# Correlation Technologies for Emerging Wireless Applications
## 3 Correlation-Based Power and Energy-Density Metrics for OTA-Testing of Mobile Devices
Energy-Geometry-Entropy trade-offs will foster new signal processing solutions taking advantage of the stochastic resonance (SR) concept for low-power operating communication systems. Although noise is generally considered a nuisance, in certain conditions, by exploiting the concept of stochastic resonance [40; 41; 42], noise can enhance the detection of weak signals. The attribute of stochastic resonance is that the signal-to-noise ratio (SNR), following the general formula [40] in Equation (28), is zero for zero added noise (that is, D\(\rightarrow\)0: no noise implies no switching or threshold crossings, thus no output), increases sharply to a maximum at some optimal noise intensity. In Equation (28), \(\epsilon\) is the input signal strength, D is the input noise intensity, and \(\Delta\)U is a constant related to the barrier height or the threshold. Figure 4:6  
The idea of using a narrow-band random noise as a carrier for information transmission was proposed by [PERSON] in his papers [43; 44] published in 1964. With stochastic carriers, a high SNR can be obtained when the modulation bandwidth is relatively limited. The optimization of SNR for stochastic signals requires proper extraction of spectral power density. Since the power spectral density and autocorrelation are a Fourier transform pair, there is a strong link between correlations and SNR. Although Fourier transforms cannot be established rigorously for random processes (infinite energy), they can nevertheless be derived for the autocorrelation and cross-correlation functions which are non-periodic energy signals. The Fourier transforms of the correlation is called power spectrum or spectral density function (SDF). In the practical case of the 3-Port power-splitter structure in Figure 12a, the temporal power of the combined signal \(S_{3}(t)\) in Figure 12a can be cast in the following form assuming a time delay of \(\tau_{D}\) in the presence of noise:  
\[ \begin{array}{c}P_{S_{3}}=\lim\limits_{t\to\infty}\frac{1}{2T}\int_{-T}^{T} \big{(}S_{3}^{2}(t)\big{)}dt=\frac{1}{2}\lim\limits_{t\to\infty}\frac{1}{2T} \int_{-T}^{T}[(S_{1}(t))+S_{1}(t-\tau_{D})+n_{1}(t)+n_{2}(t)]^{2}dt\\ =\frac{1}{2}\lim\limits_{t\to\infty}\frac{1}{2T}\int_{-T}^{T}\{[S_{1}(t)+S_{1 }(t-\tau_{D})]^{2}+\\ 2[(S_{1}(t))+S_{1}(t-\tau_{D})][n_{1}(t)+n_{2}(t)]+\\ [n_{1}(t)+n_{2}(t)]^{2}\}dt\end{array}  \tag{29}\]  
\(n_{1}(t)\) and \(n_{2}(t)\) represent the noise in the two channels. The accuracy of the broadband power power-splitters is essential for the avoidance of squint and impairments in phased-array systems. Integration of the binomial for the signal and noise contributions leads to the following expressions:  
\[ \begin{array}{c}\left[S_{1}+S_{1}(t-\tau_{D})\right]^{2}=S_{1}^{2}(t)+2S_{1 }(t)S_{2}(t)+S_{1}^{2}(t-\tau_{D})\end{array}  \tag{30}\]  
\[ \begin{array}{c}\left[n_{1}(t)+n_{1}(t)\right]^{2}=n_{1}^{2}(t)+n_{2}^{2}(t) +2n_{1}(t)n_{2}(t)\end{array}  \tag{31}\]  
Figure 11: Technology trend (**a**) of dissipated energy per operation expressed in reference to Landauer limit against evolution over years showing saturation effects, power consumption versus bit rate (**b**). The resulting power of the combined signal is twice the power of the reference signal \(S_{1}(t)\) plus twice its autocorrelation power:  
\[P_{S_{3}}(\tau_{D})=\alpha^{2}\big{(}\,2P_{S_{1}}+2R_{Autocorr}(\tau_{D})\big{)} \tag{32}\]  
where \(\alpha\) is relative to the power splitting factor.