4.5.1 Summary of definitions

• Cross-correlation function of voltage outputs $v_{\rm i}$ and $v_{\rm j}$from antenna pair (i,j):
  

  $$R_{ij}({\tau}) = \langle v_{\rm i}(t) v_{\rm j}(t+\tau) \rang... ...\infty}{\frac{1}{T}\int_0^T{v_{\rm i}(t) v_{\rm j}(t+\tau)dt}}$$ (4.18)

  
  

• Covariance of two jointly random variables:
  

  $$\mu = \langle xy \rangle - \langle x \rangle \langle y \rangle$$ (4.19)

  
  
  For signals of zero mean, and again identifying $x = v_{\rm i}(t)$ and $y = v_{\rm j}(t+\tau)$,
  

  $$\mu = R_{ij}({\tau})$$ (4.20)

  
  

• Cross-correlation coefficient of two jointly random variables x,yof variance $\sigma_{\rm x}^2$ and $\sigma_{\rm y}^2$:
  

  $$\rho = \frac{\mu}{\sigma_{\rm x}^2 \sigma_{\rm y}^2}$$ (4.21)

  
  
  For jointly normal random variables of zero mean and of variance $\sigma^2 = \langle x^2 \rangle -\langle x \rangle ^2 = \langle x^2 \rangle$, and with $x = v_{\rm i}(t)$ and $y = v_{\rm j}(t+\tau)$, the cross-correlation function R_ij(t) and the cross-correlation coefficient are related by
  

  $$R_{ij}({\tau}) = \rho\sigma^2$$ (4.22)

  
  

• Bivariate Gaussian Probability Distribution:
  Assume two Gaussian random variables x and y, both of zero mean, and variance $\sigma^2$. The probability $p(x,y)\,dx\,dy$ that the value of x is between x₀ and x₀+dx, and that simultaneously the value of y between y₀ and y₀+dy, is given by the jointly gaussian probability distribution
  

  $$p(x_0,y_0) = \frac{1}{2\pi\sigma^2\sqrt{1-\rho^2}}\, \exp{\le... ...rac{-(x_0^2+y_0^2-2\rho x_0 y_0)}{2\sigma^2(1-\rho^2)}\right)}$$ (4.23)

  
  
  In our case, the variable x is assigned to the output voltage of antenna i at time t, and y the output voltage of antenna j at time $t+\tau$.