# Correlation Technologies for Emerging Wireless Applications
## 3 Correlation-Based Power and Energy-Density Metrics for OTA-Testing of Mobile Devices
The combination of eV-Technologies RF and mmWave correlator modules with optically synchronized dual-port instruments (Figure 10b) opens new possibilities for OTA-based interferometric [22; 23] measurement (including correlation-based EVM testing [24]) of MIMO applications using minimally invasive probing solutions [20]. Figure 4:2 64-Channels correlator (**a**) module combined with 64 broadband sensor-array elements. Optically synchronized VNA modules for broadband extraction of correlation functions (**b**). Figure 4:3 Extracted power-density (**a**), (**b**) as function of probing distance. of the physical system. The variable z is a label for the set of possible messages, and the probability over this set, s is a particular value from the set. Equation (25) being valid in the case of non-equilibrium systems, for a well-defined ensemble probability distribution, Pz(s), several conceptual difficulties arise from the physical interpretation of system complexity in link with equilibrium entropy. These conceptual difficulties reveal the dependence of entropy on the scale [30] of analysis. The concept of a scale-dependent entropy dates back to the \(\varepsilon\)-entropy of [PERSON] [32], which can be linked to [PERSON] noisy-channel information theory [33]. Multi-scale analysis of the structure of the state space relates the energy, entropy, and geometry at different levels of resolution for both equilibrium and non-equilibrium conditions. For dynamic systems in non-equilibrium conditions, the use of entropy-based [34] metrics for evaluating the attributes of chaotic regimes will open new possibilities. In [35], it is suggested that for any bounded system with entropy Sentropy and rest energy ERest, there exists a universal upper limit on the entropy-to-energy ratio, which leads to the following inequality:  
Figure 4:4  
where R represents the radius of the sphere circumscribing the system. For topologically compact systems, R is to be defined in terms of the system's volume. In the derivation of (26) we have assumed \(h/2\pi=k=G=1\) without loss of generality. In [36], the implications of the entropy-to-energy ratio on an upper bound of the entropy-to-surface-area ratio are discussed. The resulting entropy-to-surface-area ratio is expressed in the form:  
Figure 4:5  
The limit of (26) leads to the equality \(\mathrm{S}_{\mathrm{Entropy}}=\mathrm{A}_{\mathrm{Surface}}/4\), which is consistent with the expression for black holes. The derived entropy-to-energy and entropy-to-surface-area bounds open new possibilities in communication theory for properly coupling IT and PT into a unified approach. In reference to computational energy dissipation, [PERSON], in 1961, has stated the principle that erasing one bit of information entails an energy loss of kTln(2) (the thermodynamic threshold), where k is the Boltzmann constant and T is the temperature in Kelvin. This principle has generated rich comments [37] and motivated new directions of ambitious research solutions (e.g., applied superconductivity [38]) with less dissipation than the Landauer limit. Figure 11a shows technology trends [39] expressed in terms of dissipated energy per operation in reference to the Landauer limit. Figure 3a presents the power consumption as a function of bit rates and computing power versus dissipated energy in J/bit. Following [PERSON]'s exponential trend, the energy efficiency of a transistor and a logic gate has improved by a factor of one hundred in ten years, corresponding to a factor of two every 18 months. The trends of [PERSON]'s prediction are starting to show some saturation as the exponential decrease in the physical size of the transistor is constrained by the physical limits. At the lower scales, the switching energy is approaching the thermal noise spectral density. In addition, the cooling capacity in terms of energy per unit time for a given area will lead to a bottleneck imposing energy-geometry-entropy trade-offs [39].