2.3 Delay Tracking and Frequency Conversion

$\parbox{120mm}{ \caption{Figure 2.4: 2-element heterodyne interferometer with delay tracking after frequency conversion}}$

To compensate for the geometrical delay variations, delay lines with mirrors (as in optics...) would be completely impractical given the required size of the mirrors. The compensating delay is thus performed electronically after one (or several) frequency conversion(s), as illustrated in Fig.2.4. This can be implemented either by switching cables with different lengths, or in a more sophisticated way, by using shift memories after digital sampling of the signal in the correlator. Apart for a few details (see R.Lucas lecture, Chapter 7), the principle remains identical.

In the case presented in Fig.2.4, for USB conversion, the phase changes of the input signals from antenna 1 and 2 before reaching the correlator are respectively

$$\Phi_1 = 2 \pi \nu \tau_G =  2 \pi (\nu_{LO} + \nu_{IF}) \tau_G$$ (2.23)

$$\Phi_2 = 2 \pi \nu \tau_I + \Phi_{LO}$$ (2.24)

Introducing $\Delta\tau = \tau_g-\tau_I$ as the delay tracking error, the correlator output is

$$r  = A_o \vert V\vert \cos(\Phi_1 - \Phi_2 - \Phi_V)$$

$$USB r  = A_o \vert V\vert \cos(2\pi(\nu_{LO}\tau_G+ \nu_{IF}\Delta\tau) - \Phi_V -\Phi_{LO})$$

$$LSB r  = A_o \vert V\vert \cos(2\pi(\nu_{LO}\tau_G- \nu_{IF}\Delta\tau) - \Phi_V -\Phi_{LO})$$ (2.25)

When the two sidebands are superposed, we can just sum the USB and LSB outputs, which yields (after usual re-arrangement of the cosine expressions)

$$DSB r  = 2 A_o \vert V\vert \cos(2\pi(\nu_{LO}\tau_G - \Phi_V - \Phi_{LO})) \cos(2\pi\nu_{IF}\Delta\tau)$$ (2.26)

This shows that the amplitude is modulated by the delay tracking error. The tolerance can be exceedingly small. For example, at Plateau de Bure, the IF frequency $\nu_{IF}$ is 3 GHz, and a 1 % loss is obtained as soon as the delay tracking error would be 7.5 picoseconds, i.e. a geometrical shift of 2.2 mm only. Due to Earth rotation, the geometrical delay changes by such an amount in 0.1 s for a 300 m baseline. Hence, delay tracking would have to be done quite fast to avoid sensitivity losses. To avoid this problem, it is common to use sideband separation. The delay tracking error should then be kept small compared to the bandwidth of each spectral channel, $\Delta \tau_G« 1/\Delta \nu$, and the delay can then be adjusted much less frequently.