9.2 Bandpass calibration

In the previous section 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:

$$\ensuremath{\widetilde{V}}_{ij}(\nu,t) = \ensuremath{\mathcal{G}}... ...\ensuremath{V}_{ij}(\nu,t) = g_i(\nu,t) g_j^*(\nu,t) \ensuremath{V}_{ij}(\nu,t)$$ (9.11)

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

$$\ensuremath{\mathcal{G}}_{ij}(\nu,t) = \mbox{\ensuremath{\mathcal{G}_\mathrm{\scriptscriptstyle B}}} _{ij}(\nu) \mbox{\ensuremath{\mathcal{G}_\mathrm{\scriptscriptstyle C}}} _{ij}(t)$$ (9.12)

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

$$g_i(\nu,t) = \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}} _i(\nu) \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}}} _i(t)$$ (9.13)

$\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}}_i(\nu)$ 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 subband. 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. O$_2$ at 118 GHz) or if a weaker line (e.g. O$_3$) happens to lie in the band.

9.2.1 Bandpass measurement

Bandpass calibration usually relies on observing a very strong source for some time; the bandpass functions are obtained by normalizing the observed visibility spectra by their integral over frequency. It is a priori not necessary to observe a point source, as long as its visibility can be assumed to be, on all baselines, independent on frequency in the useful bandwidth. If there is some dependence on frequency, then one should take this into account.

9.2.2 IF passband calibration

In many cases the correlator can be split in several independent subbands that are centered to different intermediate frequencies, and thus observe different frequencies in the sky. In principle they can be treated as different receivers since they have different anti-aliasing filters and different delay offsets, due to different lengths of the connecting cables. Thus they need independent bandpass calibrations, which can be done simultaneously on the same strong source.

At millimeter wavelengths strong sources are scarce, and it is more practical to get a relative calibration of the subbands by switching the whole IF inputs to a noise source common to all antennas (Fig. 9.1). The switches are inserted before the IFs are split between subbands so that the delay offsets of the subbands are also calibrated out. This has several advantages: the signal to noise ratio observed by observing the noise source is higher than for an astronomical source since it provides fully correlated signals to the correlator; then such a calibration can be done quite often to suppress any gain drift due to thermal variations in the analogue part of the correlator. Since the sensitivity is high, this calibration is done by baseline, so that any closure errors are taken out.

When such an ``IF passband calibration'' has been applied in real time, only frequency dependent effects occurring in the signal path before the point where the noise source signal is inserted remain to be calibrated. Since at this point the signal is not yet split between subbands, the same passband functions are applicable to all correlator subbands.

At Plateau de Bure an ``IF passband calibration'' is implemented. Of course when the noise source is observed the delay and phase tracking in the last local oscillators (the one in the correlator IF part) are not applied.

Figure 9.1: IF passband calibration scheme

The precision in phase is $360/\sqrt{\Delta \nu \Delta t} = 0.5^\circ$ at 100 kHz resolution which is sufficient for most projects.

9.2.3 RF bandpass calibration

To actually determine the functions $\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}}_i(\nu)$ we observe a strong source, with a frequency-independent visibility. The visibilities are

$$\ensuremath{\widetilde{V}}_{ij}(\nu,t) = \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}} _i(\nu) \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}} _j^*(\nu) \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}}} _i(t) \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle C}}} _j^*(t) \ensuremath{V}_{ij}(t)$$ (9.14)

Then

$$\mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}} _i(\nu) \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}} _j^*(\nu) = \frac{\ensuremath{\widetilde{V}}_{ij}(\nu,t)}{\int{\ensuremath{\widetilde{V}}_{ij}(\nu,t) d\nu}}$$ (9.15)

since the frequency independent factors cancel out in the right-end side. One then averages the measurements on a time long enough to get a sufficient signal-to-noise ratio. One solves for the antenna-based coefficients in both amplitude and phase; then polynomial amplitude and phase passband curves are fitted to the data.

9.2.3.0.1 Applying the passband calibration

The passband calibrated visibility data will then be:

$$\mbox{\ensuremath{\widetilde{V}_\mathrm{\scriptscriptstyle C}}} _{ij}(\nu,t) = \ensuremath{\widetilde{V}}_{ij}(\nu,t) / \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}} _i(\nu) \mbox{\ensuremath{g_\mathrm{\scriptscriptstyle B}}} _j^*(\nu)$$ (9.16)

the amplitude and phase of which should be flat functions of frequency.

9.2.3.0.2 Accuracy

The most important here is the phase precision: it sets the uncertainty for relative positions of spectral features in the map. A rule of thumb is:

$$\Delta \theta /\theta_{B}= \Delta \phi /360$$ (9.17)

where $\theta_{B}$ is the synthesized beam, and $\Delta\theta$ the relative position uncertainty. The signal to noise ratio on the bandpass calibration should be better than the signal to noise ratio of the spectral features observed; otherwise the relative positional accuracy will be limited by the accuracy of the passband calibration.

The amplitude accuracy can be very important too, for instance when one wants to measure a weak line in front of a strong continuum, in particular for a broad line. In that case one needs to measure the passband with an amplitude accuracy better than that is needed on source to get desired signal to noise ratio. Example: we want to measure a line which is $10\%$ of the continuum, with a SNR of 20 on the line strength; then the SNR on the continuum source should be 200, and the SNR on the passband calibration should be at least as good.

9.2.4 Sideband calibration

A millimeter-wave interferometer can be used to record separately the signal in both sidebands of the first LO (see Chapter 7). If the first mixer does not attenuate the image sideband, then it is useful to average both sidebands for increased continuum sensitivity, both for detecting weaker astronomical sources and increasing the SNR for calibration.

However the relative phases of the two sidebands can be arbitrary (particularly at Plateau de Bure where the IF frequency is variable since the LO2 changes in frequency in parallel with the LO1). This relative phase must be calibrated out. This it is done by measuring the phases of the upper and lower sidebands on the passband calibrator observation. These values can be used later to correct each sideband phase to compensate for their phase difference.

During the passband calibration one calculates:

$$e^{i\phi_U} = \frac{\int{\ensuremath{\widetilde{V}}_{ij,USB}(\nu,t) d\nu}}{\vert\int{\ensuremath{\widetilde{V}}_{ij,USB}(\nu,t) d\nu}\vert}$$ (9.18)

and

$$e^{i\phi_L} = \frac{\int{\ensuremath{\widetilde{V}}_{ij,LSB}(\nu,t) d\nu}}{\vert\int{\ensuremath{\widetilde{V}}_{ij,LSB}(\nu,t) d\nu}\vert}$$ (9.19)

Then at any time the double sideband visibility is:

$$\ensuremath{\widetilde{V}}_{ij,DSB}(\nu,t) = e^{-i\phi_U} \ensure... ...lde{V}}_{ij,USB}(\nu,t)+e^{-i\phi_L} \ensuremath{\widetilde{V}}_{ij,LSB}(\nu,t)$$ (9.20)

As a result the two terms on the right hand side have zero phase at the time of the pass band calibration and they keep the same phase during the whole observing session.

At observing time, offsets on the first and second LOs can be introduced so that both $\phi_U$ and $\phi_L$ are very close to zero when a project is done. This actually done at Plateau de Bure, at the same time when the sideband gain ratio is measured (see Chapter 12).