Signal leakage and interference in ALMA

Leakage signals in the cubes

The effect of the leakage in the image cubes is a bit more subtle because of the following reasons:

• As far as we have seen, the leakage affects the phase calibrator and the source scans. The effect on the cross-correlations is qualitatively similar on all fields, which means that the effect is partially canceled out by calibration, specifically, by the bandpass calibration.
• During image deconvolution, we combine the cross-correlation signals from many antennas, affected in various degrees by the leakage problem :from heavily affected to not at all.
• Because the extraneous leakage signal affects the cross-correlation only through the auto-correlation normalization, in the visibilities this is mainly a calibration problem affecting some channels. This means that the cross-correlation amplitudes that are multiplied by wrong calibration factors. A consequence of this is that continuum subtracted cubes (such as the ones analyzed in the weblogs) should not display any spurious "line emission," but such a feature may be observable in the RMS spectrum or in non continuum-subtracted cubes.

We evaluate the effect of the leakages on (non-continuum subtracted) image cubes of the phase calibrators. We evaluated these for several sources observing the following:

• The RMS measured in regions away from the source (see Figure 5) show an increase in the leakage region consistent with what is shown by Figure (4) in the source cube.
• The integrated flux in a region around the source seem to follow the same behavior as the RMS, increasing near the leakage frequency. However, this is not observed in the spectra of the smaller region nor in the spectra taken at the peak flux density of the source.

Simple theoretical model

A simple theoretical picture of the leakage will be presented. Consider the radio signal received from a quasar (the bandpass calibrator) coming from a point source whose electric (or magnetic) field complex amplitude can be represented by: \begin{equation} E(t) = a e^{i(2\pi \nu t -\varphi)}~~,\label{eq-1} \end{equation} where the frequency \(\nu\) is a random variable with probability density function \(f_\nu\propto S_\nu\). This function will not be used in the following, but, if necessary, we may assume (as it is observed) it is a negative power law of frequency over the frequency range of interest. Because emission from the quasar comes from many electrons emitting synchrotron incoherently, we add a random phase \(\varphi\) distributed uniformly in \([0,2\pi)\) and independent from \(\nu\). The constant \(a\) is related with the total integrated spectrum through \(a^2=\int_{-\infty}^\infty S_\nu d\nu\).

The autocorrelation of \(E(t)\) is (Papoulis, §10): \begin{equation} R(\tau) = a^2\,\mathbb{E}(e^{2\pi i \nu \tau})=a^2\,\int_{-\infty}^\infty f_\nu(\nu)e^{2\pi i \nu \tau}d\nu~~.\label{eq-2} \end{equation} We assume that the leakage injects a monochromatic signal on each antenna of the form \[ L_j e^{2\pi i \bar{\nu}_j t }~~, \] for each antenna \(j=1,2\), with constant \(\bar{\nu}_i\). In general, as mentioned above, \(\bar{\nu_1}\neq\bar{\nu_2}\).

Based on Equation (\ref{eq-1}), the signals received at antenna \(j=1,2\) is : \begin{equation} E_j(t) = a e^{i(2\pi \nu t -\varphi)}+L_j e^{2\pi i \bar{\nu}_jt}~~.\label{eq-3} \end{equation} Note that \(E_j(t)\) is no longer an stationary process, but cyclostationary with a deterministic component, i.e., the constant phase leakage signal. The explicit dependence on \(t\) will be removed by averaging in time after the correlator, as indicated in the diagram in Figure (7).

The auto-correlation of \(E_j\) is: \begin{equation} R_{jj}(\tau) = \langle\mathbb{E}\left(E_j(t+\tau)E_j^*(t)\right)\rangle_t = \lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T} \mathbb{E}\left(E_j(t+\tau)E_j^*(t)\right)dt = a^2 \,\mathbb{E}\left(e^{2\pi i \nu \tau}\right) + L_j^2 \,e^{2\pi i \bar{\nu_j} \tau}~~, \label{eq-autocorr} \end{equation} because \(\mathbb{E}\left(e^{i(2\pi \nu t-\varphi)}e^{-2\pi i \bar{\nu}_j}\right)=\mathbb{E}\left(\mathbb{E_\varphi}\left(e^{i(2\pi \nu t-\varphi)}e^{-2\pi i \bar{\nu}_j}|\nu\right)\right)=0\), that is, the cross terms between the leakage signal and the astronomical signal do not correlate. This lack of correlation (even at \(\nu=\bar{\nu}_j\)) comes explicitly in our model from the fact that the leakage signal has constant phase, in contrast with the random phase of the astronomical signal. Even if the leakage signal has a random non-zero phase, it would be characterized by a phase independent of the astronomical signal, not changing the fact that the cross terms in the correlations would be zero. The autocorrelation spectrum (the Fourier transform of the autocorrelation) from \eqref{eq-autocorr} is \begin{equation} S_{jj}(\nu) = a^2 f_\nu + L_j^2 \phi(\nu-\bar{\nu}_j)~~,\label{eq-5} \end{equation} where \(\phi\) represents an unresolved line feature normalized to one at peak, i.e., \(\phi(0)=1\).

To cross-correlation between the signals from antennae 1 and 2 is \[ R_{12}(\tau)=\langle\mathbb{E}\left(E_1(t+\tau)E_2^*(t)\right)\rangle_t~~. \] Note that the term \(L_1e^{i(2\pi \bar{\nu}_1(t+\tau))}L_2e^{-i(2\pi \bar{\nu}_2t)}\) is removed by the time averaging assuming that the time interval over which the signal is integrated is large compared with \(|\bar{\nu_1}-\bar{\nu_2}|^{-1}\). Therefore: \begin{equation} \begin{split} & & R_{12}(\tau) &= a^2 \,\mathbb{E}\left(e^{2\pi i \nu \tau}\right)\\ &\longrightarrow & S_{12}(\nu) &= a^2\,f_\nu \\ \end{split}\label{eq-6} \end{equation}

The normalized cross-correlation spectrum between the two antennas is given by \begin{equation*} \mathcal{V}_{12}(\nu) = \frac{a^2f_\nu}{\sqrt{a^2 f_\nu + L_1^2 \phi(\nu-\bar{\nu}_1)}\sqrt{a^2 f_\nu + L_2^2 \phi(\nu-\bar{\nu}_2})}~~. \end{equation*} If we consider \(\bar{\nu_1}\approx\bar{\nu_2}\approx\bar{\nu}\) within the spectral resolution, the cross-correlation normalized spectrum will show an "absorption" dip with a depth \begin{equation} \mathcal{V}_{12}(\bar{\nu}) = \frac{a^2f_\bar{\nu}}{\sqrt{a^2 f_\bar{\nu} + L_1^2}\sqrt{a^2 f_\bar{\nu} + L_2^2 }}~~.\label{eq-vis} \end{equation}

The observed auto-correlations in ALMA

The highly idealized Equation \eqref{eq-vis} is not really applicable to ALMA data because the auto-correlations are heavily influenced by atmospheric and instrumental signal, with only a minor component arising from the astronomical source. A more applicable form of the normalized visibility would be \begin{equation} \mathcal{V}_{12}(\bar{\nu}) = \frac{a^2f_\bar{\nu}}{\sqrt{\mathcal{A}_{1,\bar{\nu}} + L_1^2}\sqrt{\mathcal{A}_{2,\bar{\nu}} + L_2^2 }}~~,\label{eq-vis2} \end{equation} where \(\mathcal{A}_1\) and \(\mathcal{A}_2\) are the auto-correlations associated with antennas 1 and 2.

Bandpass calibrations, then, would compensate the factors in the denominators in Equation \eqref{eq-vis2}. Fortunately, the raw auto-correlations are apparently somewhat stable through the executions. Figure (8) shows a montage of the XX auto-correlation for the bandpass and phase calibrator in an execution block: differences between the two fields are minor. The practical consequence of this is that the bandpass corrections for the leakages are adequate to a first approximation for the phase calibrator and presumably the science fields as well. The bandpass correction compensates the effect of the leakage and there is no important calibration problem in the image tests we examined, as shown by the lack of leakage effect on the total flux (peak) spectrum in the phase calibrator cubes.

Nevertheless, the auto-correlations are not completely stable and there are a few differences. These differences may increase the auto-correlation signal in the phasecal scan respect to the bandpass scans, or vice-versa. The final effect is a slight miscalibration in the leakage channels and affected antennas and their associated baselines, which translates to an increase in image RMS in those channels. That is, the increase in RMS observed is not due to an overestimation of the visibility signal in some antennae due to miscalibration, but rather an increase of RMS due to miscalibration (under- or over-estimation) producing poorer image fidelity and lower dynamic range. We can further test this hypothesis by examining the results of interpolating the leakage affected channels in the bandpass table. The effect is that the bandpass amplitude is much larger in the affected channels than the original bandpass. This creates a dip or decrease in the flux amplitude calibration as shown in Figure (9) (that is, bandpass solution is over-estimated), but at the same time, an _increase_ in the RMS. The poorer image fidelity even compensates the decrease in flux density scaling in the leakage channels.