# Correlation Technologies for Emerging Wireless Applications
## 2 Correlation Functions for Stochastic Fields
In [12; 13] the challenges of measuring both field and intensity correlations at the same time are addressed for a direct assessment of the Siegert relation highlighting the conditions where the relation fails. In the situations where the Siegert relation fails, measuring \(g^{(2)}(\tau)\) convey quantum effects and helps in distinguishing different types of sources in the perspectives of properly classifying and reconstructing them in practical tomographic and imaging applications. An implication of Equation (6) is that there is an excess of intensity correlation at zero delays, \(g^{(2)}(0)>g^{(2)}(\tau\rightarrow\infty)=1\) following in reference to \"_photon bunching_\" or the \"_Hanbury Brown and Twiss_\" effect. For deterministic noise power density distribution, the challenge of energy detection of unknown signals in the presence of noise is discussed in [12]. For stochastic signals, it is established that numerical values of noise amplitudes cannot be specified. Thus, for modeling and measuring stochastic signals, it is required to deal with energy and power spectra through the extraction of correlation functions. The energy density can be written as the sum of electric and magnetic energy densities [14]:  
\[W(\rho)=W_{E}(\rho)+W_{H}(\rho) \tag{7}\]  
\[W_{E}(\rho)=\frac{\epsilon}{2}|E(\rho)|^{2}\;and\;\;W_{H}(\rho)=\frac{\mu}{2}| H(\rho)|^{2} \tag{8}\]  
The correlation function of the electric or magnetic field is defined as:  
\[C_{X}^{FF}\equiv\frac{X(\rho_{1})\cdot X^{*}(\rho_{2})}{\sqrt{\left\langle \left|X(\rho_{1})\right|^{2}\right\rangle\left\langle\left|X(\rho_{2})\right| ^{2}\right\rangle}} \tag{9}\]  
where \(\left\langle X\right\rangle\) refers to ensemble average (expectation) applied to stochastic variable \(X\) and \({}^{*}\) stands for complex conjugate. In (9) the square-root in the denominator is introduced for normalization purposes without loss of generality. The correlation function of the electric and magnetic energies can be deduced as:  
\[|E(x)|=\sqrt{\Gamma(x,x)} \tag{2}\]0  
\[|E(x)|=\sqrt{\Gamma(x,x)} \tag{2}\]1For stationary stochastic signals, the spatial correlation functions for the total field \(X_{t}\) exhibit a \(SinC(k\rho)\) law. \[|E(x)|=\sqrt{\Gamma(x,x)} \tag{2}\]2  
The spatial correlation functions of the transverse components \(X_{t}\) can be expressed as:  
\[|E(x)|=\sqrt{\Gamma(x,x)} \tag{2}\]3  
where it can be established that \(\kappa=cos\bigg{(}\frac{k\rho}{2}\bigg{)}\). The \(SinC(k\rho)\) law can be implemented using advanced signal processing convolutional accelerators based on broadband expansions:  
\[|E(x)|=\sqrt{\Gamma(x,x)} \tag{2}\]4  
\[|E(x)|=\sqrt{\Gamma(x,x)} \tag{2}\]5  
The proposed SinC-based convolutional signal processing enables effective unification [4] of modeling and measurement techniques into a coherent framework. The benefits of unifying and aligning measurement (instrumentation) and modeling (EDA tooling) platforms go beyond the simple need for productivity enhancement, and they open new perspectives for transforming apparent antagonists between theory and experiment into necessary complementarities [2; 15]. Such complementarities will render possible introducing in the universe of simulations the notions of uncertainties usually considered as specific to the world of measurement. Although significant research efforts have been directed towards building accurate numerical algorithms dealing with modeling of uncertainties with controlled numerical errors, lack of knowledge in exact numerical values to assign to input parameters (design parameters, boundary and environmental conditions, etc.) has put constraints in their effective usage. The uncertainties are not only the results of our partial or insufficient knowledge of the mechanisms underlying the true physics; they also reflect various types of errors. In order to adopt variation-aware modeling methodologies, it is essential to include uncertainty analysis in simulations from the beginning and not in a posttreatment procedure.