Cross Correlations

We have so far thought of the signal as a function of time, after it enters the antenna. Let us now liberate ourselves from one dimension (time) and think of the electric field as existing in space and time, before it is collected by the antenna. In this view, one can obtain a delayed version of the signal by moving along the longitudinal direction (direction of the source). Thus, the frequency content is obtained by Fourier transforming a longitudinal spatial correlation. As explained in Chapter 2, the spatial correlations transverse to the direction of propagation carry information on the angular power spectrum of the signal, i.e. the energy as a function of direction in the sky. With hindsight, this can be viewed as a generalisation of the Wiener- Khinchin theorem to spatial correlations of a complex electric field which is the sum of waves propagating in many different directions. Historically, it arose quite independently (and about at the same time!) in the context of optical interference. This is the van Cittert-Zernike theorem of Chapter 2. Since one is now multiplying and averaging signals coming from different antennas, this is called a ``cross correlation function''. To get a non-vanishing average, one needs to multiply $E_1(x,t)$ by $E_2^*(y,t)$. The complex conjugate sign in one of the terms ensures that this kind of product looks at the phase difference. Writing out each signal as a sum with random phases, the terms which leave a non-zero average are the ones in which an $e^{i\varphi_n}$ in an $E$ cancels a $e^{-i\varphi_n}$ in an $E^*$. An (ill-starred?) product of two complex $E$'s with zero (or two!) complex conjugate signs would average to zero.