Position Information from Single-Bounce Reflections
Anastasios Kakkavas, Mario H. Castañeda García, Gonzalo Seco-Granados, Henk Wymeersch, Richard A. Stirling-Gallacher, Josef A. Nossek
11 Position Information from Reflecting Surfaces
Anastasios Kakkavas,
Student Member, IEEE,
Mario H. Castañeda García,
Member, IEEE,
Gonzalo Seco-Granados,
Senior Member, IEEE,
Henk Wymeersch,
Senior Member, IEEE,
Richard A. Stirling-Gallacher,
Member, IEEE, and Josef A. Nossek,
Life Fellow, IEEE
This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer beaccessible.
Abstract —In the context of positioning a target with a single-anchor, this contribution focuses on the Fisher information aboutthe position, orientation and clock offset of the target providedby single-bounce reflections. The availability of prior knowledgeof the target’s environment is taken into account via a priordistribution of the position of virtual anchors, and the rank,intensity and direction of provided information is studied. Weshow that when no prior knowledge is available, single-bouncereflections offer position information in the direction parallel tothe reflecting surface, irrespective of the target’s and anchor’slocations. We provide a geometrically intuitive explanation ofthe results and present numerical examples demonstrating theirpotential implications.
Index Terms —positioning, localization, NLOS, reflection,single-bounce
I. I
NTRODUCTION
Although the majority of practical positioning systems relyheavily or even exclusively on line-of-sight (LOS) propaga-tion, the role of non-LOS (NLOS) propagation in wirelesspositioning has been widely studied. Traditionally, the focushas been on the mitigation of the negative impact of NLOSpaths on positioning accuracy [1], with some approachescompletely disregarding NLOS links and others aiming tocorrect the NLOS-induced bias in the range estimate [2]. Analternative approach is to treat the NLOS paths as additionalsources of position information. An early work in this directionwas [3], where it was shown that given distance, angle ofdeparture (AOD) and angle of arrival (AOA) measurementsof a single-bounce NLOS path, the receiver (Rx) can lie on aline segment, and an algorithm exploiting this observation waspresented. A similar approach for mobile targets was presentedin [4].
This work was supported in part by the EU-H2020 project Fifth GenerationCommunication Automotive Research and Innovation (5GCAR), and in partby the ICREA Academia program and the Spanish Ministry of Science,Innovation and Universities project TEC2017-89925-R.A. Kakkavas is with the Munich Research Center, Huawei TechnologiesDuesseldorf GmbH, 80992 Munich, Germany, and also with the Department ofElectrical and Computer Engineering, Technical University of Munich, 80333Munich, Germany (e-mail: [email protected]).M. H. Castañeda García and R. A. Stirling-Gallacher are with the MunichResearch Center, Huawei Technologies Duesseldorf GmbH, 80992 Munich,Germany (e-mail: [email protected]; [email protected]).G. Seco-Granados is with the Department of Telecommunications andSystems Engineering, Universitat Autonoma de Barcelona, Spain (UAB) (e-mail: [email protected]).H. Wymeersch is with the Department of Electrical Engineering,Chalmers University of Technology, 412 58 Gothenburg, Sweden (email:[email protected]).J. A. Nossek is with the Department of Electrical and Computer Engi-neering, Technical University of Munich, 80333 Munich, Germany (e-mail:[email protected]).
Such approaches became much more relevant for fifthgeneration (5G) networks [5]. The upcoming exploitation ofthe large chunks of available bandwidth at millimeter-wave(mm-Wave) frequencies and the use of antenna arrays witha large number of elements, enable the possibility of highlyaccurate temporal and angular measurements, and improvethe separability of multipath components [6]. The increasedtemporal and angular resolution has made single-anchor posi-tioning [7] an attractive option when links to multiple anchorsmay not be available. Algorithms for single-anchor localizationand mapping with a single snapshot have been presented in [8],[9] and [10] among others. In [11] it was shown that, in a two-dimensional (2D) setup, the set of time of arrival (TOA), AODand AOA measurements from a single-bounce reflection offersrank-1 information for a receiver with unknown position andorientation. The corresponding eigenvalue of the position andorientation equivalent FIM (EFIM) was computed analytically,showing that all 3 measurements are required for extracting ad-ditional position information from NLOS components. In [12]it was shown that single-bounce NLOS components can behelpful in resolving the clock offset between an imperfectlysynchronized transmitter (Tx)-Rx pair, allowing for accuratesingle-anchor positioning.In this letter, considering flat reflecting surfaces, which werefer to as reflectors, we extend the work of [3], as well as [11]and [12], as follows: • We show that, when no prior information about thereflector is available, the direction of position informationis parallel to the reflector and independent of the Tx andRx position. Hence, the line segment where the receivercan lie given the measurements of a NLOS path, asidentified in [3], is always orthogonal to the reflectingsurface. • By encoding prior information about reflectors as priordistribution of the location of their corresponding virtualanchors (VAs), we study the effect of the accuracy of priorinformation on the intensity and direction of positioninformation offered by single-bounce reflections.II. S
YSTEM M ODEL
A. System Model
The Tx consists of an array with 𝑁 T antennas and referencepoint located at p T = [ 𝑝 T,x , 𝑝
T,y ] T ∈ R , where (·) T denotestransposition. For the 𝑗 -th element of the Tx array, 𝑑 T , 𝑗 and 𝜓 T , 𝑗 are its distance and angle from the Tx array’s referencepoint as shown in Fig. 1. The position of the 𝑗 -th element ofthe Tx array is given by p T , 𝑗 = 𝑑 T , 𝑗 u ( 𝜓 T , 𝑗 ) ∈ R , 𝑗 = , . . . , 𝑁 T − , where u ( 𝜓 ) = [ cos ( 𝜓 ) , sin ( 𝜓 )] T . The Rx a r X i v : . [ c s . I T ] D ec p T p T , 𝑗 𝜓 T , 𝑗 − 𝜋𝑑 T , 𝑗 p R 𝛼 R 𝜃 T , p s , p VA , ˜ 𝜃 R , 𝜃 R , − 𝜋𝜃 ref , 𝜃 ref , − 𝜋 / Δ 𝜃 Δ 𝜃 u ( 𝜃 ref , ) Fig. 1. Geometric model, example with uniform linear arrays (ULAs) at theTx and the Rx. consists of an array with 𝑁 R antennas, orientation 𝛼 R (withrespect to the Tx array’s orientation) and reference pointlocated at p R = [ 𝑝 R,x , 𝑝
R,y ] T ∈ R . The position of antennaelements at the Rx array are defined similarly as for the Tx.Between each Tx-Rx pair of antennas there are 𝐿 discretepropagation paths, where the first path ( 𝑙 = ) is the LOS pathand the rest ( 𝑙 = , . . . , 𝐿 − ) are single-bounce NLOS paths.The 𝑙 -th single-bounce NLOS path results from a reflectionon a flat surface with normal vector u ( 𝜃 ref ,𝑙 ) and point ofincidence p s ,𝑙 = [ 𝑝 s ,𝑙, x , 𝑝 s ,𝑙, y ] T . Each single-bounce NLOScan be viewed as a direct path resulting from a VA, with the 𝑙 -th VA located at p VA ,𝑙 = [ 𝑝 VA ,𝑙, x , 𝑝 VA ,𝑙, y ] T , 𝑙 = , . . . , 𝐿 − .The length of the 𝑙 -th path is given by 𝑑 𝑙 , i.e. 𝑑 𝑙 = (cid:107) p R − p T (cid:107) for 𝑙 = and 𝑑 𝑙 = (cid:107) p R − p VA ,𝑙 (cid:107) for 𝑙 ≠ , with (cid:107) · (cid:107) beingthe Euclidean norm. The AOAs are defined as 𝜃 R ,𝑙 = (cid:40) atan2 ( 𝑝 T , y − 𝑝 R , y , 𝑝 T , x − 𝑝 R , x ) , 𝑙 = , atan2 ( 𝑝 VA ,𝑙, y − 𝑝 R , y , 𝑝 VA ,𝑙, x − 𝑝 R , x ) , 𝑙 ≠ , (1)with atan2 (cid:0) 𝑦, 𝑥 (cid:1) being the four-quadrant inverse tangent func-tion. The AOAs in the local frame of the Rx are defined as ˜ 𝜃 R ,𝑙 = 𝜃 R ,𝑙 − 𝛼 R , 𝑙 = , . . . , 𝐿 − . With the observation that fora single-bounce reflection it holds that 𝜃 ref ,𝑙 = ( 𝜃 T ,𝑙 + 𝜃 R ,𝑙 )/ ,the AODs can be expressed as 𝜃 T ,𝑙 = (cid:40) 𝜃 R , − 𝜋, 𝑙 = , ( 𝑝 VA ,𝑙, y − 𝑝 T , y , 𝑝 VA ,𝑙, x − 𝑝 R ,𝑙, x ) − 𝜃 R ,𝑙 , 𝑙 ≠ . (2)The array dimensions are much smaller than the distancesbetween Tx, Rx and reflectors. Thus, the delay of the 𝑙 -thpath from Tx element 𝑗 to Rx element 𝑖 can be approximatedby 𝜏 𝑙,𝑖, 𝑗 ≈ 𝜏 𝑙 − 𝜏 T , 𝑗 ( ˜ 𝜃 T ,𝑙 ) − 𝜏 R ,𝑖 ( ˜ 𝜃 R ,𝑙 ) , where 𝜏 T , 𝑗 ( ˜ 𝜃 T ,𝑙 ) = 𝑑 T , 𝑗 u T ( 𝜓 T , 𝑗 ) u ( ˜ 𝜃 T ,𝑙 )/ 𝑐 , 𝜏 R ,𝑖 ( ˜ 𝜃 R ,𝑙 ) = 𝑑 R ,𝑖 u T ( 𝜓 R ,𝑖 ) u ( ˜ 𝜃 R ,𝑙 )/ 𝑐 and 𝜏 𝑙 = (cid:40) ((cid:107) p R − p T (cid:107) + 𝑑 clk )/ 𝑐, 𝑙 = ((cid:107) p R − p VA ,𝑙 (cid:107) + 𝑑 clk )/ 𝑐, 𝑙 ≠ , (3)where 𝑑 clk = 𝑐𝜖 clk with 𝜖 clk as the clock offset between Txand Rx and 𝑐 as the speed of light. An orthogonal frequency-division multiplexing (OFDM)waveform with 𝑁 subcarriers and subcarrier spacing Δ 𝑓 isconsidered. The set of used subcarriers is given by P . Anarrowband signal model is assumed, i.e. 𝐵 / 𝑓 c (cid:28) 𝜆 c / 𝐷 max ,where 𝐵 ≈ Δ 𝑓 ( max (P) − min (P)) is the signal bandwidth, 𝑓 c and 𝜆 c are the carrier frequency and wavelength, and 𝐷 max isthe largest dimension of the Tx and Rx arrays. The receivedsignal at the 𝑝 -th subcarrier (for 𝑝 ∈ P ) is y [ 𝑝 ] = m [ 𝑝 ] + η [ 𝑝 ] , (4)where η [ 𝑝 ] ∼ N C ( , 𝜎 𝜂 I 𝑁 R ) is the additive white Gaussiannoise (AWGN) at the 𝑝 -th subcarrier and m [ 𝑝 ] = 𝐿 − ∑︁ 𝑙 = ℎ 𝑙 e − j 𝜔 𝑝 𝜏 𝑙 a R ( ˜ 𝜃 R ,𝑙 ) a T T ( ˜ 𝜃 T ,𝑙 ) x [ 𝑝 ] , (5)where ℎ 𝑙 ∈ C is the gain of the 𝑙 -th path, 𝜔 𝑝 = 𝜋 𝑝 Δ 𝑓 , and x [ 𝑝 ] ∈ C 𝑁 T is the reference signal of the 𝑝 -th subcarrier. With 𝜔 c = 𝜋 𝑓 c , the Tx array steering vector a T ( ˜ 𝜃 T ,𝑙 ) is given a T ( ˜ 𝜃 T ,𝑙 ) = [ 𝑒 j 𝜔 𝑐 𝜏 T , ( ˜ 𝜃 T ,𝑙 ) , . . . , e j 𝜔 𝑐 𝜏 T ,𝑁 T ( ˜ 𝜃 T ,𝑙 ) ] T ∈ C 𝑁 T , (6)with the Rx steering vector a R ( ˜ 𝜃 R ,𝑙 ) defined similarly.III. C RAMÉR -R AO L OWER B OUND
We first define the channel parameter vector φ ∈ R 𝐿 as φ = [ 𝜏 , ˜ 𝜃 T , , ˜ 𝜃 R , , h T0 , · · · , 𝜏 𝐿 − , 𝜃 T ,𝐿 − , 𝜃 R ,𝐿 − , h T 𝐿 − , ] T . (7)and the position parameter vector ˜ φ = [ p R , 𝛼 R , 𝑑 clk , p VA , , · · · , p VA ,𝐿 − ] T ∈ R 𝐿 + . (8)According to the Cramér-Rao lower bound (CRLB), the co-variance matrix C ˆ˜ φ of any unbiased estimator ˆ˜ φ of ˜ φ satisfies C ˆ˜ φ − J − φ (cid:23) [13], where (cid:23) denotes positive semi-definiteness and J ˜ φ ∈ R ( 𝐿 + )×( 𝐿 + ) is the hybrid Fisherinformation matrix (FIM) of ˜ φ given by J ˜ φ = J ( p ) ˜ φ + J ( o ) ˜ φ , with J ( p ) ˜ φ and J ( o ) ˜ φ accounting for the prior information andobservation-related information on ˜ φ , respectively. We notethat the hybrid FIM and the corresponding CRLB characterizethe estimation performance for a deterministic φ , where twosources of information are used: the received signal and theprior 𝑝 ( ˜ φ ) [14]. The observation-related FIM J ( o ) ˜ φ can beobtained based on the FIM J ( o ) φ of the channel parametervector φ as J ( o ) ˜ φ = T J ( o ) φ T T . The entries of J ( o ) φ ∈ R 𝐿 × 𝐿 and T ∈ R 𝐿 + × 𝐿 are given by (cid:104) J ( o ) φ (cid:105) 𝑖, 𝑗 = 𝜎 𝜂 ∑︁ 𝑝 ∈P (cid:60) (cid:26) 𝜕 m H [ 𝑝 ] 𝜕𝜙 𝑖 𝜕 m [ 𝑝 ] 𝜕𝜙 𝑗 (cid:27) , 𝑖, 𝑗 = , . . . , 𝐿, (9) [ T ] 𝑖, 𝑗 = 𝜕𝜙 𝑗 / 𝜕 ˜ 𝜙 𝑖 , 𝑖 = , . . . , 𝐿 + , 𝑗 = , . . . , 𝐿, (10)where m H is the conjugate transpose of m and (cid:60){ m } isits real part. Details on the required derivatives can be foundin [12]. The position error bound (PEB) for the Rx is definedas Rx PEB = √︃(cid:2) J − φ (cid:3) , + (cid:2) J − φ (cid:3) , (11)and the PEB for the VAs is defined in a similar manner. The Rx has prior information on the clock offset 𝑝 ( 𝜖 (cid:48) clk ) = N ( 𝜖 (cid:48) clk ; 𝜖 clk , 𝜎 clk ) and the VAs’ locations 𝑝 ( p (cid:48) VA ,𝑙 ) = N ( p (cid:48) VA ,𝑙 ; p VA ,𝑙 , Σ VA,pr ,𝑙 ) , which encode map information aboutreflectors available at the Rx, with N ( x ; µ , 𝚺 ) denoting that x follows a Gaussian distribution with mean µ and covariance 𝚺 . The hybrid FIM of the position parameter vector is J ˜ φ = T J ( o ) φ T T + (cid:20) J VA,pr (cid:21) + 𝑐𝜎 clk e e T4 , (12)where J VA,pr = 𝚺 − VA,pr , . . . 𝚺 − VA,pr ,𝐿 − ∈ R ( 𝐿 − )× ( 𝐿 − ) (13)where 𝚺 VA,pr ,𝑙 ∈ R × for 𝑙 = , · · · , 𝐿 − is the covariancematrix of the 𝑙 -th VA’s location given by 𝚺 VA,pr ,𝑙 = [ u (cid:0) 𝜃 R ,𝑙 (cid:1) u ⊥ (cid:0) 𝜃 R ,𝑙 (cid:1) ] (cid:34) 𝜎 𝑙, (cid:107) 𝜌 𝑙 𝜎 𝑙, (cid:107) 𝜎 𝑙, ⊥ 𝜌 𝑙 𝜎 𝑙, (cid:107) 𝜎 𝑙, ⊥ 𝜎 𝑙, ⊥ (cid:35) (cid:20) u T (cid:0) 𝜃 R ,𝑙 (cid:1) u T ⊥ (cid:0) 𝜃 R ,𝑙 (cid:1)(cid:21) . (14)where u ⊥ ( 𝜃 ) = u ( 𝜃 − 𝜋 / ) .We employ the EFIM [15], to focus on the availableinformation on the paramters of interest. Splitting T as T = [ T T poc T T VA ] T , with T poc ∈ R × 𝐿 comprising the first four rowsof T corresponding to the position and orientation parametersand clock offset and T VA ∈ R ( 𝐿 − )× 𝐿 including the restof the rows of T , the EFIM for the position and orientationparameters and clock offset is given by J poc = T poc J φ T T poc − T poc J φ T T VA J − VA T VA J φ T T poc + e e T4 ( 𝑐𝜎 clk ) , (15)where J VA = T VA J φ T T VA + J VA,pr .Making use of the fact that for large bandwidth and numberof antennas the paths become asymptotically orthogonal [6], J ( o ) φ becomes a diagonal matrix. Indexing the diagonal el-ements of J ( o ) φ by the parameter they correspond to, e.g. 𝐽 𝜏 = [ J ( o ) φ ] , , it can be shown that (15) can be written as J poc = 𝐽 𝜏 𝑐 z 𝜏 z T 𝜏 + 𝐽 𝜃 T , 𝑑 z 𝜃 T , z T 𝜃 T , + 𝐽 𝜃 R , 𝑑 z 𝜃 R , z T 𝜃 R , + 𝐿 − ∑︁ 𝑙 = J 𝑙 + e e T4 ( 𝑐𝜎 clk ) where the EFIM J 𝑙 of the 𝑙 -th NLOS path is J 𝑙 = (cid:12)(cid:12) J VA ,𝑙 (cid:12)(cid:12) [ z 𝜏 𝑙 , z 𝜃 T ,𝑙 , z 𝜃 R ,𝑙 ] M 𝑙 [ z 𝜏 𝑙 , z 𝜃 T ,𝑙 , z 𝜃 R ,𝑙 ] T (16)with z 𝜏 𝑙 = (cid:2) − u T (cid:0) 𝜃 R ,𝑙 (cid:1) , , (cid:3) T (17) z 𝜃 T ,𝑙 = (cid:2) u T ⊥ (cid:0) 𝜃 R ,𝑙 (cid:1) , , (cid:3) T (18) z 𝜃 R ,𝑙 = (cid:2) u T ⊥ (cid:0) 𝜃 R ,𝑙 (cid:1) , − 𝑑 𝑙 , (cid:3) T (19)and u ⊥ ( 𝜃 ) = u (cid:16) 𝜃 − 𝜋 (cid:17) . The entries of M 𝑙 ∈ C × and | J VA ,𝑙 | in (16) are given in the Appendix. p T p R p s , 𝜃 ref , p VA , p R p s , p VA , Fig. 2. Potential solutions for 𝑝 R , 𝑝 s ,𝑙 and 𝑝 VA ,𝑙 explaining the measurementsfor a single-bounce reflection. IV. G
EOMETRIC I NTERPRETATION OF P OSITION I NFORMATION
It is interesting to carefully examine and obtain geometricintuition on the position information for the cases of perfectand no knowledge of the VA’s position. The former case isstraightforward: from (16) for 𝜎 𝑙, (cid:107) , 𝜎 𝑙, ⊥ → we get J 𝑙 = 𝐽 𝜏 𝑙 𝑐 z 𝜏 𝑙 z T 𝜏 𝑙 + 𝐽 𝜃 T ,𝑙 𝑑 𝑙 z 𝜃 T ,𝑙 z T 𝜃 T ,𝑙 + 𝐽 𝜃 R ,𝑙 𝑑 𝑙 z 𝜃 R ,𝑙 z T 𝜃 R ,𝑙 . (20)As expected, in this case the NLOS path acts in the same wayas LOS path. Using (20) and (17)-(19) we can see that therank of J 𝑙 is equal to 3, with each of the measurements pro-viding position and orientation information independently: theTOA provides position information in the radial direction, theAOD and AOA provide position information in the tangentialdirection and the AOA provides orientation information.In the case of no knowledge of the VA’s location, i.e. 𝜎 𝑙, (cid:107) , 𝜎 𝑙, ⊥ → ∞ , it can be shown that (16) becomes J 𝑙 = 𝑗 𝑙 z 𝑙 z T 𝑙 (21)where 𝑗 𝑙 = 𝐽 𝜏 𝑙 𝐽 𝜃 T ,𝑙 𝐽 𝜃 R ,𝑙 | J VA ,𝑙 | 𝑐 𝑑 𝑙 𝑑 T,s ,𝑙 cos ( Δ 𝜃 𝑙 / ) (22) z 𝑙 = [ u T ⊥ ( 𝜃 ref ,𝑙 ) , − 𝑑 R,s ,𝑙 cos ( Δ 𝜃 𝑙 / ) , sin ( Δ 𝜃 𝑙 / )] T , (23)with 𝑑 T,s ,𝑙 = (cid:107) p s ,𝑙 − p T (cid:107) and 𝑑 R,s ,𝑙 = (cid:107) p R − p s ,𝑙 (cid:107) . Wecan observe from (21) that, as first noted in [11], J 𝑙 hasrank . Furthermore, from (23) and Fig. 1, we conclude thatthe direction of position information is always parallel to thereflecting surface and independent of the Tx and Rx location.At first glance, this is a surprising result, since for LOS pathsand NLOS paths with perfect knowledge of the correspondingVAs’ location the direction of position information dependson p R and p T . A geometrically intuitive explanation of thisresult can be obtained from Fig. 2. In Fig. 2 we considera single-bounce reflection and plot two potential geometries { p R , p 𝑠, , p VA , } and { p (cid:48) R , p (cid:48) 𝑠, , p (cid:48) VA , } that would produce thesame TOA, AOD and AOA. In fact, there are infinitely manysuch geometries, parametrized as p R = p T − 𝑐 · 𝜏 𝑙 u ( 𝜃 R ,𝑙 ) + 𝜆 cos ( Δ 𝜃 𝑙 / ) u ( 𝜃 ref ,𝑙 ) (24) p VA ,𝑙 = p T + 𝜆 cos ( Δ 𝜃 𝑙 / ) u ( 𝜃 ref ,𝑙 ) (25) p s ,𝑙 = p T + 𝜆 u ( 𝜃 T ,𝑙 ) , (26)
100 10 20 30 40 50 60 − − − p T p R p s , p VA , p s , p VA , p s , p VA , refl. surf. 2refl. surf. 1 refl. surf. 3AB x in m y i n m Fig. 3. Simulation scenario. with < 𝜆 < 𝑐 · 𝜏 𝑙 . As can be seen in (24), the locus of p R is a line segment normal to the reflecting surface. Hence, theNLOS path associated with the reflection provides positioninformation only in the direction that is perpendicular to thisline segment, i.e. in the direction parallel to the reflectingsurface. An implication of this result is that informationfrom single-bounce paths from parallel (or close to parallel)reflecting surfaces may not suffice for target localization.V. N UMERICAL R ESULTS
A. Simulation setup
We set 𝑓 c =
38 GHz , 𝑁 = , Δ 𝑓 =
120 kHz , P = {− , . . . , − , . . . , } and 𝐵 ≈
100 MHz . Theentries of x [ 𝑝 ] have constant amplitude and random phase,with E [(cid:107) x [ 𝑝 ] (cid:107) ] = . The noise variance is 𝜎 𝜂 = . ( 𝑛 Rx + 𝑁 ) 𝑁 Δ 𝑓 , where 𝑁 = −
174 dBm Hz − is the noisepower spectral density and 𝑛 Rx = is the Rx noise figure.We consider the scenario depicted in Fig. 3, where the Txlies at the origin and the Rx at p R = [ . , ] T m . The Txhas a ULA with 32 antennas and the Rx has a UCA with 16antennas and orientation 𝛼 R . The VAs resulting from single-bounce reflections at the rooms’ walls are located at 𝑝 VA , = [ , − ] T m , 𝑝 VA , = [ , ] T m and 𝑝 VA , = [ , ] T m . Inorder to concentrate on the potential implications of the resultspresented in Sec. IV, we assume that the Rx orientation 𝛼 R is known and the Tx and Rx are perfectly synchronized. Weconsider two NLOS-only cases: • case A : the paths corresponding to the 1st and 2nd VAsare received; • case B : the paths corresponding to the 1st and 3rd VAsare received.The amplitude of the complex path gain of the 𝑙 -th path is | ℎ 𝑙 | = √ 𝛾 𝑙 𝜆 /( 𝜋𝑑 𝑙 ) , where 𝛾 𝑙 = . ∀ 𝑙 , is the reflectioncoefficient, and the phase is uniformly distributed. B. Results
From the analysis in Sec. IV we have a clear picture aboutthe position information offered by single-bounce NLOS pathsunder perfect or no prior knowledge of their corresponding − − ( J θ T , + J θ R , )/ d J τ / c j σ ref in m E i g e nv a l u e s (a) Eigenvalues of J . − − − − θ ref , + π / θ R , + πθ R , + π / σ ref in m E i g e nv ec t o r d i r ec ti on i n ◦ (b) Direction eigenvectors of J .Fig. 4. Eigenvalues and directions of eigenvectors of the EFIM J of VA 1as function of the prior VA position error 𝜎 ref . VAs locations. In order to gain more insight about the inter-mediate cases, setting 𝜌 = and 𝜎 , (cid:107) = 𝜎 , ⊥ = 𝜎 ref /√ ,we plot the eigenvalues and the directions of the eigenvectorsfor varying 𝜎 ref in Fig. 4. We see that, as expected, whenknowledge about the VA’s position is accurate ( 𝜎 ref → ), forknown orientation and perfect synchronization (in this case z 𝜃 T ,𝑙 = z 𝜃 R ,𝑙 = u T ⊥ ( 𝜃 R ,𝑙 ) ), J has two strong eigenvalues,with the eigenvectors pointing in the radial and the tangentialdirection. As 𝜎 ref increases, the strongest eigenvalue decreases,starting from 𝐽 𝜏 / 𝑐 and converges to 𝑗 , while the secondeigenvalue vanishes, resulting in a rank-1 J . The directionof the eigenvector corresponding to the strongest eigenvaluegradually changes from 𝜃 R , + 𝜋 , which corresponds to rangeinformation, to 𝜃 ref , + 𝜋 / , that is parallel to the reflectingsurface.In Fig. 5 we plot the PEB of the Rx and VA 1 for the twoconsidered cases as functions of 𝜎 ref . We set again 𝜌 𝑙 = and 𝜎 𝑙, (cid:107) = 𝜎 𝑙, ⊥ = 𝜎 ref /√ , 𝑙 = , , . We see that for 𝜎 ref → the PEB of VA 1 converges to 0, while the Rx PEBconverges to its lowest value as the two paths behave as LOSpaths, providing position information in linearly independentdirections. In case A, as 𝜎 ref increases the two paths provideposition information in almost the same direction, as they arisefrom parallel reflecting surfaces, with the available informationin the orthogonal direction decreasing with increasing 𝜎 ref . Asa result, for high values of 𝜎 ref (i.e. less accurate prior) the PEBof the Rx and VA 1 grows linearly with 𝜎 ref . In case A, forhigh values of 𝜎 ref (i.e. less accurate prior) the PEB of the Rxand VA 1 grows linearly with 𝜎 ref . This is due to the fact that − − − − σ ref in m P E B i n m Case A, Rx PEBCase A, VA 1 PEBCase B, Rx PEBCase B, VA 1 PEB
Fig. 5. Rx and VA 1 PEB as function of the prior VA position error 𝜎 ref . the two paths provide position information in almost the samedirection, as they arise from parallel reflecting surfaces, withthe available information in the orthogonal direction decreas-ing with increasing 𝜎 ref . For moderate values of 𝜎 ref ( < ),good positioning accuracy is achievable as, the directions ofthe strongest eigenvectors of J and J are sufficiently distinct.On the contrary, in case B, the PEB of the Rx and VA 1saturates for high values of 𝜎 ref , as the two paths provideposition information in different directions, resulting from thetwo perpendicular walls. Therefore, combining the two NLOSpaths the Rx position can be resolved and, consequently, theposition of the VA. VI. C ONCLUSION
We provided an analysis of the Fisher information on po-sition, orientation and clock offset provided by single-bounceNLOS paths. The effect of prior map information on the posi-tion information was studied. It was shown that when no priorknowledge is available, the direction of position informationis always parallel to the reflecting surface, independent of theRx target position. We also provided a geometrically intuitiveexplanation of the result. Numerical examples considering apractical room geometry showed that, as a consequence of theaforementioned analysis, the availability of different multipathcomponents can have a significant impact on the achievablepositioning accuracy. A
PPENDIX E NTRIES OF M 𝑙 AND | J VA ,𝑙 | IN (16)The entries of M 𝑙 and | J VA ,𝑙 | in (16) are given by [ M 𝑙 ] , = 𝐽 𝜏 𝑙 𝑐 𝐽 𝜃 T ,𝑙 𝐴 𝐹 + ( − 𝜌 𝑙 ) 𝜎 𝑙, (cid:107) (cid:32) 𝐽 𝜃 T ,𝑙 𝐵 + 𝐽 𝜃 R ,𝑙 𝑑 𝑙 + 𝐺𝜎 𝑙, ⊥ (cid:33) [ M 𝑙 ] , = 𝐽 𝜃 T ,𝑙 𝑑 𝑙 (cid:32) 𝐽 𝜏 𝑙 𝑐 𝐹 + ( − 𝜌 𝑙 ) 𝜎 𝑙, (cid:107) 𝑃 (cid:33) [ M 𝑙 ] , = 𝐽 𝜃 R ,𝑙 𝑑 𝑙 (cid:32) 𝐽 𝜃 T ,𝑙 𝐵 𝑄 + ( − 𝜌 𝑙 ) 𝜎 𝑙, ⊥ (cid:32) 𝐽 𝜏 𝑙 𝑐 + 𝐽 𝜃 T ,𝑙 𝐴 + 𝐺𝜎 𝑙, (cid:107) (cid:33)(cid:33) [ M 𝑙 ] , = [ M 𝑙 ] , = 𝐽 𝜏 𝑙 𝑐 𝐽 𝜃 T ,𝑙 𝑑 𝑙 (cid:32) 𝐴𝐹 + 𝐵 𝜌 𝑙 ( − 𝜌 𝑙 ) 𝜎 𝑙, (cid:107) 𝜎 𝑙, ⊥ (cid:33) [ M 𝑙 ] , = [ M 𝑙 ] , = 𝐽 𝜏 𝑙 𝑐 𝐽 𝜃 T ,𝑙 𝑑 𝑙 (cid:32) 𝜌 𝑙 ( − 𝜌 𝑙 ) 𝜎 𝑙, (cid:107) 𝜎 𝑙, ⊥ − 𝐽 𝜃 T ,𝑙 𝐴𝐵 (cid:33) [ M 𝑙 ] , = [ M 𝑙 ] , = 𝐽 𝜃 T ,𝑙 𝑑 𝑙 𝐽 𝜃 R ,𝑙 𝑑 𝑙 (cid:32) 𝐵𝑄 + 𝐴𝜌 𝑙 ( − 𝜌 𝑙 ) 𝜎 𝑙, (cid:107) 𝜎 𝑙, ⊥ (cid:33) | J VA ,𝑙 | = 𝐽 𝜏 𝑙 𝑐 (cid:0) 𝐽 𝜃 T ,𝑙 𝐵 + 𝐹 (cid:1) + 𝑃 + 𝐽 𝜃 T ,𝑙 𝐵 ( 𝐵 + 𝜌 𝑙 𝐴𝜎 𝑙, (cid:107) / 𝜎 𝑙, ⊥ )( − 𝜌 𝑙 ) 𝜎 𝑙, (cid:107) + 𝐽 𝜃 T ,𝑙 𝐴 𝐹, where 𝐴 = tan (cid:0) Δ 𝜃 𝑙 (cid:1) / 𝑑 T,s ,𝑙 , 𝐵 = / 𝑑 𝑙 − / 𝑑 T,s ,𝑙 , Δ 𝜃 𝑙 = 𝜃 R ,𝑙 − 𝜃 T ,𝑙 , 𝐹 = 𝐽 𝜃 R ,𝑙 𝑑 𝑙 + ( − 𝜌 𝑙 ) 𝜎 𝑙, ⊥ , 𝐺 = + 𝜌 𝑙 𝜎 𝑙, (cid:107) 𝜎 𝑙, ⊥ 𝐽 𝜃 T ,𝑙 𝐴𝐵 , 𝑃 = 𝐽 𝜃 R ,𝑙 𝑑 𝑙 + 𝜎 𝑙, ⊥ and 𝑄 = 𝐽 𝜏𝑙 𝑐 + ( − 𝜌 𝑙 ) 𝜎 𝑙, (cid:107) .R EFERENCES[1] Li Cong and Weihua Zhuang, “Nonline-of-sight error mitigation inmobile location,”
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