4.3 The Correlator in Practice

In order to numerically evaluate the cross-correlation function $R_{\rm ij}$, the continuous signals entering the cross correlator need to be sampled and quantized. According to Shannon's sampling theorem [Shannon 1949], a bandwidth-limited signal may be entirely recovered by sampling it at time intervals $\Delta t \le 1/(2\Delta\ensuremath{\nu_\mathrm{\scriptscriptstyle IF}} )$ (also called sampling at Nyquist rate). The discrete Fourier transform of the sufficiently sampled cross-correlation function theoretically yields the cross-power spectrum without loss of information. However, in practice, two intrinsic limitations exist:

• In order to discretize a signal, it is not only sampled, it also has to be quantized. The cross-correlation function, as derived from quantized signals, does not equal the cross-correlation function of continuous signals. Moreover, the sampling theorem does not hold anymore for quantized signals. The reasons will become clear below.
• Eq.4.7 theoretically extends from $-\infty$ to $+\infty$. In practice (Eq.4.8), only a maximum time lag can be considered: limited storage capacities and digital processing speed are evident reasons, another limiting factor are the different timescales mentioned before. The abrupt cutoff of the time window affects the data.

These ``intrinsic'' limitations are discussed in Sections 4.3.1 and 4.3.2. The system-dependent performance will be addressed in Section 4.3.3.