XF Correlator

Figure 9.5: Block diagram of a XF correlator.

Eq. 9.0.1 for a broadband signal after delay compensation and integration (time average) can be written as

$$<r_R> = \mbox{Re}\bigl[\int_{-\infty}^{+\infty}<v_1(\nu,t)v^*_2(\nu,t)>\mbox{d}\nu\bigr]\;,$$ (9.2.13)

where $v_1(\nu,t)$ and $v_2(\nu,t)$ can be considered as the spectral components of the signals from the antennas. Introducing a delay of $\tau$ to one of the signals $v_1(\nu,t)$ modifies the above equation to

$$<r_R(\tau)> = \mbox{Re}\bigl[\int_{-\infty}^{+\infty}<v_1(\nu,t)v^*_2(\nu,t)>e^{-j2\pi\nu\tau}\mbox{d}\nu\bigr]$$ (9.2.14)

The above equation is a Fourier Transform equation; the Fourier Transform of the cross spectrum $<v_1(\nu,t)v^*_2(\nu,t)>$ (averaging over $t$). Thus $<r_R(\tau)>$ is the cross correlation measured as a function of $\tau$. Since $v_1(\nu,t)$ and $v^*_2(\nu,t)$ are Hermitian functions, as they are spectra of real signals, their product is also hermitian. Therefore $<r_R(\tau)>$ is a real function and hence it can be measured with a simple correlator (not a complex correlator). Thus the second method of measuring spectral visibility is to measure $<r_R(\tau)>$ for each pair of antennas as a function of $\tau$ and later perform an Fourier Transform to get the cross spectrum. The digital implementation of this method is called an XF correlator.

A block diagram of an XF correlator is shown in Fig. 9.5. In this diagram, fractional delays are compensated for by changing the phase of the sampling clock. After delay compensation, the cross correlations for different delay are measured using delay lines and multipliers, which are followed by integrator. Since the cross correlation function in general is not an even function of $\tau$, the delay compensation is done such that the correlation function is measured for both positive and negative values of $\tau$ in the correlator. The zero lag autocorrelations of the signals are also measured, which is used to normalize the cross correlation. The quantization correction (block marked as F) is then applied to the normalized cross correlations. The cross spectrum is obtained by performing a DFT on the corrected cross correlation function. A peculiarity of this implementation is that the correlations are measured first and the Fourier Transform is taken later to get the spectral information. Hence it is called an XF correlator.