Digital Delay

In interferometry the geometric delay suffered by a signal (see Chapter 4) has to be compensated before correlation is done. In an analog system this can be achieved by adding or removing cables from the signal path. An equivalent method in digital processing is to take sampled data that are offset in time. Mathematically, $s(n-m)$ is the sample delayed by $m \times 1/f_s$ with respect to $s(n)$ (where $f_s$ is the sampling frequency). In such an implementation of delay it is obvious that the delay can be corrected only to the nearest integral multiple of $1/f_s$.

A delay less than $1/f_s$ (called fractional delay) can also be achieved digitally. A delay $\tau$ introduced in the path of a narrow band signal with angular frequency $\omega$ produces a phase $\phi = \omega\tau$. Thus, for a broad band signal, the delay introduces a phase gradient across the spectrum. The slope of the phase gradient is equal to the delay or $\tau = \frac{\mbox{d}\phi}{\mbox{d}\omega}$. This means that introducing a phase gradient in the FT of $s(t)$ is equivalent to introducing a delay is $s(t)$. Small enough phase gradients can be applied to realize a delay $< 1/f_s$. In the GMRT correlator, residual delays $\tau < 1/f_s$ is compensated using this method. This correction is called the Fractional Sampling Time Correction or FSTC.