Performance Analysis for AOA-based Localization under Millimeter-Wave Wireless Networks

Millimeter-Wave (mmWave) communication is a promising solution for achieving high data rate and low latency in 5G wireless networks. Since directional beamforming and antenna arrays are exploited in the mmWave networks, accurate angle-of-arrival (AOA) measurements can be obtained and utilized for localization purposes. In this work, we consider the AOA-based positioning for the mmWave networks using stochastic geometry and analyze how the Cramér-Rao lower bound (CRLB) is affected by the spatial distribution of nodes, including the target and participating anchors. In order to apply the CRLB on a network setting with random node locations, we propose an accurate approximation of the CRLB using the (cid:100) L / 4 (cid:101) -th value of the ordered distances where L is the number of participating anchors. These ﬁndings provide us deep insight into optimum network design that meets speciﬁed localization requirements.


I. INTRODUCTION
Positioning techniques received considerable attention due to the emergency of internet-of-things, which can be utilized to ameliorate the user experience of location-based services, including indoor navigation, asset tracking, and simultaneous localization and mapping (SLAM) [1]- [3].In the 5G wireless networks, accurate angle-of-arrival (AOA) information can be obtained by leveraging the massive antenna array and highly directional transmission [4].Thus, AOA-based localization is regarded as a promising candidate to achieve high-precision localization for 5G networks.

A. MOTIVATION
Millimeter-Wave (mmWave) is a promising technology for 5G wireless networks to meet the requirements of large bandwidth and high carrier frequency [5].Due to its high transmission rate, mmWave signals enable us to efficiently detect and resolve the multipath components, which provide us an opportunity to achieve submeter level localization accuracy [6], [7].However, the coverage area of the mmWave signals is limited due to its fast attenuation.Blockage and multipath effects also take significant roles in the mmWave signal propagation, greatly increasing localization error [8].In addition, the cost of the mmWave-based system is The associate editor coordinating the review of this manuscript and approving it for publication was Qinghua Guo .exorbitant due to the deployment of the massive antenna array and a large number of mmWave anchors.The experimental verification for the mmWave-based system is complicated and time-consuming, which brings challenges to examining the localization performance for the mmWave-based system.
Motivated by these limitations, we attempt to provide an analytical tool to quantify the localization performance for the AOA-based positioning and examine how its performance is affected by the mmWave channel over diverse channel propagation conditions.To characterize the fundamental limits of the mmWave-based localization system, Cramér-Rao lower bound (CRLB) is applied, which indicates the optimal localization performance for any unbiased estimator.However, the CRLB is generally assessed by given specified anchor geometry, which cannot reflect how the random network configurations affect this metric.Thus, stochastic geometry is applied to allow random spatial node locations, including the target and anchor locations.Based on our devised result, the system designers can examine how different network parameters and the overall setup influence localization performance without prolix and complicated experimental verification.methods, mmWave based localization can be categorized into proximity, fingerprinting, and geometry-based [9].A singleanchor localization method for mmWave wireless networks is presented in [10], which leverages time delay and angle information to locate the target of interest.In [11], the authors utilized the received signal strength (RSS)-based fingerprinting algorithm to estimate the target location using commercial mmWave Wi-Fi.Due to the fast attenuation of mmWave signals, the localization performance under mmWave networks is subject to none-line-of-sight (NLOS) components, and the impact of the NLOS propagation is introduced in [12].Furthermore, a neural networks-enhanced hybrid localization framework for the mmWave wireless networks is proposed in [13] to achieve high-precision localization purposes.
To quantify the localization error, L-localizability and CRLB are utilized to analyze the performance of AOAbased positioning.The L-localizability is first introduced in [14] to identify whether a target is localizable with a sufficient number of participating anchors.In [14], the authors modeled a cellular network with a homogeneous Poisson point process (HPPP) and applied a dominant interferer analysis to derive an expression for L-localizability.With an emphasis on mmWave networks, a reflection hyperbola model is proposed in [15] to analyze the target localizability, where the authors considered how the first-order reflection path affects this metric.Considering the CRLB, a performance benchmark is provided to quantify localization performance for the mmWave-based positioning system [16].The impacts of the massive antenna array and multipath propagation on the CRLB under mmWave wireless network are examined in [17].However, these works assumed that the locations of the target and anchors are fixed.To examine the localization performance with random network settings, the authors of [18] derived an accurate approximation for the time-of-arrival (TOA)-based CRLB by utilizing the second-largest inter-nodal angle, which introduced a new paradigm to analyze the localization performance in the area of stochastic geometry.For RSS-based localization, [19] applied u-statistics to approximate CRLB, which is only accurate when numerous anchors participate in the localization procedure.In contrast to TOA/RSS-based localization, a comprehensive analysis of AOA-based localization using stochastic geometry cannot be found in the open literature, which motivates us to extend the CRLB paradigm in [18] to the AOA-based positioning.

C. CONTRIBUTION
We aim to exploit the localization error in a random mmWave network using stochastic geometry and provide insight into how the network parameters affect the localization performance.The stochastic geometric framework in [18] is applied to analyze the impact of random anchor placement on CRLB.The main challenges of this paper are how to utilize a single random variable to approximate the AOA-based CRLB and how to bridge the gap between this metric and mmWave channels.In addition, the NLOS propagation takes a significant role in the mmWave signal transmission.Thus, a suitable mmWave model is required to analyze how the NLOS effect influences the localization performance.The main contributions of this paper are listed as follows:

1) L-LOCALIZABILITY
A tractable expression of L-localizability is derived for the mmWave networks, where the directional antenna, Nakagami fading, and the NLOS effect's impacts on mmWave-based localization systems are assessed.An asymptotic bound for the distribution of the L-localizability is provided to track the performance of AOA-based localization under mmWave networks.Thus, the system designers can have insight into how to optimize the network environment to improve the localization performance.

2) DISTRIBUTION OF AOA-BASED CRLB
In [18], the authors applied stochastic geometry to TOA-based localization, which open a new paradigm to evaluate the localization performance under random wireless networks.To the best of our knowledge, there is no prior work that investigates the impact of random geometry on AOA-based localization.Motivated by [18], we accurately approximate the AOA-based CRLB using the L/4 -th ordered distance, and merge the impacts of mmWave channels into the devised result.The obtained distribution provides an average performance bound for the system designers how to achieve the stated localization requirements by selecting the most appropriate network parameters.
The rest of this paper is organized as follows.The system model is presented in Section II, localization performances are analyzed in Section III, and numerical results are provided in Section IV.Finally, Section V concludes the paper.

II. SYSTEM MODEL
In this section, we present the underlying model for analysis.

A. NETWORK MODEL
We consider downlink transmission in a mmWave network where the locations of anchors follow an HPPP1 with anchor density λ.Let us denote the locations of the anchors as ψ = [x, y] ∈ R2 , and the distance between an arbitrary anchor and target as r = ||ψ||, where • is the Euclidean norm.Since the channel gains of NLOS paths are typically weaker than those from the line-of-sight (LOS) [23], we make the following assumption.
Assumption 1: The LOS region of the target can be approximated by a LOS ball b(O, r L ), which only contains the LOS anchors.The anchors outside the LOS ball may be a LOS anchor or a NLOS anchor with a certain probability. 2 Remark 1: The number of anchors inside the LOS ball, i.e., L, is also regraded as the average number of anchor the target

can be observed. Conditioned on the number of participating anchors, coverage area of the LOS ball can be computed based on the Theorem 2 from [26].
As illustrated in Fig. 1, we assume that the target is located at the origin O and the anchors are randomly distributed in the R 2 plane.Given that L anchors participate on the localization, we will denote the nearest and furthest distances between the associated anchor and target in the LOS ball as r 1 and r L , respectively.Based on the model, the distance distribution of the anchors residing in the localization ball is given by [27] where r is an unordered parameter with r 1 ≤ r ≤ r L .
With the particular interest of the distribution of l-th nearest anchor, we sort the distance between the anchor and target in an ascending order.Consequently, the probability density function (PDF) and cumulative distribution function (CDF) of the l-th nearest anchor are given by [28]: where r l is an ordered parameter with 1 ≤ l ≤ L.

B. CHANNEL MODEL
We consider a widely adopted mmWave channel model, where the probability of the associated anchor to be a LOS transmitter, referred to as LOS probability, is given by [29] where r is the link distance between the target and the anchor, the NLOS probability is P NLOS (r) = 1 − P LOS (r), A = 18m and B = 63m for urban environment.Remark 2: In Fig. 2, we plot the LOS/NLOS probability across different link distances.It is evident that as the link distance between the target and anchor increases, the corresponding anchor is more likely to be a NLOS transmitter.It indicates that the signal transmitted over a long-distance is more susceptible to blockage and multi-path effects.Furthermore, we can observe that P LOS is close to 1 when the distance r is less than 20m, which justifies Assumption 1.
The target has a single antenna, whereas each anchor is equipped with a directional antenna array with N t elements and all anchors transmit with a constant power P t .We use a subscript q to denote LOS/NLOS link, i.e., q = {L, N } for LOS and NLOS link, respectively. 3The received signal power from the l-th anchor is given by N t G T h l r α q l , where G T is the antenna gain at the anchor, h l is the small-scale fading gain that follows a Gamma distribution, i.e., h l ∼ (M q , 1 M q ) with parameter M q , and α q denote the path-loss exponent; α L for the LOS link and α N for the NLOS link, respectively.The antenna gain G T is modeled by the flat-top antenna radiation pattern as follows [30] G T = G 1 for main lobe with prob.

C. AOA MEASUREMENT
Let us denote the AOA between the l-th anchor θ l and the location of target where we used subscript t and l to indicate the target and the l-th anchor, respectively.The AOA measurement at l-th anchor is modeled by where n AOA,l imitates the measurement error with variance σ 2 AOA,l .The vector representation of the AOA measurements is given by where r AOA , n AOA , and f AOA (ψ t ) are respectively defined by

III. PERFORMANCE ANALYSIS
In this section, we characterize the L-localizability and CRLB of AOA-based localization for mmWave networks.

A. L-LOCALIZABILITY
L-localizability was first introduced in [14], which is defined as the probability of having at least L participating anchors in the localization procedure.We assume that the associated anchor can align the beam by the implementation of sophisticated beam training protocols [31].The received signal-tointerference-plus-noise ratio (SINR) at the target from the L-th anchors located in the LOS ball can be expressed as where σ 2 n = σ 2 t βP t N t is the normalized noise power, σ 2 t is the thermal noise power, β = λ w 4π 2 is the path-loss intercept with antenna wavelength, λ w , and I is the aggregate interference.In [14], the L-localizability is defined for a given SINR threshold τ as The aggregate interference I in (9) consist of two components; I in and I out , which is the interference from the nodes inside and outside the LOS ball respectively.Then we have where the indicators a i , b j ∈ {0, 1} represent the network load, the probability P (a i = 1) = q, P b j = 1 = p correspond to the anchor activation probability, and the indicators a i , b j are fixed throughout the localization procedure.To simplify the derivation, we will express the interference inside the LOS ball as follows Let us denote Based on the dominant interference analysis [14], I 1 and Îin can be approximated by the following Lemma.

Lemma 1: I 1 and Îin can be approximated by its mean
where L is the number of active anchors inside the LOS ball, and E[h] is the mean channel gain.Note that (13) are functions of r 1 and r L .Proof: See Appendix A. Based on the Lemma 1, the denominator of SINR L can be expressed as where step (a) follows from (11), step (b) follows from (12), and we substitute σ 2 = I 1 + Îin + σ 2 n in step (c).Based on Assumption 1, I out consists of interference generated by both LOS or NLOS anchors as expressed below [23]- [26] where I L and I N represents the interference generated by the LOS and NLOS anchors, respectively, that are located outside the LOS ball.Then the Laplace transform of I out is where ω s|r L , α q is defined below The term E h,G T [exp (−shG T r α q )] is derived as follows where we applied (4) in step (a) and utilized the Laplace transform of the Gamma random variable in step (b) [29].
To conclude, the L-localizability in (10) can be evaluated as where 14) and Alzer's inequality [32] are used in step (a), a change of variable is used in step (b), i.e., µ = ηr α L G 1 , and E I out e −iτ µI out = L I out (iτ µ) in step (c).Note that L I out (s) in ( 16) is a function of r L , whereas σ 2 = I 1 + Îin + σ 2 n is a function of both r 1 and r L .Remark 3: When the parameter M L is equal to 1, (19) provides the localization performance for Rayleigh fading.Different from the approximation method in [14], we only approximate the interference inside the LOS ball by its mean value.We can observe in Fig. 3 that our devised method is slightly better than that in [14].

B. APPROXIMATION OF CRAMER-RAO LOWER BOUND
We derive the AOA-based random CRLB using the FIM, denoted by I AOA (ψ t ), as [9] where C AOA represents the noise covariance matrix and f AOA (ψ t ) is the angle vector with respect to ψ t , given by Remark 4: In contrast to the AOA-based CRLB, the FIM of TOA-based localization can be represented by angular information θ i as [18] However, the AOA-based CRLB cannot be simplified as (22) because the sufficient statistics of ( 21) requires both x and y coordinates.In order to resolve this issue, [19] used the u-statistics to compute the joint distribution of AOA-based CRLB.However, this method is accurate only when the number of participating anchors is sufficiently large.
To evaluate the average localization performance, we assume σ AOA to be known and identical for each anchor [18], [19] Then, the standard deviation of AOA measurement is computed as [16], [33] σ AOA = E(r) 2 10 E(SNR)/10 , where E(r) is the mean distance from all participating anchors, E(SNR) is the average signal-to-noise-ratio (SNR) for the mmWave network, E[G] is the mean channel gain, N 0 is the spectral density of the white Gaussian noise, and W TOT is the total system bandwidth.

Remark 5:
The assumption in (23) has been validated and justified in [14], [18], [19].In this paper, we attempt to compute the average localization error using the AOA-based positioning, thus we assume that the AOA measurement is statistically independent with its means and same variance.
By using I AOA (ψ t ) in (20), the CRLB of AOA-based localization can be expressed as [18], [19] where σ 2 AOA is the approximated AOA variance; Q 1 and Q 2 are given by In order to characterize the distribution of the CRLB in ( 25), (26) needs to be simplified.In the following Lemma, we use asymptotic bounds to approximate Q 1 and Q 2 , which enable us to represent the AOA-based CRLB using a single variable.Lemma 2: Q 1 and Q 2 in ( 26) can be approximated as Proof: See Appendix B. Based on Lemma 2, (25) can be expressed as In the following Proposition, we will further simplify (28).Proposition 1: Let us assume that the link distances are sorted in an ascending order, i.e., r where L is the number of participating anchors and r L/4 is the L/4 -th link distance in the ordered set R = [r 1 , . . ., r L ].
In the following subsections, we employ two methods to justify the approximation in ( 29), namely, mutual information and root-finding.Then in Section IV, we use Monte Carlo simulation to validate the accuracy of (29).

1) MUTUAL INFORMATION
To justify (29), we will approximate D Q 1 − Q 2 by an optimized approximation of r k , denoted as r * , given by where the term r * minimizes the mean square error (MSE) Since R is a sorted set, the expression in ( 29) is bounded by In [18], the authors introduced an approximation scheme where the best approximation was achieved by maximizing the mutual information.Furthermore, feature selection by maximizing the mutual information is a well know technique in machine learning and statistics [34]- [36].Motivated by these approaches, we adopt a heuristic method and perform an iterative search to find the k-th term r k , 1 ≤ k ≤ L, as where I (D; r k |L) is the mutual information (MI) between D and r k for a given L. The MI is defined in terms of the differential entropies h(D|L) and h(D|r k , L) as [18] I (D; where the supports of r k and d are denoted by R k and D, Through extensive simulation, we observed that the L/4th distance achieves the maximum mutual information as illustrated in Fig. 4, thus we can use the L/4 -th distance to approximate D in (30), justifying Proposition 1. Furthermore, we observed that as the number of anchors increases, the mutual information trends to saturate for L ≥ 9. 2) ROOT-FINDING Method [37] We may use root-finding algorithm as an alternative approach to approximate (29), where the optimized approximation of r k is determined as the root of the following function Since the range of r is bounded between r 1 ≤ r ≤ r L and (33), the following property holds; f (r L ) • f (r 1 ) < 0. By applying Newton's iterative method, we found that the roots of ( 36) are located at the 5L 16 -th and L/4 -th distances, which validates (29) and Proposition 1.
Based on Proposition 1, the CRLB in (28) of the AOA-based localization can be approximated as follows We note that the CRLB in ( 37) is a function of r L/4 , which is a random variable.In the following proposition, we provide the distribution of CRLB based on the distance distribution of r L/4 [27].To remain consistency with the root mean square error, we derive the distribution of the squared-root CRLB, instead of CRLB.Proposition 2: Let us assume that the number of participating anchors L and the standard deviation of AOA measurement σ AOA are known.Based on [27], the CDF of a squared-root CRLB is given by where F r n (r) is the CDF of the n-th distance [27, eq. ( 9)].Remark 6: In Fig. 5, we plot the CRLB curve evaluated by (25) (labeled as 'original CRLB'), (37) (labeled as ' L/4 -th distance'), the 1-st and the L-th ordered distances.Note that there is a wide gap between the approximations of CRLB using the L-th, 1-st distance and the original CRLB.However, the approximation based on (37) accurately matches the exact CRLB, which justifies Proposition 1 and Proposition 2.

IV. SIMULATION RESULTS
In this section, we evaluate the L-localizability and AOA-based CRLB for mmWave networks, compare simulation results to numerical results, and investigate the impact of network parameters on the localization performance.We used MATLAB to simulate a realization of the node deployment 10 6 times.We assumed that the anchors are randomly distributed by a HPPP with density λ = 2/ √ 3 × 500 2 m 2 , bandwidth W TOT = 1 GHz, transmit power P T = 1 Watt, antenna spacing d = λ w /4, path-loss intercept β = (λ w /4π) 2 , and antenna gain G 1 = 1 and G 2 = 0.2 with its associate probability p 1 = 0.4 and p 2 = 0.6, respectively.
In Figs.6-7, we assessed the impact of network parameters on the performance of L-localizability.Specifically, we first   plotted the L-localizability versus the SINR threshold across different number of participating anchors L in Fig. 6.Then we fixed L and compared the performance of L-localizability across different Nakagami fading parameter M L in Fig. 7.The simulation results are plotted by circle markers with a line, whereas the analytical results are illustrated by dashed curves.We observed that the numerical results are very close to the analytical results, justifying the analysis.In Fig. 6, we note that reducing the number of participating anchors increases the L-localizability P L (τ ).This phenomenon is understandable as a large number of participating anchors leads to a higher interference, degrading the overall performance.In Fig. 7, we observed that the L-localizability performance escalates with a higher M L value, which represents the Nakagami parameter of the LOS link.As M L increases, the LOS channel becomes more deterministic while the NLOS channel condition remains fixed, which results in a higher SINR in (9) and better P L (τ ) in (10).Note that the case of M L = 1 corresponds to Rayleigh fading environment.
In Figs.8-9, we evaluated the distribution of the squared-root CRLB P( √ CRLB ≤ s) and compared its perfor-mance across various configuration of network parameters.
The numerical results are evaluated by using (38) in Proposition 2 and plotted by solid curves, whereas the simulation results are illustrated by dotted curves.We used the approximation based on the L/4 -th distance and observed that the numerical results closely match simulation results, validating the accuracy of Proposition 2. In Fig. 8, we observed that P( √ CRLB ≤ s) is an increasing function of L, indicating that the localization error declines for a large number of participating anchors.Hence, a network designer looking to improve the localization accuracy needs to optimize the network environment to ensure a sufficient number of anchors participate in the localization procedure.For example, a joint optimization problem can be designed that maximizes the CRLB distribution P( √ CRLB ≤ s) given that the L-localizability P L (τ ) is above certain threshold.Fig. 9 compares the localization performance for various Nakagami parameter M L and the number of antenna elements N t .We observed that increasing M L parameter improves the localization performance, which provides similar result in Fig. 7. Furthermore, we consider the impact of the number of antenna N t on the localization performance.For a larger N t , the normalized noise power σ 2 n = σ 2 T βP t N t will decrease, resulting in a higher P( √ CRLB ≤ s).This indicates that the localization performance can be enhanced by adding more antenna elements in the anchors, which raises the implementation cost for each anchor.

V. CONCLUSION
This paper presents L-localizability and random AOA-based CRLB for mmWave wireless networks, where we used stochastic geometry to account for all possible positioning scenarios.We derived the L-localizability and random CRLB for AOA localization while considering the flat-top antenna radiation pattern and Nakagami fading.We provided numerical results to validate the analytical derivation and investigated the impact of various network parameters, e.g., network load, fading parameters, number of anchors, number of antenna elements, on the localization performance.The analytical framework developed in this paper offers an accurate tool to evaluate the localization performance of mmWave wireless networks without relying on extensive simulation.The network operators can use the asymptotic bounds to optimize the network parameters and find the best deployment of the anchors to ensure localization performance.

APPENDIX A
The interference inside the LOS ball except for the nearest anchor is approximated as [14]: +(y i −y t ) 2 ≥ 2(x i −x t )(y i −y t ) is applied to step (a) and Cartesian coordinates is converted to polar coordinate in step (b).As the polar coordinate of (x i , y i ) is given by we derive the upper bound of Q 1 as where we will maximize Q 1 with respect to the phase {θ i } for a given distance {θ i }.Then, (42) can be expressed as and the second order derivative of Q 1 has a negative value at ξ * as follows Thus, Q 1 is upper bounded by This completes the proof.
B. RELATED WORKLocalization based on the mmWave signals is thoroughly investigated in the existed literature.Based on the processing VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/

FIGURE 1 .
FIGURE 1. System model of mmWave wireless network.

FIGURE 2 .
FIGURE 2. The LOS and NLOS probability in mmWave wireless networks.

FIGURE 4 .
FIGURE 4. Impact of distance selection on mutual information.

FIGURE 8 .
FIGURE 8. Impact of number of anchors on the distribution of √ CRLB when α L = 2.1, N t = 64, M L = 5 and q = 0.75.