# Measurement Grid Optimization for OTA Testing of 5G Smart Watches
## 2 Measurement Uncertainty Analysis with Grid Point Reduction  
For OTA testing, the TRP (TIS) is the weighted sum of Effective Isotropic Radiated Power, EIRP (Effective Isotropic Sensitivity, EIS\({}^{-1}\)) values sampled over specified points (theta/phi) across the integration sphere. For a complete sphere with N theta intervals and M phi intervals, respectively, both are measured with even angular spacing, the TRP and TIS can be numerically calculated as follows:  
\[TIS\cong\frac{2M}{\sum_{i=0}^{N}\omega(\theta_{i})\cdot\sum_{j=0}^{M-1}\left[ \frac{1}{EIS_{\theta}(\theta_{i},\varphi_{j})}+\frac{1}{EIS_{\varphi}(\theta_{ i},\varphi_{j})}\right]}. \tag{1}\]  
\[TRP\cong\frac{1}{2M}\cdot\sum_{i=0}^{N}\sum_{j=0}^{M-1}\left[EIRP_{\theta}( \theta_{i},\varphi_{j})+EIRP_{\varphi}(\theta_{i},\varphi_{j})\right]\cdot \omega(\theta_{i}) \tag{2}\]  
\[\theta_{i}=i\Delta\theta\text{ where }\Delta\theta=\frac{\pi}{N} \tag{3}\]  
\[\varphi_{j}=j\Delta\varphi\text{ where }\Delta\varphi=\frac{2\pi}{M} \tag{4}\]  
where EIS and EIRP are Effective Isotropic Sensitivity and Effective Isotropic Radiated Power values sampled at each polarization and grid point, respectively, while \(w(\theta_{i})\) is the theta-dependent weighting function. In this work, the Clenshaw-Curtis quadrature in Equation (5) is utilized for TRP/TIS calculations [14]. Since TIS testing takes significantly more test time compared to that of the TRP, prioritizing the test-time reduction for TIS is crucial. Moreover, the proposed analysis and related conclusions for the TIS also apply to the TRP due to reciprocity. \[W(\theta_{i})=\frac{c_{i}}{N}\left[1-\sum_{j=1}^{int(\frac{N}{2})}\frac{b_{j} }{4j^{2}-1}cos(2j\theta_{i})\right]\]  
with  
\[b_{j}=\left\{ \begin{aligned} & 1,j=N/2\\ & 2,j<N/2\end{aligned} \right\},c_{i}=\left\{ \begin{aligned} & 1,i=0\text{ or }N\\ & 2,\text{ }otherwise\end{aligned} \right\}. \tag{5}\]  
The standard TRP measurement grid with 15-degree intervals in both theta and phi is illustrated in Figure 2.  
The samples and weights for the Clenshaw-Curtis quadrature with this reference grid configuration (\(\Delta\theta\) = 15\({}^{\circ}\)) are illustrated in Table 1.  
Figure 2: Illustration of the standard 15-degree TRP grid as reference. Currently, the fast TIS approaches being considered in the industry for test-time reduction include the following:  
1. TIS based on the device-reported RSS (received signal strength) values [14; 15];
2. Spiral scan method [14];
3. The theta-dependent phi optimization [14];
4. Device rotation during measurement [14];
5. Single or multi-point offset in Anechoic Chambers (ACs) [14];
6. Continuous-mode stirring in Reverberation Chambers [16; 17; 18];
7. Grid points reduction. Among all the aforementioned potential methods, the grid points reduction is the most effective and direct method, applicable to all bands and wireless technologies-- that is, the original intention of this study, i.e., to determine optimal measurement grids with grid point reduction for OTA testing of 5G smart watches. Apparently, from (1) and (2), the accuracy of OTA testing, specifically the calculation of TRP/TIS, depends heavily on measurement grid configurations. The more points and the more even the distribution, the higher the precision, and consequently, the longer the test time. To reduce test time, a compromise should be made between accuracy and the number of grid points required. Importantly, analyzing the influence of different grid configurations on OTA testing accuracy, as well as quantitatively evaluating relevant MU terms, are prerequisites. However, relevant work in this area has not been conducted or published yet and remains a major concern of the industry. For the sake of simplicity, this study provides a quantitative influence analysis of measurement grid configurations on the accuracy of TRP calculations. The proposed approach and related conclusions presented here also apply to TIS testing.