# Correlation Technologies for Emerging Wireless Applications
## 2 Correlation Functions for Stochastic Fields  
We introduce correlation technologies implemented using integrated front-end modules for effective, simple, and robust methods for extracting _correlation functions_ defined in (1) as \(\Gamma(x,x^{\prime})\) where the angle brackets refer to an average over the ensemble of realizations of the (dimensionless) analytic signal \(E(x)\) :  
\[\Gamma\big{(}x,x^{\prime}\big{)}=\big{\langle}E^{*}(x)E\big{(}x^{\prime}\big{)}\big{\rangle} \tag{1}\]  
In Figure 1a, an adaptive front-end module (AFEM), implementing a unified RF/mmWave, and baseband correlation technologies are presented. The front-end module provides an adjustable 80 dB dynamic range for efficiently characterizing user-equipment (UE) mobile devices using power-density and energy-density metrics. The AFEM is combined with Mosaic-based array technologies built using wafer-level chip-scale-packaging (WLCSP) compliant with antenna-in-package (AiP) solutions. The resulting AFEM-AiP co-design enables radical vision: replacing antenna arrays with energy-efficient and affordable 'Smart Integrated Electromagnetic Emitting and Receiving surfaces and volumes', conformal to the shape of the object they are installed on. Assuming fully coherent EM distributions, the CF can be factorized to yield the field amplitude:  
\[|E(x)|=\sqrt{\Gamma(x,x)} \tag{2}\]  
The phase of the field can be extracted relative to a constant phase reference \(\varphi_{ref}\):  
\[Arg[E(x)]=\tan^{-1}\bigg{[}\frac{Im(\Gamma(x,x^{\prime}))}{Re(\Gamma(x,x^{ \prime}))}\bigg{]}+\varphi_{ref} \tag{3}\]For random fields, it is required to deal with energy and power spectra [2] through the extraction of stochastic correlation functions. [PERSON] and [PERSON] [10] have indicated possibilities to extract correlation functions with the paradigm of two-beam interferometers, [PERSON]'s double slit, by measurement of the fringe visibility and position for a large number of slit positions and separations. Interferometric techniques have been mainly aimed at a partial reconstruction of the field based on a priori assumptions such as space-shift invariance of the correlation functions or complete coherence of the source. Such measurement setup is thought to be challenging and ill-suited to the measurement of arbitrary correlation functions. Key coherence related quantities such as the temporal field correlation function \(g(\tau)\) referenced as first-order correlation function:  
\[g^{(1)}(\tau)=\frac{\langle E^{*}(t)E(t+\tau)\rangle}{\langle I(t)\rangle} \tag{4}\]  
With \(I(t)=\langle E^{*}(t)|E(t)\rangle\) being the intensity associated with the field \(E(t)\). From the first-order correlation function, coherence-based quantities such as optical spectrum,  
Figure 1: (**a**) Wireless adaptive front-end modules including advanced auto and cross-correlation signal processing and multi-beam MIMO antennas. (**b**) Smart connectors with embedded front-end modules. (**c**) Dual-polarization probe-array using Vivaldi antennas. frequency noise power spectral density (PSD) can be measured. Nevertheless, following the work of [PERSON] [11], it is established that full characterization of the coherence properties of a light source requires the measurement of the correlation functions \(g^{(k)}\) at all orders \(k\). Thus, the second-order temporal correlation function, also called temporal intensity correlation function, can be derived as:  
\[g^{(2)}(\tau)=\frac{\langle I(t)I(t+\tau)\rangle}{\left\langle I(t)^{2}\right\rangle} \tag{5}\]  
For fields that present properties of Gaussian processes, it happens [12; 13] that the correlation functions \(g^{(k)}(\tau)\) are related to each other. In the case of a spatially-coherent polarized chaotic field, the second-order correlation function \(g^{(2)}(\tau)\) is related to the modulus of the temporal first-order correlation function \(g^{(1)}(\tau)\) as follows:  
\[g^{(2)}(\tau)=1+\left|g^{(1)}(\tau)\right|^{2} \tag{6}\]  
Equation (6) is commonly called the Siegert relation.