Evaluation of 5G Positioning Performance Based on UTDoA, AoA and Base-Station Selective Exclusion

. 2021 Dec 24;22(1):101. doi: 10.3390/s22010101

Algorithm 1 NLoS BS detection and exclusion. Procedure from 1 to 20 performs the algorithm with all the available BS, while procedure from 22 to 27 performs the algorithm excluding one BS at a time when the computed test statistic with all the available BS exceeds the given threshold

Input: $x_{B S, n}$ , $\hat{x}$ , $Cov$ , $γ$ , $W_{T D o A}$ , $\hat{ρ}$ , where $n = 1, 2, \dots, M_{B S_{T D o A}}$

Output: ${\hat{x}}_{l}$

1:
Set $NrIter = 10$ to perform GN solution
2:
${\hat{x}}_{l} = \hat{x}$
3:
Set $l = 0$
4:
while $l \leq NrIter$ do
5:
$l = l + 1$
6:
Compute $G_{TDoA, (n, 1 : 3)} = [\begin{matrix} \frac{x_{BS, 1} - {\hat{x}}_{l}}{‖ x_{BS, 1} - {\hat{x}}_{l} ‖} - \frac{x_{BS, (n + 1)} - {\hat{x}}_{l}}{‖ x_{BS, (n + 1)} - {\hat{x}}_{l} ‖} \end{matrix}]$
7:
Compute the pseudo-range error $ρ ({\hat{x}}_{l - 1})$ .
8:
Compute the UE estimated location ${\hat{x}}_{l}$ :
9:
${\hat{x}}_{l} = {\hat{x}}_{l - 1} + {(G_{TDoA, l - 1}^{T} W_{TDoA}^{- 1} G_{TDoA, l - 1})}^{- 1} G_{TDoA, l - 1}^{T} W_{TDoA}^{- 1} (ρ ({\hat{x}}_{l - 1}) - \hat{ρ})$
10:
end while
11:
Compute $G_{TDoA}$ as in 6, where ${\hat{x}}_{l}$ is from 9
12:
Decompose $G_{TDoA}$ into $U_{S}$ and $U_{N}$ through QR factorization
13:
Compute the residuals $ε = U_{N}^{T} Δ ρ$ , where $Δ ρ = (ρ ({\hat{x}}_{l - 1}) - \hat{ρ})$ from 9
14:
Compute the covariance matrix of the range residual vector ${Cov}_{r e s} = U_{N} Cov U_{N}^{T}$
15:
Perform eigen decomposition of ${Cov}_{r e s} = V D V^{T}$
16:
Whiten the range residual vector as $\hat{ϵ} = {(V \sqrt{D})}^{- 1} ϵ$
17:
Compute the sum of squared residuals: $SSR = {\hat{ϵ}}^{T} \hat{ϵ}$
18:
Compute $T = \sqrt{SSR / (M_{B S_{T D o A}} - 4)}$
19:
if $T \leq γ$ then
20:
Use the solution ${\hat{x}}_{l}$ in 9
21:
else
22:
A bias might be present
23:
Create subsets of BS by setting the 1st BS as serving and remove one BS from the rest of BS at a time, ${x_{B S, n^{'}}}_{i}$ , where $i \in {1, 2, \dots, M_{B S_{T D o A}} - 1}$ and $n^{'} \in n - {i + 1}$
24:
For each of the subsets compute $T_{i}$ following the procedure from 1 to 18
25:
Choose ${x_{B S, n^{'}}}_{i}$ with minimum $T_{i}$ that does not exceeds $γ$
26:
if $T_{i}$ found then
27:
Use solution ${\hat{x}}_{l}$ computed by the corresponding ${x_{B S, n^{'}}}_{i}$
28:
else
29:
Choose the minimum $T_{i}$ among test statistics computed in 18 and 24
30:
Use solution ${\hat{x}}_{l}$ computed by the corresponding group of BSs
31:
end if
32:
end if