A Fisher Information Analysis of Joint Localization and Synchronization in Near Field
AA Fisher Information Analysis of Joint Localizationand Synchronization in Near Field
Henk WymeerschDepartment of Electrical EngineeringChalmers University of Technologye-mail: [email protected]
Abstract —In 5G communication, arrays are used for bothpositioning and communication. As the arrays become larger, thefar-field assumption is increasingly being violated and curvatureof the wavefront should be taken into account. We explicitlycontrast near-field and far-field uplink localization performancein the presence of a clock bias from a Fisher informationperspective and show how a simple algorithm can provide acoarse estimate of a user’s location and clock bias.
I. I
NTRODUCTION
Cellular localization has largely relied on measurements oftime-difference-of-arrival in either uplink or downlink. Suchmeasurements can cope with the clock bias of the user, butrequire multiple base stations (BSs) [1]. With 5G, where largearrays are used to provide improved spectral efficiency, anglemeasurements have become possible [2], [3]. Estimating theangle-of-arrival (AOA) at different BSs, the user’s location isdetermined by a set of bearing lines, so that localization can beperformed without any stringent synchronization requirements[4]. When both user and base station are equipped with largearrays, the user’s position and orientation can be inferred[5]. Extensive studies have been performed to assess thefundamental performance of array-based positioning [6]–[8](through the Cram´er-Rao bound (CRB)), as well as to developpractical algorithms [5], [9]. All these works assume far-fieldpropagation (plane wave assumption), where the user is faraway from the BS. With communication systems beyond 5Gtargeting novel technological enablers, such as large intelligentsurfaces and extreme aperture arrays [10], the far-field prop-agation condition may be violated, requiring us to revisit themodels, performance characterization, and algorithm design.Such activities have now started in communication [11], [12]and radio localization [13] and are the main topic of this paper.Near-field localization dates back around 30 years in thecontext of source localization. The early work [14] studied theimpact of an imperfectly calibrated array on near-field sourcelocalization and a calibration method was proposed, while [15]estimated the direction-of-arrival (DOA) of multiple sourcesusing the MUSIC algorithm and a maximum likelihood (ML)approach. The latter was shown to be superior in low SNRconditions, though comes at a significant complexity cost.In [16], an ESPRIT-based method was proposed and theperformance was theoretically determined. In [17], a multi-source CRB was derived for stochastic sources, highlightingthe benefits of centro-symmetric arrays. An overlapping sub- array approach was proposed in [18] for low complexity rangeand bearing estimation. In [19], time-varying sources werestudied in the narrowband regime in terms of the CRB. Rangeand bearing estimation were also treated in [20], based onsparse recovery techniques. The localization of near- and far-field sources was proposed in [21]. A simplified CRB for near-field positioning was derived in [22], as well as an algorithmthat directly exploits the wavefront curvature for positioning.In contrast to the above works in the narrowband regime, [23]considered spatial wideband signals (where signals arrive atdifferent antenna elements with different resolvable delays)and derived the CRB and an ML estimator. The extension[24] relied on an expectation-maximization method, which iscomputationally less demanding than ML. Positioning usinglarge intelligent surface was considered in [13], which showedthat the CRB reduces quadratically in the size of the array.Most of these works rely on second-order statistics and arethus data-intensive.In this paper, in contrast to the above works, we con-sider not only the position but also the clock bias of thetransmitter to be unknown. This leads us to investigate jointlocalization and synchronization in near-field. Moreover, wedo not rely on second-order statistics and instead exploit thecommunication signal directly. Our main contributions are:(i) a Fisher information analysis of uplink near-field jointlocalization and synchronization with a linear array; (ii) asimple joint localization and synchronization method usingsub-array processing.II. S
YSTEM M ODEL
We consider a 2D scenario with a single-antenna userequipment (UE) with unknown location x = [ x, y ] T (or [ d, θ ] T in polar coordinates, with d = (cid:107) x (cid:107) and θ = arccos( x/ (cid:107) x (cid:107) ) )and a small BS (e.g., indoors in close proximity to the UE)with an N + 1 -element linear array with element spacing ∆ , with locations x n = [ n ∆ , T , n ∈ {− N/ , . . . , N/ } .The UE has an unknown clock bias B (expressed in meters)and sends a known OFDM signal with transmit power P t ata high carrier frequency f c (28 GHz or higher) and a totalbandwidth W = ( K + 1)∆ f , where ∆ f is the subcarrierspacing and K + 1 is the number of subcarriers. For nota-tional convenience, but without any loss of generality, we let The analysis can easily be modified for an array with N elements, withlocations x n = [∆ / n ∆ , T , for n = − N/ , . . . , N/ − . a r X i v : . [ ee ss . SP ] J a n hasereference d
The phase − πξ n [ k ] /λ is shown in Fig. 2, for ∆ = λ/ , where the left figure shows the evolution of thephase with n for various distances d as a function of theantenna index n and the right figure the evolution of the phaseas a function of the subcarrier index k , for different antennaelements. We see from Fig. 2 (left) that when d is small, thephase exhibits a nonlinear behavior as a function of n , whileFig. 2 (right) illustrates that for small d , different antennaelements see different absolute delays δ n (note that this effectis only visible when we have a large bandwidth W ). The complex channel gain at antenna n is α n = ρ n e jψ with ρ n = λ/ (2 πd n ) and ψ = − πd /λ . Similarly, α n,l isthe complex gain of non-line-of-sight (NLOS) path l , ξ n,l [ k ] is a phase increasing with k due to the delay of the NLOSpath l . We will make several assumptions, in order to facilitatecompact closed-form expressions: we assume that α n is notused directly for localization (so it is treated as a separate -50 0 50-200-1000100200 -100 -50 0 50 100-200-150-100-500 Fig. 2. Left: Plot of the signal phase − πξ n [ k ] /λ as a function of theantenna index n for different distances d between transmitter and the phasereference of the receiver. Right: Plot of the signal phase − πξ n [ k ] /λ as afunction of the subcarrier index k for different antenna indices, for d = 0 . m.Parameters: K = 257 , N = 129 , f c = 28 GHz, W = 1 . GHz, ∆ = λ/ . unknown) [5], [25]; | α n | (cid:29) | α n,l | , ∀ l so that the line-of-sight(LOS) path is dominant [11], [17], [19], [23] (the robustnessof the proposed method to multipath will be evaluated inSection V-B); the transmitted signal spectrum is symmetric( | s [ k ] | = | s [ − k ] | ). Our goal is to determine x and B fromthe observation Y ∈ C ( N +1) × ( K +1) , though the proposedmethods can be combined with tracking as in [25] to accountfor user mobility and clock drift [26]. Terminology:
We distinguish the following operating con-ditions. • Far-field vs near-field:
When (cid:107) x (cid:107) > N ∆) /λ , the far-field regime with plane wave assumption holds. When . (cid:112) ( N ∆) /λ < (cid:107) x (cid:107) < N ∆) /λ , we operate inthe radiative near-field zone, where wavefront curvatureis non-negligible [21]. • Narrowband vs wideband:
When
W < c/ ( N ∆) , thesignals at the different antennas are not resolvable in thedelay domain and communication is narrowband. When W > c/ ( N ∆) , we consider the signals to be spatiallywideband in the sense that they are resolvable in the delaydomain at different antennas [23]. • Beam squint:
When
W > f c / , the wavelength ofthe signal varies significantly over its bandwidth, leadingto beam squint. Generally, beam squint implies spatialwideband operation, but not vice versa.Throughout this paper, we assume that the bandwidth issufficiently small to ignore beam squint (i.e., W (cid:28) f c ).III. S TANDARD F AR -F IELD AND N ARROWBAND C ONDITION
A. Model and Fisher Information Matrix
In far-field and narrowband condition, the standard modelreverts to [5], [15] y n [ k ] = α s [ k ] e − j πλ ξ n [ k ] + w n [ k ] , (3)where ξ n [ k ] = − n ∆ cos θ − k ( d − B ) r f , (4)with r f = ∆ f /f c . This model is derived by taking a first-order Taylor series expansion of d n − d around n ∆ /d = 0 [15] in combination with δ n ≈ δ and α n ≈ α . The Fisherinformation matrix (FIM) of η = [ ρ, ψ, d, θ, B ] T is composedf the sum of the FIM for each subcarrier and each antenna J S ( η ) = (cid:80) N/ n = − N/ (cid:80) K/ k = − K/ J n [ k ] where [27] J n [ k ] = (5) N | s [ k ] | (cid:60) (cid:110) ∇ H η ( α e − j πλ ξ n [ k ] ) ∇ η ( α e − j πλ ξ n [ k ] ) (cid:111) in which the derivatives are given by ∇ T η ( α e − j πλ ξ n [ k ] ) = e − j πλ ξ n [ k ] e jψ jα − j πλ α ∂ξ n [ k ] ∂d − j πλ α ∂ξ n [ k ] ∂θ − j πλ α ∂ξ n [ k ] ∂B , (6)where ∂ξ n [ k ] /∂d = − kr f , ∂ξ n [ k ] /∂θ = n ∆ sin θ , ∂ξ n [ k ] /∂B = kr f . We find that, since (cid:60){ j } = 0 , J ,i (cid:54) =1 = 0 (here J i,i (cid:48) refers the entry in J ( η ) on row i , column i (cid:48) ).Hence, we can ignore ρ when determining the FIM of η =[ ψ, d, θ, B ] T . We introduce e i as an all-zero vector with a 1on index i , b = [1 , , − T , and γ = | α | (2 π/λ ) /N . Then J S ( η ) = γ J S + γ J S + γ J S , (7)where J S = (cid:18) λ π (cid:19) E K, E N, e e T (8) J S = E K, E N, r f (cid:20) b T b 0 × (cid:21) (9) J S = E K, E N, ∆ sin θ e e T , (10)in which E K,i = (cid:80) K/ k = − K/ k i | s [ k ] | and E N,i = (cid:80) N/ n = − N/ n i . The directions in which we obtain informationare radially (along the line from the center of the BS array tothe UE) and tangentially (orthogonal to the line between BSarray center and UE). Transformation to the position domain isachieved as follows. With x = d cos θ and y = d sin θ , the FIMof J ( x , B ) is given by J ( ψ, x , B ) = T T J ( η ) T with Jacobian T . Since ψ does not depend on the other parameters, J ( x , B ) = (11) γE K, E N, r f e x e T x + γE K, E N, ∆ y (cid:107) x (cid:107) e x , ⊥ e T x , ⊥ . where e x = [ x/d, y/d, T and e x , ⊥ = [ − y/d, x/d, T . Since e x is orthogonal to e x , ⊥ , this decompositionshows that delay estimation provides radial informationwith intensity γE K, E N, r f and AOA estimation pro-vides tangential information with location-dependent intensity γE K, E N, ∆ y / (cid:107) x (cid:107) . Hence, AOA information is only use-ful for short distances. Moreover, the matrix J ( x , B ) is rank2, so B and x are not identifiable. The Jacobian is given by T =[ e T ; 0 , x/d, y/d,
0; 0 , − y/d , x/d , e T ] in which the “ ; ” operator separates rows in a matrix. B. Localization Algorithm
We organize the observations y n [ k ] in a ( N + 1) × ( K + 1) matrix: Y = α a N +1 (cos θ ) a H K +1 ( δ r f ) S + W (12)where S is a diagonal matrix containing the K + 1 pilotsymbols and a M +1 ( · ) is a vector of length M + 1 withentries [ a M +1 ( β )] m = exp ( j πβm/ ( M + 1)) , for m = − M/ , . . . , M/ . Similar to [5], we exploit the sparse natureof Y by applying a 2D-FFT Z = F N +1 YS H ( SS H ) − F K +1 (13)to the observation YS H ( SS H ) − , where the impact of the pilotsymbols has been removed. This also allows multiple usersto be treated independently if there is no pilot contamination[28]. Here, F M denotes the M × M discrete Fourier transformmatrix. Higher accuracy can be achieved by zero-padding Y and applying larger FFT matrices. The peak of | Z | directlyprovides us an estimate of cos θ and δ r f . As indicated bythe FIM, the parameters are not identifiable, so we can onlylocalize the user when the bias B is known. The complexityof this method is of order O ( N K log KN ) .IV. N EAR -F IELD AND S PATIAL W IDEBAND C ONDITIONS
A. Narrowband Near-field Model and FIM
When signals arriving at different antennas are not resolv-able in delay, the model (1) applies with (2) specialized to ξ n [ k ] = d n + ( kr f − d − kr f B. (14)We can now state the following result. Theorem 1.
In the case of narrowband near-field operation,the FIM of the parameter η = [ ψ, d, θ, B ] T is J N ( η ) = γ A (0)0 E N, J S + γ A (0)0 E N, J S + γ A (2)2 E N, J S + γ J N + γ J N , where A ( j ) i = (cid:80) n ( d/d n ) j +2 n − i and J N = λ π E K, (cid:20) j T j 0 × (cid:21) J N = E K, T T with j = [ − ∆ d cos θA (1)1 + A (1)0 − A (0)0 , A (1)1 ∆ sin θ, T , C , = A (0)0 + A (2)0 − ∆ d cos θA (2)1 + A (1)0 − ∆ d cos θA (1)1 ) + ∆ d A (2)2 cos θ , C , = C , = ∆ sin θA (2)1 − ∆ sin θA (1)1 , C , = 0 .Proof. We readily find that ∂ξ n [ k ] /∂d = ( d − n ∆ cos θ ) /d n − kr f , ∂ξ n [ k ] /∂θ = dn ∆ sin θ/d n , ∂ξ n [ k ] /∂B = − kr f .Substituting these derivatives in (6) and then in (5), we obtainthe desired result.We observe that the first 3 components are similar to thosein the standard case (7), up to a scaling. On the other hand, J N nd J N are due to the near-field propagation. In particular, J N couples the channel phase ψ with the UE distance d and theAOA θ . The diagonal element C , in J N provides additionalinformation on the distance, which allows J N ( η ) to becomefull rank. This information is due to the dependence of thecurvature on the UE location, but not on the bias. B. Spatial Wideband Far-field Model and FIM
Under spatial wideband far-field communication, the model(1) applies with (2) specialized to ξ n [ k ] = − n ∆ cos θ + k ( d n − B ) r f . (15)We can then state the following result. Theorem 2.
In the case of spatial wideband far-field opera-tion, the FIM of the parameter η = [ ψ, d, θ, B ] T is J W ( η ) = γ A (0)0 E N, J S + γ J W + γ A (2)2 E N, J S + γ J W , where J W = E K, r f A (2)0 − A (1)0 − A (1)0 A (0)0 J W = E K, r f ∆ cos θd A (2)2 ∆ d cos θ − A (2)1 A (1)1 A (1)1 Proof.
We readily find that ∂ξ n [ k ] /∂d = kr f ( d − n ∆ cos θ ) /d n , ∂ξ n [ k ] /∂θ = n ∆ sin θ , ∂ξ n [ k ] /∂B = − kr f .Substituting these derivatives in (6) and then in (5), we obtainthe desired result.We observe that the radial information in J W is nowscaled and that there is an additional term J W thatprovides distance information with positive information E K, r f A (2)2 ∆ cos θ/d , which is important for large ∆ /d .The information is larger near the end-fire ( θ ≈ ), as this iswhere the delay spread is maximized. Note that the amountof information due to large bandwidth is generally less thanthe amount of information due to near-field. C. Localization and Synchronization Algorithm
From the Fisher information analysis, we find that for small ∆ (e.g., ∆ = λ/ ) the amount of information increase due tonear-field is more pronounced than due to spatial wideband.This is also confirmed by Fig. 2, where the nonlinear curvatureof the phase across antennas (left Figure) is more significantthan the difference in slope for different antennas (rightFigure). Based on this finding, we focus on the near-field case.The observation model in near-field is no longer of theform (12) so that a 2D-FFT will lead to multiple peaks. Apure maximum likelihood approach can be formulated, butleads to many local optima. Instead, we propose to extend themethod from Section III-B with a simple sub-arrays approach as in [11], [18], without aiming for an optimal solution. Wedivide the array (equivalently, the rows of YS H ( SS H ) − ) intonon-overlapping sub-arrays with ˜ N elements. The value of ˜ N should be chosen to satisfy the following conditions: (i) far-field condition: ˜ N ≤ √ ¯ dλ/ (2∆ ) , so that the far-field assump-tion is valid per sub-array (here, ¯ d is an expected distanceto the UE); (ii) narrowband condition: ˜ N (cid:28) c/ ( W ∆) , sothat paths are unresolved in the delay domain per sub-array.With these conditions (and ensuring that ˜ N ≥ ), the methodfrom Section III-B can be applied to each sub-array, providing (cid:98) ( N + 1) / ˜ N (cid:99) estimates ˆ θ ˜ n = arccos (cid:18) x − ˜ x ˜ n (cid:107) x − ˜ x ˜ n (cid:107) (cid:19) + w θ, ˜ n , (16) ˆ δ ˜ n = (cid:107) x − ˜ x ˜ n (cid:107) − B + w δ, ˜ n , (17)where w θ, ˜ n and w δ, ˜ n are measurement errors with variances σ θ, ˜ n and σ δ, ˜ n , due to the background noise and the finiteresolution of the FFTs. From these sub-array estimates, wecan recover the UE position by intersecting the bearing linesand then the clock bias from the delay estimates. The completeprocedure can be found in Algorithm 1. The complexity of themethod is of order O ( N K log ˜
N K ) . Algorithm 1
Sub-array Localization and Synchronization procedure L OCALIZE -N EAR -F IELD ( Y ) Determine ˜ N ∈ N ≥ : ˜ N ≤ (cid:112) ¯ dλ/ (2∆ ) and ˜ N (cid:28) c/ ( W ∆) Partition rows of Y into blocks of size ˜ N for ˜ n = 1 : (cid:98) N + 1 / ˜ N (cid:99) do Denote block ˜ n by Y ˜ n Estimate θ ˜ n and δ ˜ n from Y ˜ n as in Section III-B end for Solve for x : ˆ x = arg min x (cid:98) N +1 / ˜ N (cid:99) (cid:88) ˜ n =1 (cid:16) ˆ θ ˜ n − arccos (cid:16) x − ˜ x ˜ n (cid:107) x − ˜ x ˜ n (cid:107) (cid:17)(cid:17) σ θ, ˜ n Solve for B : ˆ B = arg min B (cid:98) N +1 / ˜ N (cid:99) (cid:88) ˜ n =1 (cid:16) ˆ δ ˜ n − (cid:107) ˆ x − ˜ x ˜ n (cid:107) − B ) (cid:17) σ δ, ˜ n (18) return ˆ x , ˆ B end procedure Note that in the narrowband far-field regime, ˜ N = N + 1 ,so that the method reverts to the one from Section III-B.V. N UMERICAL R ESULTS
We consider a nominal scenario at a carrier f c of 28 GHz( λ ≈ . cm), a bandwidth W of 100 MHz, c = 0 . m/ns, Sub-array ˜ n corresponding to the observations at antenna (˜ n −
1) ˜ N + 1 through ˜ n ˜ N , with array center ˜ x ˜ n = x − N/ +[∆((˜ n −
1) ˜ N +1+ ˜ N/ , T .Here indexing ˜ n starts at . -5 Fig. 3. PEB as a function of UE distance for known and unknown clock bias B , with ∆ = λ/ . N = 4 . × − mW/GHz, a transmit power P t of 1mW (with E {| s [ k ] | } = P t /W ) and K + 1 = 257 subcarrierswith QPSK pilots. The UE has bias B = 20 m. The array has N + 1 = 129 elements spaced at λ/ , corresponding to a totalsize of 69.11 cm and a far-field distance of 89 m. A. Fisher Information
We will evaluate the position error bound (PEB ) for severalmodels: the general (correct) model (1), and three approximatemodels: the standard model (3), the narrowband near-fieldmodel (14), and the spatial wideband far-field model (15),for known and unknown clock bias B . In Fig. 3 we showthe PEB as as a function of the distance d . As expected, thefar-field model is correct for distances larger than about 8 m,while for shorter distances the near-field models provide lowerPEB. Moreover, at short distances, the PEB of the generalmodel does not depend on whether we know B , while for largedistances, the PEB quickly increases when B is unknown. Theresults clearly show that joint synchronization and positioningin near-field can give good performance.In Fig. 4, we show the PEB as a function of the inter-antennaspacing. In this case, the PEB under the standard model doesnot depend on ∆ , as it is mainly limited by the estimation ofthe distance. The general model leads to lower PEB for largeantenna spacing, and larger PEB for small antenna spacing(for the case of unknown B ). Note that for very large ∆ ,the PEB of the general model increases due to the path loss.For both figures, we see that the main benefit for small d orlarge ∆ comes from the near-field information, not the spatialwideband information. B. Algorithm
We now evaluate the performance of the method describedin Section IV-C, whereby the AOAs are computed using a2DFFT with N points in the spatial domain and K + 1 points in the frequency domain. We vary d with random θ ∼ U ( π/ , π/ and place a scatterer with radar cross Source code is available at https://tinyurl.com/y3jybhdp. The PEB is defined from the × FIM J ( ψ, x , B ) as (cid:112) trace [ J − ( ψ, x , B )] , and is expressed in m. -1 -6 -4 -2 Fig. 4. PEB as a function of inter-antenna spacing for known and unknownclock bias B , with x = [1 m , m ] T . The legend is the same as in Fig. 3. -1 -3 -2 -1 Fig. 5. Joint localization and synchronization performance using sub-arrayprocessing vs standard processing with LOS only and with multipath (LOS+ NLOS). Bounds are not included, as they are loose since the algorithm isdeveloped for proof-of-concept, not for ultimate accuracy. section of m uniformly in the plane (this correspondsto a large scattering object). This enables us to evaluate therobustness to multipath. For comparison purposes, the methodfrom Section III-B is also evaluated, assuming known bias.From Fig. 5, we observe low position RMSE for distancesbelow 3 m. The non-LOS (NLOS) path increases the RMSEcompared to the LOS-only case, as it causes large outliers.Note that multipath appears as a second peak in the 2D-FFTand can thus be recovered and separated from the LOS path.However, which path corresponds to LOS is often harder todetermine due to the poor delay resolution (the resolutionat W = 100 MHz is only 3 m). Beyond 15 m distance, ˜ N → , so that the problem is no longer identifiable. Thelocalization performance is worse than the PEB from Fig. 3,as the method has not been optimized for accuracy. Moreover,the bias estimate has orders of magnitude larger error, as it isbased on low-quality range estimates. This error can be furtherreduced by using larger FFTs along the frequency dimension.The far-field method from Section III-B with known bias islimited by the bandwidth and thus leads to worse performancefor all d .I. C ONCLUSIONS
When large arrays are used for positioning, near-field prop-agation must be taken into account. This presents challengesand opportunities for the development of localization systemsbeyond 5G. We have performed a Fisher information analysisand proposed a simple joint localization and synchronizationmethod for this regime. Our results show that near-fieldpropagation can be exploited in uplink and that the Fisherinformation provided from wavefront curvature is more pro-nounced than from spatial wideband. Immediate suggestionsfor future research are the inclusion of hybrid combining atthe BS, as in [11], the study of downlink localization with asingle receive antenna [29], as well as the inclusion of a morerealistic propagation model [30], accounting for coupling [31]and electromagnetic theory, as well as jointly localizing andsynchronizing multiple mobile users [25].A
CKNOWLEDGMENT
This research was supported, in part, by the Swedish Re-search Council under grant No. 2018-03701.R
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