7.2 Bandpass calibration

In the previous section we have we have considered a monochromatic system. We have actually finite bandwidths and in principle the gain coefficients are functions of both frequency and time. We thus write:

$$\ensuremath{\widetilde{V}} _{ij}(\nu,t) = \ensuremath{\mathca... ...}(\nu,t) = g_i(\nu,t) g_j^*(\nu,t) \ensuremath{V} _{ij}(\nu,t)$$ (7.11)

If we make the assumption that the passband shape does not change with time, then we should have for each complex baseline-based gain:

$$\ensuremath{\mathcal{G}} _{ij}(\nu,t) = \mbox{\ensuremath{\ma... ...nsuremath{\mathcal{G}_\mathrm{\scriptscriptstyle C}} }_{ij}(t)$$ (7.12)

The same decomposition can also be done for the antenna-based gains:

$$g_i(\nu,t) = \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B... ...\nu) \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}} }_i(t)$$ (7.13)

$\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}} }_i(\nu)$ is the antenna complex passband shape, which by convention we normalize so that its integral over the observed bandpass is unity; then $\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}} }_i(t)$ describes the time variation of the complex gains.

Frequency dependence of the gains occurs at several stages in the acquisition chain. In the correlator itself the anti-aliasing filters have to be very steep at the edges of each sub band. A consequence is that the phase slopes can be high there too. Any non-compensated delay offset in the IF can also be seen as a phase linearly dependent on frequency. The attenuation in the cables strongly depends on IF frequency, although this is normally compensated for, to first order, in a well-designed system. The receiver itself has a frequency dependent response both in amplitude and phase, due the IF amplifiers, the frequency dependence on the mixer conversion loss. Antenna chromatism may also be important. Finally the atmosphere itself may have some chromatic behavior, if we operate in the vicinity of a strong line (e.g. O₂ at 118 GHz) or if a weaker line (e.g. O₃) happens to lie in the band.