define the correlator branches in the frequency domain as phase rotations of the OFDM subcarriers. A delay offset e τ of the received signal causes additional rotation and can be expressed at a given time symbol as [14] R (e τ , ∓δ) = 1 P where R (e τ , −δ) represents the output of the early branch, R (e τ , +δ) represents the output of the late branch, P is the number of pilots, and n p is the OFDM index of the pilot from (4). The power of the received pilot from (3) is assumed to be equal on each subcarrier and can be denoted as |Υ p | 2 = C. The power of each pilot replica at the receiver is assumed to be unitary and simplifies to |ϒ p | 2 = 1. The normalized non-coherent EMLP delay discriminator is then defined as [sample], (6) where the unit of samples denotes the estimated delay offset as a fractional number of samples that can be converted to the unit of seconds by dividing it with the signal sampling frequency f s , and k EMLP (δ) is the normalization factor to keep D EMLP (e τ , δ) ≈ e τ when e τ ≈ 0. The normalization factor can be expressed for an arbitrary correlator spacing as [16] k where β = P P N fft is the OFDM waveform factor that corresponds to the ratio between the usable signal bandwidth spanned by the pilots and the IFFT/FFT length.

1) DELAY ERROR VARIANCE
In the AWGN channel without multipath, the linear region of the EMLP discriminator function in (6) approximates to D EMLP (e τ , δ) ≈ e τ + n e τ , where n e τ is approximately the zero-mean white complex Gaussian noise with the variance σ 2 e τ that can be expressed as [15] σ 2 e τ = 4 (1 − sinc (2πβδ)) sinc 2 (πβδ) where the SNR is defined as where S is the power of the signal over N sc subcarriers, σ 2 n is the power of the noise over N sc subcarriers, and w 2 is the power of the noise over a single subcarrier. The expected performance in the multipath-free AWGN channel is relevant to understand the impact of the signal parameters on the lower bound of the discriminator error.

2) CRAMÉR-RAO LOWER BOUND
The Cramér-Rao lower bound (CRLB) describes the minimum achievable variance of an unbiased estimator in sufficiently high SNR and is defined for an arbitrary OFDM delay estimator in the AWGN channel without multipath as [30] CRLB = N 2 fft 8π 2 SNR where n p is the OFDM index of the pilot from (4).

3) DELAY TRACKING THRESHOLD
The tracking threshold determines the maximum delay offset that can still be tracked by the discriminator. Delay offsets above this threshold will likely cause the tracking loop to lose the lock of the signal. In the case of the EMLP discriminator, it is defined as the delay offset whose deviation from the ideal linearity does not exceed a given threshold α as As a rule of thumb, the value ζ EMLP determines the maximum acceptable 3-sigma of the total jitter and dynamic stress error of a delay-locked loop (DLL) implementing the EMLP discriminator (3 σ DLL ≤ ζ EMLP ) [2]. To enable the analytical evaluation of the EMLP discriminator function, the non-coherent early and late branch outputs of the EMLP discriminator can be approximated for large FFT sizes N fft 1 in a noiseless multipath-free channel as [16] |R (e τ , ±δ)| 2 ≈ C sinc 2 (πβ (e τ ± δ)) , where sinc (x) = sin(x) x . By substituting (12) into (6), the closed-form expression of the discriminator function becomes The function of the EMLP discriminator (13) evaluated under various correlator spacings is shown in Fig. 3. The shape of the discriminator function gives a name to the so-called S-curve. Outside the linear region, the discriminator output no longer corresponds to the delay offset and the performance of the tracking loop degrades. Fig. 3 shows that the correlator spacing has a negligible impact on the shape of the discriminator function around the linear region.

B. ATAN2 PHASE DISCRIMINATOR
The phase discriminator function estimates the phase offset between the received signal and the local replica at the receiver by applying the ATAN2 operator to the estimated CFR and can be expressed as [17], [18]

1) PHASE ERROR VARIANCE
In the AWGN channel without multipath, the ATAN2 discriminator function becomes D ATAN2 = e φ +n e φ , where n e φ is approximately the zero-mean white complex Gaussian noise with the variance σ 2 e φ that can be expressed as [18] σ 2 e φ = 2) PHASE TRACKING THRESHOLD The ATAN2 tracking threshold is limited by the 2π pull-in range of the argument operator used in (14). As a conservative rule of thumb, the tracking threshold ζ ATAN2 determines the maximum acceptable 3-sigma of the total jitter and dynamic stress error of a phase-locked loop (PLL) implementing the ATAN2 discriminator (3 σ PLL ≤ ζ ATAN2 ) [2]. The threshold is set to one-fourth of the pull-in range of the ATAN2 discriminator resulting in

IV. JOINT DELAY AND PHASE DISCRIMINATOR BASED ON ESPRIT
The ESPRIT algorithm is commonly used to estimate directly delays and phases of paths in the radio channel during the signal acquisition [18]. In this section, the use of ESPRIT as a discriminator replacing the traditionally used EMLP and ATAN2 discriminators is proposed. At the input, the algorithm takes the CFR with the removed delay and phase estimates from the previous iteration of the tracking loops as shown in Fig. 2.
Although the remainder of the section assumes that the line-of-sight (LOS) component is always present in the radio channel, this assumption does not invalidate the usability of ESPRIT in the non-LOS conditions. In the case the LOS is not present or is lost, the ESPRIT algorithm will naturally track the first arriving multipath component. The decision on whether the receiver is tracking the LOS or non-LOS is usually taken by the navigation processor and is beyond the scope of this paper.

A. DELAY OFFSET ESTIMATION
The ESPRIT discriminator estimates the delay offsets of paths using the rotational invariance property of the estimated CFR at the receiver. Assuming P received pilots at a given time symbol, the CFRĥ can be expressed aŝ VOLUME 9, 2021 whereĥ(p) is the CFR from (3). The CFR can be organized into snapshot vectors x(q) of length M as where M ≤ P. The parameter m ∈ [0, 1] is a design parameter of the ESPRIT discriminator that determines the length of snapshots M = m·P . Higher values of m increase the multipath resolution at the cost of reduced noise averaging. The configuration of this parameter should consider the number of expected paths in the channel, denoted as L, as it is necessary that M ≥ L. The number of snapshots is determined as N = P − M . The data matrix X can be constructed as The singular value decomposition (SVD) of X is where U ∈ C M ×M and V ∈ C N ×N are unitary matrices and where σ j ≥ σ j+1 . The standard ESPRIT approach can be used to estimate the delays of L paths from the snapshot matrix X.
The ESPRIT discriminator steps are defined as follows: Step 1: Extract the signal column subspace as Step 2: Split U s into two sub-matrices defined as Step 3: Estimate the ESPRIT rotational matrix as Step 4: Compute the eigenvalues ψ l of and their complex arguments as Step 5: Compute the delays of the paths and sort them in ascending order aŝ It is noted that the index l ofτ l from (27) does not necessarily identify the same path as the index of ψ l from (26) due to the sorting operation between steps 4 and 5.
The ESPRIT discriminator takes the delay of the earliest path as the estimated delay offset as D ESPRIT =τ 0 . The ESPRIT delay tracking threshold is limited by the range of the argument operator (ψ l ) ∈ (−π, π] from (26) and constrains the delay offset estimation to where ζ ESPRIT = 1 2 P f sc is the delay tracking threshold.

B. PHASE OFFSET ESTIMATION
To estimate the phase offsets, the complex amplitudes of paths need to be determined from the CFR. In the AWGN channel without multipath, the CFR at a given time symbol can be expressed as are the complex amplitudes of the paths, w ∈ C P×1 is the additional zero-mean complex Gaussian noise, and G ∈ C P×L is the delay matrix defined as where n p is the OFDM index of the pilot from (4). The estimated delay matrixĜ is obtained by substituting (27) into (30). The complex amplitudes can then be estimated using the least-squares solution aŝ The phase of the l-th path can be determined aŝ The ESPRIT discriminator takes the phase of the earliest path as the estimated phase offset as D (φ) ESPRIT =φ 0 . The phase tracking threshold of ESPRIT is the same as that of the ATAN2 discriminator from Section III-B2.

C. CHANNEL ORDER ESTIMATION
To properly estimate the delay and phase offsets in multipath environments, the ESPRIT algorithm requires knowledge of the channel order L. Channel order estimators usually rely on the information included in the eigenvalues λ 0 ≥ λ 1 ≥ . . . ≥ λ M −1 of the autocorrelation matrix obtained from the data matrix asR x = 1 N XX H ∈ C M ×M . The eigenvalues can be more conveniently determined from the singular values in (21) as λ j = σ 2 j N .

1) MINIMUM DESCRIPTIVE LENGTH METHOD
The MDL method is based on the information-theoretic criteria and can be computed as [23] MDL

2) EFFICIENT CHANNEL ORDER DETERMINATION METHOD
The ECOD method estimates the channel order by utilizing the ratio of two consecutive eigenvalues and can be computed as [26] ECOD The MDL method may overestimate the channel order that can cause the ESPRIT algorithm to produce delay outliers in the form of the false paths arriving before the true earliest path [18], [21], [24]. To avoid the impact of the delay outliers on positioning, the proposed ESPRIT discriminator applies a path selection criterion. Before defining the criterion, the effect of the channel order overestimation on the ESPRIT delay output is studied. For this purpose, the proposed discriminator is coupled with the MDL and simulated in the AWGN channel without multipath in 10 5 realizations. The evaluation is performed using the CSI-RS pilots of the 5G NR downlink evaluation waveform from Section II-A. The ESPRIT design parameter is set to m = 0.24 resulting in M = 819 · 0.24 = 196 eigenvalues. The estimated delayτ 0 andŜNR 0 of the earliest path in each channel realization are shown for the evaluated SNR equal to 12 dB and −6 dB in Figs. 4a and 4b, respectively. The SNR of the l-th path is estimated aŝ whereŵ is the noise vector estimated using the complex amplitudes in (31) asŵ =ĥ −Ĝˆ . Figs. 4a and 4b show that the MDL method may indeed overestimate the channel order as valuesL = 2 andL = 3 are observed. The overestimations may result in false earliest paths that are characterized by large negative delays and low SNRs, both of which are unrelated to the SNR of the overall signal. Although the negative delay outliers only appear when the channel order is overestimated, the overestimation itself does not always lead to negative delay outliers.
To remove the false earliest paths and allow the usage of the MDL for positioning, Driusso et al. [20] applied a filtering threshold to the ESPRIT delay measurements, exploiting their large negative values. However, this approach requires estimation of the delay and its confidence margin, and might miss the small negative values. Figs. 4a and 4b show that instead of the delay measurements, the false earliest paths can be filtered based on their SNRs as these remain low for both large and small negative delays.
The proposed path selection criterion selects the earliest path whose SNR exceeds a configurable threshold γ [dB]. Using the estimated SNR from (37), the index of the earliest path l γ exceeding the threshold is determined as The proposed ESPRIT discriminator uses the l γ -th path to determine the delay offset from (27) as and the phase offset from (32) as The threshold γ can be derived empirically using the results of the outlier analysis of the discriminator outputs for a given configuration and various SNRs. The threshold depends on the number of pilots P and configured data snapshot size M . Since the outliers are well distinguishable in the evaluation shown in Fig. 4, the threshold is set to be 6 dB above the maximum SNR of the outlier paths (−16 dB) resulting in γ = −10 dB. Although this threshold may also potentially discard a very weak LOS component, signals with such low SNRs are not expected to be tracked in practical positioning scenarios.

V. PERFORMANCE OF JOINT DELAY AND PHASE DISCRIMINATOR BASED ON ESPRIT
The performance of the proposed joint delay and phase discriminator based on ESPRIT coupled with the MDL and ECOD methods is studied. The discriminator is evaluated in two configurations. In the first configuration, denoted as ESPRIT, no path selection criterion is applied to neither MDL nor ECOD and the resulting discriminators are referred to as ESPRIT/MDL and ESPRIT/ECOD. In the second configuration, denoted as ESPRIT-S, a path selection threshold γ = −10 dB derived from the results of Section IV-D is applied to the MDL method. The resulting discriminator is referred to as ESPRIT-S/MDL. No path selection is performed for the ECOD. The EMLP delay and ATAN2 phase discriminators are considered as references in the study. Delay and phase errors are estimated by tracking the CSI-RS pilots of the 5G NR waveform from Section II-A. The ESPRIT design parameter is set to m = 0.24 resulting in data snapshots of length M = 819 · 0.24 = 196 providing a sufficient multipath resolution for the simulations. The correlator spacing of the EMLP discriminator is set to δ = 0.1. The received signal power is assumed to be perfectly known at the receiver.

A. DELAY TRACKING THRESHOLD AND COMPUTATION COMPLEXITY
The delay tracking thresholds ζ of the ESPRIT and EMLP discriminators are determined using (28) and (11), respectively. To derive ζ EMLP , a linearity threshold α = 0.05 is assumed. The results summarized in Table 2 show that the ESPRIT discriminator has three orders of magnitude larger tracking threshold than the EMLP. Such a large threshold allows to either reduce the update rate of the tracking loop or to track delay offsets with high variances without causing the tracking loop to lose the lock of the signal.
The main drawback of the ESPRIT discriminator is its computational complexity that is higher than that of the EMLP due to numerous matrix operations involving inversion, singular value decomposition, and eigenvalue computation [24]. To evaluate the computational overhead, the average times needed to process the CSI-RS pilots of a single slot of the 5G NR evaluation waveform are measured. The algorithms are implemented in the Python programming language and executed in a computer running the Intel Core i5 processor at 2.4 GHz. Table 2 shows that ESPRIT requires three orders of magnitude more processing time than the EMLP. The processing time of the ESPRIT discriminator (97.11 ms) is in this implementation also much larger than the duration of the 5G radio slot (0.5 ms). This limits the real-time usage of ESPRIT in the receivers if a high loop update rate is desired. However, given that the ESPRIT discriminator provides a large delay tracking threshold, the update rate might be reduced. The performance of the algorithm could also be optimized with tailored hardware and software.

B. DELAY AND PHASE ERRORS IN ABSENCE OF MULTIPATH
The delay and phase errors of the discriminators are studied in the AWGN channel in the absence of multipath. This evaluation helps to understand the performance of the discriminators in the multipath-free environment and study the lower bounds of the errors. In the case of the delay error, the results are compared to the CRLB in (10). The evaluated SNR ranges from −18 dB to 18 dB. Although the SNR range goes lower than the expected values in a vicinity of a cell, it helps to understand the performance limits of the discriminators and generalize the results. Each discriminator is simulated 10 4 times for a given SNR. For comparison, the ESPRIT discriminator fixed to the true number of paths (L = 1) is used in the evaluation. The standard deviations of delay and phase errors of the evaluated discriminators are shown in Figs. 5a and 5b, respectively. The non-zero probabilities of the channel orders estimated by the MDL and ECOD methods are shown in Figs. 6a and 6b, respectively. Values reported asL = 0 indicate that the estimator is unable to identify any path and that ESPRIT produces neither delay nor phase outputs.
The standard deviation of the delay error of the ESPRIT discriminator fixed to the true number of paths approaches the CRLB for the SNR above −12 dB as shown in Fig. 5a. Fig. 5b shows a similar behavior of the phase error of the ESPRIT discriminator fixed to the true number of paths as it approaches the ATAN2 discriminator representing the lower bound of the error.
Figs. 5a and 5b show that the standard deviations of neither delay nor phase errors of the ESPRIT/MDL discriminator approach the CRLB or ATAN2. This is due to the presence of the delay outliers studied in Section IV-D. The MDL method overestimates the channel order with values as high asL = 3 regardless of the SNR as shown in Fig. 6. When using the MDL, taking the ESPRIT delay and phase outputs without applying a path selection criterion is not recommended for positioning.
The ESPRIT-S/MDL discriminator employing the proposed path selection criterion from Section IV-D correctly removes the phase and delay outliers of the ESPRIT/MDL as the standard deviations of the delay and phase errors converge to the CRLB and ATAN2 for the SNR above −12 dB as shown in Figs. 5a and 5b. The ESPRIT-S/MDL discriminator does not produce any estimates for the SNR below −12 dB as no paths above the configured threshold γ = −10 dB are detected. This is not considered a problem as the removed estimates are in the SNR range in which the performance of the ESPRIT discriminator fixed to the true  number of paths does not approach the CRLB and ATAN2 anyways.
Figs. 5a and 5b show that the standard deviations of the delay and phase errors of the ESPRIT/ECOD discriminator also converge to the CRLB and ATAN2 for the SNR above −12 dB. Fig. 6b shows that although the ECOD method may overestimate the channel order for the SNR below −10 dB by identifying the majority of eigenvalues as signal components (L ≥ 195), the method remains stable in a sufficiently high SNR. The path selection criterion is not needed by ESPRIT when the ECOD is used.

C. DELAY AND PHASE ERRORS IN PRESENCE OF MULTIPATH
To study the delay and phase errors of the discriminators in the presence of multipath, a multipath error envelope is evaluated by simulating a two-ray channel model containing the LOS component and a single reflection. Although the two-ray channel is a very particular model, its use is widely accepted by the GNSS community as a generally representative way to assess the robustness and sensitivity of a given estimator to multipath. Likewise, in the case of 5G NR positioning, this model is specific but beneficial to generally understand the fundamental impact of a single path on the discriminator performance. The power of the reflected path is configured to −3 dB of the power of the LOS component. The excess path delay ranges from 0 to 3 samples. In the delay error evaluation, the phase of the reflection is set to 0 rad and π rad corresponding to constructive and destructive multipath combination, respectively. In the phase error evaluation, the phase of the reflection is set to ± 3 4 π rad. The multipath error envelope of the EMLP and ATAN2 discriminators is usually evaluated in a noiseless channel as the mean error is independent of the noise. However, this cannot be done with the ESPRIT discriminator as the channel order estimation depends on the SNR. As such, the mean and root-mean-square (RMS) of the delay and phase errors are evaluated in the SNR equal to 12 dB. Due to the unsatisfactory performance of the ESPRIT/MDL discriminator in the AWGN channel without multipath presented in Section V-B, the MDL without the path selection criterion is not considered in this evaluation.
The observed multipath delay error envelopes of the EMLP, ESPRIT/ECOD, and ESPRIT-S/MDL discriminators are shown in Fig. 7a. The related root-mean-square delay errors are shown in Fig. 7b. The observed multipath phase error envelopes of the ATAN2, ESPRIT/ECOD, and ESPRIT-S/MDL discriminators are shown in Fig. 8a. The related root-mean-square phase errors are shown in Fig. 8b. Fig. 7a shows that the ESPRIT/ECOD and ESPRIT-S/MDL delay discriminators perform similarly to the EMLP for small multipath excess delays. Above a given multipath excess delay, the ESPRIT discriminators start resolving the reflection in the channel and the mean delay errors rapidly converge to zero in both constructive and destructive cases, achieving better performances than the EMLP. Fig. 7b shows that the root-mean-square delay errors of both ESPRIT discriminators also converge to very small values once a given multipath excess delay is reached.
Similar to the delay errors, the mean phase errors of the ESPRIT/ECOD and ESPRIT-S/MDL rapidly converge to zero above a given multipath excess delay as shown in Fig. 8a, outperforming the ATAN2 discriminator. The ESPRIT-S/MDL discriminator achieves lower root-meansquare phase errors than the ATAN2 as shown in Fig. 8b.
The convergence of the mean delay and phase errors to zero occurs for the ESPRIT-S/MDL discriminator at a noticeably smaller excess delay than for the ESPRIT/ECOD as shown in Figs. 7a and 8a. To understand the convergence, the mean delay error of the ESPRIT-S/MDL is shown in Fig. 9a for the constructive multipath for the excess delay between 0.40 and 0.54 samples. The related non-zero probabilities of the total number of paths above the SNR threshold (L γ ) outputted by the discriminator are shown in Fig. 9b. It can be seen that the convergence of the mean delay error to zero coincides with the increasing probability of resolving the reflection (L γ = 2). To describe the convergence, the multipath resolution τ MR is defined as the multipath excess delay at which the probability of resolving all the paths in the channel is P(L γ = 2) = 0.5. For example, the MDL method achieves a multipath resolution of τ MR = 0.470 samples for the SNR equal to 12 dB as shown in Fig. 9b.
To compare the multipath resolution of the MDL and ECOD, τ MR of both channel order methods is evaluated for the SNR ranging from 6 dB to 18 dB. The results obtained for constructive and destructive multipath are shown in Fig. 10. The MDL method achieves a better multipath resolution than the ECOD in both cases, explaining the lower multipath excess delay needed for the convergence of the delay and phase errors in the case of the ESPRIT-S/MDL shown in Figs. 7 and 8. The ESPRIT-S/MDL discriminator provides superior performance for positioning than the ESPRIT/ECOD.

D. DELAY AND PHASE ERRORS IN TERRESTRIAL 3GPP CHANNEL MODEL
The evaluation of the delay and phase errors of the discriminators in a realistic terrestrial 5G NR multipath environment is performed using the 3GPP TDL-E channel model containing 14 taps [31]. The Rician K-factor is set to K = 0 dB retaining half of the power of the signal in the multipath components. The channel is generated 10 4 times. In each channel realization, the phase of each tap is randomly generated following a uniform distribution ϕ ∼ U (−π, π). The SNR is set to 12 dB. The root-mean-square delay and phase errors are evaluated for the values of the delay spread (DS) between 20 ns and 200 ns. The DS is used to scale the normalized path delays from the TDL-E model to obtain the desired multipath excess delays. The data snapshot length of the ESPRIT discriminator (M = 196) is considered sufficient  Figs. 11a and 11b show that the proposed ESPRIT-S/MDL discriminator achieves lower root-mean-square delay and phase errors than the EMLP and ATAN2 discriminators for all evaluated delay spreads. The exception is the phase error for a very low delay spread of DS = 20 ns as shown in Fig. 11b, where the ESPRIT-S/MDL discriminator remains more biased than the ATAN2. This is attributed to a very small path separation preventing the ESPRIT algorithm to properly distinguish the paths in the channel. The ESPRIT-S/ MDL achieves noticeably better performance in a realistic multipath environment than the traditionally used EMLP and ATAN2 discriminators.

E. COMPARISON OF 5G NR AND LTE
The performance of the ESPRIT discriminator in 5G NR is compared to the previous generation of cellular signals in the AWGN channel in the absence and presence of multipath. In the case of LTE, the CRS pilots of the LTE downlink evaluation waveform from Section II-B are tracked.   Fig. 12. The mean delay errors of the ESPRIT-S/MDL observed in the two-ray channel model are shown in Fig. 13 for the SNR equal to 12 dB. Fig. 12 shows that the discriminators achieve an order of magnitude better standard deviation of the delay error in 5G NR than in LTE when no multipath is present. The tracking is also possible for a lower SNR in 5G NR. Fig. 13 shows a better performance of 5G NR in the two-ray channel model as it achieves a sub-meter delay error. Better performances in both scenarios are obtained thanks to the higher number of available pilots and higher signal bandwidth.

VI. CONCLUSION
A joint delay and phase discriminator based on ESPRIT was proposed for positioning with 5G NR downlink cellular signals. The study of the effect of the MDL channel order overestimation on the ESPRIT algorithm led to the introduction of the path selection criterion based on the SNR. The ESPRIT-S/MDL discriminator with the path selection criterion eliminates the delay and phase outliers produced by the ESPRIT/MDL without the criterion, confirming its suitability for positioning purposes. The ESPRIT-S/MDL achieves lower mean and root-mean-square delay and phase errors than the EMLP and ATAN2 discriminators in the presence of multipath in the two-ray channel model after a given multipath excess delay. The ESPRIT-S/MDL also achieves lower root-mean-square delay and phase errors in a representative 5G channel model. The higher computational complexity of the ESPRIT algorithm could be optimized with tailored hardware and software. FRANCESCA ZANIER received the M.Sc. degree in telecommunications engineering and the Ph.D. degree from the University of Pisa, Italy. She joined the European Space Agency, ESTEC, Noordwijk, The Netherlands, in 2009, where she is currently a Radio Navigation System Engineer with the Radio Navigation Systems and Techniques Section of the Technical Directorate. Her research interests include signal processing, estimation theory, GNSS receivers, and signals and navigation applications.
LIONEL RIES is currently the Head of the Radio Navigation Systems and Techniques Section, Directorate of Technology, Engineering and Quality, ESA, the European Space Agency. He joined ESA, as a Radio Navigation System Engineer, in 2016, supporting activities on future navigation systems and concepts (both at infrastructure and user segment level) as well as on standardization of 5G positioning in 3GPP. Beforehand, he worked with CNES (French Space Agency), first supporting GNSS-related research and development activities and European Commission on Galileo, then leading CNES Section for localization/navigation signal and equipment. VOLUME 9, 2021