3. Uplink Search-Based Direction-of-Arrival (DoA) Estimation 77
4.2. Uplink Positioning based on Two DoA Estimates
4.2.1. Skew Lines in Euclidean Space
According to Fig. 4.5, it is assumed that two BSs send a DoA estimated of the same source to a central processing unit, e.g. using the LTE positioning protocol [3GP16a, 3GP16b]. These estimates consist of an azimuth and elevation angle, ˆα and ˆβ, respectively. However, each BS obtains these estimates relative to its array orientation. Therefore, additionally the absolute position and the orientation of the array are required to use these estimates for positioning. Given, that the orientation and position of the antenna arrays are static we assume both as known in the network. First, the relative DoAs are transformed to a reference coordinate system given by ¯αi and ¯βi, where the subscript i is the BS ID. The estimated DoA of source k in the reference coordinate system is obtained by
¯
αk = mod(˜αi+ ˆαk+ 180◦,360◦)−180◦
β¯k = mod( ˜βi+ ˆβk+ 90◦,180◦)−90◦, (4.6) where “mod” is the modulo operation and ˜α,β˜are the horizontal/vertical orientation angles of the BS array, respectively. Note that the vertical angle β is counted in this work in the interval [−90,90]◦, which is the
z(C)
Figure 4.6.: Three dimensional position based on two DoAs from two BS.
reason of the−90◦offset after the modulus operation. The next step in Fig. 4.5 is to obtain vector expressions of the DoAs including BS positions. Without limit to generality, the origin of the reference coordinate system is in this work assumed such that BS 1 is located at ˜p(BS)1 = hx(C)1 = 0 my1(C)= 0 mz1(C)iT and BS 2 at
˜
p(BS)2 = hx(C)2 y2(C) z2(C)iT with orientations of BS 1 as ˜α1 = 0◦,β˜1 = 0◦ and BS 2 as ˜α2,β˜2 according to Fig. 4.6. Combining the position and DoA estimate of BS 1 a linear equation is obtained according to
s(BS)=
where s(BS) ∈R+ is an auxiliary parameter defines the distance from BS 1 along the DoA in [m]. Trans-forming the DoA estimate of BS 2 into the coordinate system of BS 1 with Eq. (4.6) the linear equation for BS 2 is obtained as
where the auxiliary parameter t(BS) ∈R+ given in [m] is the corresponding distance along the DoA from BS 2.
As shown in Fig. 4.6, without estimation errors the position can be determined by setting Eq. (4.7) equal to Eq. (4.8). However, the probability that a pair of parameters s(BS) and t(BS) can be found such that s(BS) =t(BS) is zero, due to noise and the variance of any DoA estimator. Therefore, s(BS) and t(BS) are called skew lines. In [Lev15] it is stated that the source position is determined as the middle point of the smallest segment between these skew lines, but [Lev15] does not describe how the smallest segment and the centroid is obtained. The author of this thesis published this in [KTPI17]. In literature, this method is known as the “shortest distance between skew lines” technique [HCV99]. For ease of notation the notations
in Eq. (4.9) and Eq. (4.10) according to Fig. 4.7 are introduced as
m(DoA) in Eq. (4.9) is the orientation vector ofs(BS) and n(DoA) in Eq. (4.10) is the orientation vector of t(BS).
With this, Eq. (4.7) and Eq. (4.8) can be rewritten as
s(BS)=s(BS)m(DoA)+ ˜p(BS)1 =
The vector from s(BS) tot(BS) is obtained by
v(st)=t(BS)−s(BS)
According to [HCV99] the length of vector v(st) is the shortest distance if and only ifv(st) is perpendicular to the orientation vectorsm(DoA) and n(DoA) resulting in conditions
m(DoA)v(st)= 0 =m(x)(t(BS)n(x)+x(C)2 −s(BS)m(x)−x(C)1 )+
The problem of two linear equations for two unknown variabless(BS)andt(BS) has always a unique solution.
First, Eq. (4.14) is transformed to
0 =t(BS)m(x)n(x)+m(x)x(C)2 −s(BS)m2(x)−m(x)x(C)1 + t(BS)m(y)n(y)+m(y)y2(C)−s(BS)m2(y)−m(y)y1(C)+ t(BS)m(z)n(z)+m(z)z2(C)−s(BS)m2(z)−m(z)z1(C).
(4.16)
Secondly,t(BS)ands(BS)are separated and for the sake of readability other values (constants) are aggregated to auxiliary variables a(h), b(h),and c(h) below the under-braces in
0 =t(BS)(m(x)n(x)+m(y)n(y)+m(z)n(z))
Next, Eq. (4.17) is rearranged tos(BS) by
s(BS)= t(BS)a(h)+c(h)
b(h) . (4.18)
The same steps are performed on Eq. (4.15), where constants are aggregated to d(h), e(h), and f(h) in 0 =t(BS)(n2(x)+n2(y)+n2(z))
which is rearranged to t(BS) by
t(BS)= s(BS)e(h)−f(h)
d(h) . (4.20)
Eq. (4.18) is put into Eq. (4.20) and rearranged to t(BS) by t(BS)= c(h)e(h)−f(h)b(h)
d(h)f(h)−e(h)a(h) = c(h)a(h)−f(h)
1−a2(h) , (4.21)
where details can be found in the Appendix in Eq. (A.1). The re-arrangement in Eq. (4.20) utilizes b(h) = d(h) = 1 and a(h) = e(h) of which details are given in Table A.1. Finally, s(BS) is obtained by putting and ˜t(BS)are obtained, respectively, defining the shortest distance. ˜s(BS)is marked as a light blue downward pointing triangle in Fig. 4.7b and ˜t(BS) is marked as a light blue upward pointing triangle in Fig. 4.7b, respectively. Straightforward the segment which is the shortest distance between s(BS) and t(BS) is denoted as ˜v(st) and given by
illustrated in Fig. 4.7b by the deep blue diamond.
0 100 200 300 400 500
u: : :Estimated source position
~
Figure 4.7.: Illustration for DoA estimation based on “shortest distance between skew lines” in 3D space.
Figure 4.8.: Deployment of BSs and sources. The arrows indicate the orientation of the antenna arrays.
For the numerical evaluation a typical hexagonal deployment as recommended for system level evaluation in 3GPP [3GP17f] is assumed. From this hexagonal deployment two neighbor BSs are selected resulting in BS positions ˜p(BS)1 = [0 0 5]Tm and ˜p(BS)2 = [250 −422 5]Tm corresponding to an inter-site distance (ISD) of 500 m.
The corresponding orientation vectors ˜α1 = ˜β1 = 0◦for BS 1 and ˜α2 = 120◦, ˜β2 = 0◦ for BS 2 are illustrated in Fig. 4.8, where arrows represent the orientation vectors. Sources are uniformly randomly distributed in the horizontal and vertical angular range given byα∈ {0, ...,−65}◦ and β∈ {−5, ...,5}◦, respectively, from BS 1 perspective. The distance between BS 1 and the sources is in the interval {30, ...,300}m. Fig. 4.8 shows and example of 500 source positions in the above defined area. All remaining parameter assumptions are listed in Table 4.2.
In Fig. 4.9, the DoA estimation and position errors are given. A first observation is that in Fig. 4.9a the cumulative distribution functions (CDFs) of ∆(α) have a larger variance and larger values compared to the CDFs of ∆(β) for both BSs. This is because the horizontal angular spread of the sources is larger than the vertical angular spread and it is shown in Fig. 3.12 that for angles ≷±60◦ the resolution of the array becomes the limiting factor. A second observation is that the horizontal and vertical estimation errors of BS 2 are larger compared to BS 1, because the average distance of the source to BS 2 is larger than to
Table 4.2.: Parameter assumptions for simulations on positioning based on two DoAs estimates from two BSs.
Parameter Value
Simulation type Monte Carlo
Number of realization 100
Channel model QuaDRiGa [JRBT14]
Scenario Urban macro line of sight [3GP17f]
Center frequency 3.75 GHz
RicianK-Factor 10 dB
BS antenna distribution UPA
Number of BS antenna elements N 64
Number of elements in y(C)-directionn(y) 8 Number of elements in z(C)-direction n(z) 8 Antenna element spacingd(N)i,i+1 λ(c)/2
Antenna element type Patch
Patch element azimuth HPBW 65◦
Patch element elevation HPBW 65◦
Element directive gain 9.4 dBi
Bandwidth 180 kHz
BS coordinates ˜p(BS)1 ,p˜(BS)2 [0 0 5]Tm, [250 −422 5]Tm BS orientations ˜α1,β˜1,α˜2,β˜2 0◦,0◦,120◦,0◦
DoA estimator MUSIC
Target quantization of ASSQ [qS(α)qS(β)] [0.1 0.1]◦
Position calculation method Shortest distance of skew lines
Weighting factor g(p) 0.5
Source distribution Random and uniform
Horizontal angular range of sources [0,−65]◦ Vertical angular range of sources [5,−5]◦ Distance range from BS 1 to sources [30,300] m
(a) Distribution of horizontal and vertical DoA estima-tion errors.
0.124
(b) Position error distribution. 0.124 % of the position errors are below the target of 1 m.
Figure 4.9.: Localization performance in cellular deployment of two BSs with 500 m ISD.
BS. Even if both BSs observe the same K-factor9, the receive signal to noise ratio (SNR) at BS 2 is lower compared to BS 1. In Fig. 4.9b the CDF of the position error ∆(xyz)is larger compared to ∆(α)or ∆(β). One reason for this is the distance dependency of the position error additional to the DoA estimation error, see Fig. 3.15a. Another reason for the larger distribution of ∆(xyz) is caused by the geometry of the deployment, the so-called DoP effect or geometry dilution of precision (GDoP). In Fig. 4.10 the position error is plotted over the positions of the devices in the x(C)-y(C) plane. There is a region around the line of sight between BS 1 and BS 2 where position errors ∆(xyz)≤20 m are concentrated. This problem is known as dilution of precision and in more detail discussed in the following section as a challenge of DoA based positioning in cellular deployments.