In Multi-RTT positioning model, the main factors that affect the positioning performance include the number of base stations, distribution of base stations, and accuracy of observations. Assuming the estimated error of the state vector \({\varvec{X}}\) is \(\Delta {\varvec{X}}\), after ignoring and simplifying the second-order error term, the relationship between the estimated error \(\Delta {\varvec{X}}\) and the observation error \({\varvec{V}}\) can be obtained as

$$\Delta {\varvec{X}}={\left({{{\varvec{H}}}_{t}}^{\mathrm{T}}{\varvec{P}}{{\varvec{H}}}_{t}\right)}^{-1}{{{\varvec{H}}}_{t}}^{\mathrm{T}}{\varvec{P}}{\varvec{V}}$$

(15)

where \({{\varvec{H}}}_{t}\) is calculated by bringing in the true value of the user’s coordinates in the formula for calculating the transition matrix \({\varvec{H}}\). The time delay observations are assumed to be independent of each other and satisfy the zero-mean Gaussian distribution with the variance of \({{\sigma }_{t}}^{2}\). Accordingly, the accuracy of the 3D position estimation based on Multi-RTT positioning Root Mean Square (RMS) \({s}_{\mathrm{RMS}}^{\mathrm{RTT}}\) can be written as

$${s}_{\mathrm{RMS}}^{\mathrm{RTT}}=\sqrt{\mathrm{tr}\left\{\mathrm{E}\left(\Delta {\varvec{X}}\cdot \Delta {{\varvec{X}}}^{\mathrm{T}}\right)\right\}}={c\sigma }_{t}\sqrt{\mathrm{tr}\{{\left({{{\varvec{H}}}_{t}}^{\mathrm{T}}{{\varvec{H}}}_{t}\right)}^{-1}\}}$$

(16)

\({d}_{\mathrm{DOP}}=\sqrt{\mathrm{tr}\left\{{\left({{{\varvec{H}}}_{t}}^{\mathrm{T}}{{\varvec{H}}}_{t}\right)}^{-1}\right\}}\) is completely determined by the number of base stations and the geometric relationship between the user and base stations. Thus,

$${s}_{\mathrm{RMS}}^{\mathrm{RTT}}={d}_{\mathrm{DOP}}\cdot c{\sigma }_{t}$$

(17)

When the accuracy of time delay observation is the same, the smaller the \({d}_{\mathrm{DOP}}\), the higher the accuracy of position estimation will be.

In RTT/AOD positioning, vector \({\varvec{V}}\) is a column vector composed of angle and time delay observation errors. Assuming that each observation error is independent, the accuracy of the 3D position estimation RMS \({s}_{\mathrm{RMS}}^{\mathrm{RTT}/\mathrm{AOD}}\) can be written as

$${s}_{\mathrm{RMS}}^{\mathrm{RTT}/\mathrm{AOD}}=\sqrt{\mathrm{tr}\left\{\mathrm{E}\left(\Delta {\varvec{X}}\cdot \Delta {{\varvec{X}}}^{\mathrm{T}}\right)\right\}}=\sqrt{\mathrm{tr}\{{\left({{\varvec{H}}}_{t}^{\mathrm{T}}{\varvec{P}}{{\varvec{H}}}_{t}\right)}^{-1}\}}$$

(18)

The weighted matrix \({\varvec{P}}\) in (18) includes time delay accuracy and angle measurement accuracy. Thus, the relationships among \({s}_{\mathrm{RMS}}^{\mathrm{RTT}/\mathrm{AOD}}\), \({d}_{\mathrm{DOP}}\), and observation accuracy are no longer linear (Han, 2014).

In (16) and (18), the RMS represents the accuracy of the Weighted Least Square (WLS) estimated user’s position. Because WLS estimation belongs to a minimum variance estimation, when the initial coordinates of the user are equal to the true coordinates, the a posteriori estimated variance matrix of WLS is the inverse of the Fisher information matrix, which is consistent with the definition of Cramér-Rao Bound (CRB).

As shown in Section “Appendix”, \({s}_{\mathrm{RMS}}^{\mathrm{RTT}/\mathrm{AOD}}\) is smaller than \({s}_{\mathrm{RMS}}^{\mathrm{RTT}}\). Thus, the addition of AOD improves the accuracy of position estimation. One can see if the accuracy of time delay \({\sigma }_{t}\) and the \({d}_{\mathrm{DOP}}\) are the invariant, the higher the accuracy of angle measurements, the higher the accuracy of position estimation will be.

The addition of AOD reduces the influence of the distribution of base stations on positioning results. In a complex environment, the number of base stations that can be observed by the user is small, and the distribution of base stations is poor. Accordingly, the addition of AOD is conducive to the accuracy and reliability of 5G positioning. The following part will analyze the influence of the AOD on positioning accuracy and reliability.

### Influence of AOD on positioning accuracy

The influence of the distribution of base stations on positioning accuracy can be measured by \({d}_{\mathrm{DOP}}\) value. For example, in the 2D Multi-RTT positioning, assuming that the number of base stations is \(N\), and \(\left(\mathrm{cos}{\theta }_{i},\mathrm{sin}{\theta }_{i}\right),{\theta }_{i}\in (-\uppi ,\uppi ]\) represents the relative position from the \(i\) th base station to the user, then

$${{\varvec{H}}}_{t}=\left[\begin{array}{cc}-\mathrm{cos}{\theta }_{1}& -\mathrm{sin}{\theta }_{1}\\ -\mathrm{cos}{\theta }_{2}& -\mathrm{sin}{\theta }_{2}\\ \vdots & \vdots \\ -\mathrm{cos}{\theta }_{N}& -\mathrm{sin}{\theta }_{N}\end{array}\right],$$

(19)

$${d}_{\mathrm{DOP}}=\sqrt{{\mathrm{tr}\{\left({{\varvec{H}}}_{t}^{\mathrm{T}}{{\varvec{H}}}_{t}\right)}^{-1}\}}=\frac{\sqrt{N}}{\sqrt{\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}{\mathrm{sin}}^{2}({\theta }_{i}-{\theta }_{j})}}.$$

(20)

Taking 3 base station as an example, the connections between base stations and a user divide the space around the user into three sectors with the sector angles denoised as \({\alpha }_{1}\), \({\alpha }_{2}\), and \({\alpha }_{3}\). When \({\alpha }_{1}={\alpha }_{2}=60^\circ ,{\alpha }_{3}=240^\circ\), or \({\alpha }_{1}={\alpha }_{2}={\alpha }_{3}=120^\circ\),\({d}_{\mathrm{DOP}}\) has a minimum value \(\frac{2\sqrt{3}}{3}\), corresponding to the optimal distribution of three base stations, as shown in Fig. 1.

The positioning accuracy with Multi-RTT and RTT/AOD obtained by the user under the optimal distribution of three base stations (equilateral triangle distribution) and with the accuracy of RTT and AOD measurements being 1 m and 2.5°, respectively, is shown in Fig. 2, where the red triangles stand for the fixed base stations with the spacing of 100 m between them. The above values are typical for mathematical analysis, but not real in an urban area, even in an UMi scenario. For Multi-RTT positioning and RTT/AOD positioning, the users near a base station can be positioned with an accuracy better than 1.4 m. Compared with the positioning accuracy based on RTT, the area where the accuracy is better than 1.4 m is expanded for RTT/AOD positioning.

### Influence of AOD on positioning reliability

Generally, 3D positioning involves three unknown parameters. In Multi-RTT positioning, each base station can only provide one RTT observation, and thus at least three base stations are needed. The positioning principle is shown in Fig. 3(a). In RTT/AOD positioning, a base station provides three observations: distance, vertical angle, and azimuth angle. Therefore, only one base station is enough for positioning, as shown in Fig. 3(b). The addition of AOD reduces the the minimum number of base stations and can effectively improve the positioning reliability when the number of observable base stations is small.

For the details, refer to Section “Influence of AOD on positioning accuracy” When \({\alpha }_{1}={\alpha }_{2}=180^\circ ,{\alpha }_{3}=0^\circ\) or \({\alpha }_{1}=360^\circ ,{\alpha }_{2}={\alpha }_{3}=0^\circ\), \({d}_{\mathrm{DOP}}\) value is the maximum and approaches infinity, which corresponds to the worst distribution of three base stations, as shown in Fig. 4.

In the worst distribution of the three base stations, the base stations are in line with the user. The accuracy of Multi-RTT positioning and RTT/AOD positioning by users under the worst distribution of the three base stations is shown in Fig. 5. The user close to the connection of the base stations has poor accuracy in Multi-RTT positioning. Only when the user is far away from the straight line between the base stations vertically can the positioning accuracy of greater than 1.2 m be obtained. In RTT/AOD positioning, the accuracy for the users near the straight line between the base stations has been greatly improved from greater than 10 m to less than 2 m.

Some conclusions can be drawn below. In the same observation condition, the addition of AOD provides more observations resulting in higher positioning accuracy compared with Multi-RTT positioning. Moreover the addition of AOD reduces the number of base stations, lowers the influence of the poor distribution of base stations on positioning results, and improves the reliability of positioning in poor observation environments.