# Correlation Technologies for Emerging Wireless Applications
## 3 Correlation-Based Power and Energy-Density Metrics for OTA-Testing of Mobile Devices
With \(E_{i}(\theta,\varphi)\) and \(E_{j}(\theta,\varphi)\) being the radiation patterns of antennas 1 and 2, respectively, the envelope cross-correlation between the two antennas \(i\) and \(j\) expressed in the frequency-domain is given by the following equation:  
\[\rho_{E}(\omega)=\frac{\left|\int_{4\pi}d\Omega\;E_{i}(\theta,\varphi)\cdot E_{ j}^{*}(\theta,\varphi)\right|}{\sqrt{\int_{4\pi}d\Omega|E_{i}(\theta,\varphi) |^{2}}\sqrt{\int_{4\pi}d\Omega|E_{j}(\theta,\varphi)|^{2}}} \tag{16}\]  
When using the S-parameters, the envelope cross-correlation [17; 18; 19] can be expressed as:  
\[\rho_{S}(\omega)=\frac{\left|S_{ii}^{*}(\omega)S_{ij}(\omega)+S_{ji}^{*}( \omega)S_{jj}(\omega)\right|}{\sqrt{1-\left|S_{ii}(\omega)\right|^{2}-\left|S _{ji}(\omega)\right|^{2}}\sqrt{1-\left|S_{jj}(\omega)\right|^{2}-\left|S_{ij }(\omega)\right|^{2}}} \tag{17}\]  
The normalizing loss-factor in Equation (18) is linked to radiation patterns information: when \(\eta_{1}\) and \(\eta_{2}\) denote the radiation efficiencies of antennas 1 and 2, S-parameters based envelope correlation can be cast in the following form:  
\[\rho_{S}^{\eta_{i}-\eta_{j}}(\omega)=\frac{\rho_{S}(\omega)}{\sqrt{(1-\eta_{i })}\sqrt{(1-\eta_{j})}} \tag{18}\]  
Relation (17) results in inequality as a consequence of uncertainties related to the actual value of the correlation of losses. Through the power conservation principle, the pattern orthogonality coefficient can be estimated from the scattering parameters for lossless antenna arrays. For lossy antenna arrays, because of losses in a decoupling network, the scattering parameters are incomplete [17; 18; 19] to calculate pattern orthogonality. In Figure 7, S-parameters-based cross-correlation and field-field-based cross-correlation functions extracted between antenna \(i\) and antenna \(j\) are presented, showing very good agreement over a broad frequency band. When the correlations between antenna \(i\) and antenna \(j\) are extracted, all other antenna elements are loaded by 50 \(\Omega\) condition. Observed differences between S-parameters port-to-port-based cross-correlation and field-field-based cross-correlation functions in the band 20-30 GHz are attributed to imperfect load matching conditions and radiation losses [18; 19] not easy to capture in S-parameters based definitions. Figure 7: Comparisons between S-parameters port-to-port cross-correlation and field–field cross-correlation as function of frequency. Measured power-density (Figure 8) in the near-field (10 mm) using EO [20] probing (with different antenna-factor (AF) values) with variable sensitivity from the DUT at 26 GHz with 23 dBm input power are compared to 3D full-wave EM simulations (CST, EMPIRE, HFSS) in time and frequency domains. Integral representation is used to derive spatial correlation functions for the complex electric and magnetic field components and total energy densities. The propagation of correlation [21] functions discriminating longitudinal and transverse directions of electric and magnetic field components in the near- and far-field lead to new algorithmic solutions (correlation ready convolutional signal-processing at FPGA level) for proper extraction of power-density and energy-density of wireless devices accounting for DUT to probing system interactions. Figure 9,b represent the power-density as a function of distance from the DUT. Figure 8: Measured (**a**) power-density as function of electro-optical (EO) probe sensitivity (from 132 dB/m to 120 dB/m) at 26 GHz: comparison with 3D full-wave EM solvers (CST (**a**), EMPIRE (**b**) and HFSS (**c**)). In the time domain, the auto-correlation (AC) of signal \(I_{S_{i}}\)(_antenna i_) and cross-correlation (CC) functions of signals \(I_{S_{i}}\) and \(I_{S_{j}}\)(_antenna j_) can be extracted using the following expressions [21]:  
Figure 4:0  
Figure 4:1  
The extraction of the auto-correlation and cross-correlation functions can be automated using eV-Technologies RF and mmWave correlator modules (Figure 10a) with embedded FPGA processing capabilities.