Spectral Correlator

Figure 9.2: Block diagram of a complex multiplier.

The output of a simple multiplier of the two element interferometer after delay compensation can be written as:

$$r_R = \vert{\cal V}\vert\cos(\Phi_{\cal V}).$$ (9.2.7)

To separate $\vert{\cal V}\vert$ and $\Phi_{\cal V}$ a second product is measured after introducing a phase shift of $90\deg$  in the signal path (see Fig 9.2). Introducing a $90\deg$  shift in the path of one of the signals will result in (se Eq. 9.0.1)

$$r_I(\tau_g) = \vert{\cal V}\vert\cos(2\pi\nu\tau_g + \Phi_{\cal V} + \pi/2),$$ (9.2.8)

and after compensating for $2\pi\nu\tau_g$

$$r_I = \vert{\cal V}\vert\cos(\Phi_{\cal V} + \pi/2)$$ (9.2.9)

$$= \vert{\cal V}\vert\sin(\Phi_{\cal V}).$$

From these two measurement we get

$$\vert{\cal V}\vert = \sqrt{r_R^2 + r_I^2}$$ (9.2.10)

$$\Phi_{\cal V} = \tan^{-1}(\frac{r_I}{r_R}).$$ (9.2.11)

Alternatively, for mathematical convenience, the two measurements can be considered as the real and imaginary part of a complex number, i.e. 

$${\cal V} = r_R + j r_I$$ (9.2.12)

Thus the pair of multipliers together with an integrator (to get the time average) form the basic element of a complex correlator.

In the above analysis a narrow band signal (quasi monochromatic) is considered. In an actual interferometer the observations are made over a finite bandwidth $\Delta \nu$ and one requires the complex visibilities to be measured as a function of frequencies within $\Delta \nu$. This can be achieved in one of the two ways described below.