The ALMA Phasing System: A Beamforming Capability for Ultra-high-resolution Science at (Sub)Millimeter Wavelengths

VLBI depends on the fundamental ability of radio receivers to accurately preserve the phase of astronomical signals by referencing them to an independent frequency standard at each of the participating VLBI stations. Prior to the APP, the ALMA array, when operating as a connected element interferometer, depended on a common local oscillator (LO) derived from a 5 MHz rubidium (Rb) clock. However, the Rb clock does not meet the strict stability requirements for a VLBI frequency standard, namely that the rms phase fluctuations are kept well below one radian. This condition may be expressed as $\omega t{\sigma }_{y}(t)\ll 1$, where ω is the LO frequency in rad s⁻¹, ${\sigma }_{y}(t)$ is the Allan standard deviation (equal to the square root of the Allan variance), and t is the integration time in seconds (Rogers & Moran 1981).

Because of their superb stability on timescales comparable to VLBI integrations, hydrogen masers are used almost exclusively as frequency references for VLBI. At mm wavelengths, tropospheric turbulence limits typical VLBI integration times to ∼10 s. Since modern hydrogen masers achieve ${\sigma }_{y}(10\,{\rm{s}})\approx 1.5\times {10}^{-14}$, VLBI observations at wavelengths of ∼1 mm can be carried out with coherence losses of $\lt 5 \%$ (Doeleman et al. 2011). To fulfill this need at ALMA, a commercially available T4 Science iMaser was purchased and installed in a seismic rack at the ALMA AOS as part of the APP. ALMA's Rb clock was also kept in place as a hot spare for use in non-VLBI observations, but the hydrogen maser is now the default frequency standard for all ALMA observing modes.

The original APS design called for a straightforward repla-cement of the 5 MHz Rb reference with a 5 MHz output from the hydrogen maser. However, it was found that to preserve the required VLBI stability and phase noise specifications for the ALMA receiver LO signals, both the 5 MHz and 10 MHz outputs of the hydrogen maser had to be utilized. This required a minor modification of the ALMA Central Reference Generator, but the capability of switching back to the Rb clock as a hot spare for non-VLBI observations was preserved.

The hydrogen maser at ALMA is routinely monitored in two ways. With the first method, the stability of the maser's 10 MHz signal is compared with that of a high-precision quartz oscillator. The crystal is not as good as the maser over long periods, but it is of comparable stability over ∼1 s intervals (where the maser signal is also conditioned by a crystal). Results of such a comparison are shown in Figure 4.

Zoom In Zoom Out Reset image size

The left panel of Figure 4 shows a phase noise comparison between the ALMA maser and an Oscilloquartz 8607 quartz oscillator. The phase noise specification for the maser is −124 dBc Hz⁻¹ at 1 Hz from the 10 MHz reference. In a comparison of two oscillators with similar noise characteristics, one expects to measure a value that is 3 dBc Hz⁻¹ higher, and the phase noise plot confirms the maser is operating at specification or better. The right panel of Figure 4 is the measured Allan deviation, ${\sigma }_{y}(t)$, between the hydrogen maser and the same quartz oscillator, both of which should have similar stability over 1 s integrations. The measured value of the Allan deviation of $1.15\times {10}^{-13}$ at 1 s for two similar oscillators, implying that the Allan deviation for the maser alone is $8\times {10}^{-14}$ (i.e., $1/\sqrt{2}$ times the value measured for the two oscillators), which matches the required specification.

A second type of maser health check involves routinely monitoring the offset between the 1 PPS signal generated by the maser from its 10 MHz output with another 1 PPS signal generated by a global positioning system (GPS) receiver. GPS stability on 1 s timescales is poor due to variations induced by the ionosphere and will vary by a few tens of nanoseconds on a diurnal cycle due to ionospheric total electron content (TEC) changes. However, on longer timescales ($\gt {10}^{4}$ s), GPS is providing an average time rate synchronized with the United States Naval Observatory timekeeping for Coordinated Universal Time (UTC) and its precision will exceed that of a single hydrogen maser. If the maser is operating properly, a plot showing a comparison of the two signals is expected to show a linear drift at a rate characteristic of the maser, with some arbitrary offset due to the initialization of the first maser 1 PPS pulse. A plot of the residuals to such a linear fit for the ALMA maser is shown in Figure 5 for the period leading up to the 2017 April VLBI campaign at ALMA. It shows several weeks of data, with fits on ≈10 day intervals (designated in different colors). Prior to the campaign, the maser was "tuned" by adjusting the digital synthesizer which generates the 10 MHz oscillation for zero drift relative to UTC. Several fits over the final two-week period suggest that the average drift was essentially zero, with a measurement uncertainty of fs s⁻¹.

Zoom In Zoom Out Reset image size

The 64-antenna ALMA BL correlator processes four 2 GHz frequency chunks of dual-polarization spectrum from up to 64 antennas, one chunk per quadrant. Each data stream requires a PIC to serve as a VLBI backend and to transfer the ALMA signals to the VLBI recorders. In total, eight PICs are required (one per polarization in each of the four 2 GHz correlator quadrants).

The ALMA BL correlator has an "FXF" architecture (Escoffier et al. 2007). The 2 GHz input band is initially sliced into thirty-two 62.5 MHz pieces ("F") by sets of tunable digital filter bank (TFB) cards, whose design is described in Camino et al. (2008). Each 62.5 MHz "TFB channel" has an approximately flat response determined by the digital filter design implemented in the field programmable gate arrays (FPGAs) of each TFB card. These channels are overlapped slightly in frequency (15/16th of each channel), yielding an effective bandwidth is 1.875 GHz per quadrant. The TFB channels are processed in separate correlator "planes." The TFBs are followed by a hardware correlator ("X") and software Fourier transform ("F"). The correlator was designed with "hooks" for VLBI (Baudry et al. 2012), including spare card slots and rack space; extra power capability; a digital, high-resolution adjustment capability for the phases of the LOs in the TFBs (to 1/4096 of a 360° turn; see Figure 6); and FPGAs dedicated to computing a sum of a selectable set of antennas.

Zoom In Zoom Out Reset image size

The BL Correlator has two modes of operation with regard to how the antenna signals are passed through the TFBs: time division mode ("TDM") and frequency division mode ("FDM"). TDM is the default for continuum observing with ALMA during standard interferometry observations; in this mode, the TFB cards are bypassed and the correlator behaves like a pure XF rather than an FXF system (Baudry et al. 2012). However, only FDM allows for phase adjustment of the TFBs and the option to easily capture channelized data in a VLBI data set.

In practice, the spectral specification allows one to define some number of channel averages (spectral sub-regions), which correspond to sets of TFBs, for the purpose of making phase adjustments. Because the employed FDM setup generates more ALMA standard interferometry data than can readily be archived in real time, some degree of spectrum averaging (typically a factor of eight) is used to reduce the volume of archived data.

The final design of the correlator modifications required for the APS is shown in Figure 7. Parts added for VLBI are shaded blue, with additional detail shown in yellow. Because timing precision and stability are critical to successful VLBI operations, a 1 PPS distribution system (card and cables) was installed to bring signals from the hydrogen maser (Section 4.1), as well as from a GPS unit, to the PICs so that the timing of the data in the packets could be continuously recorded against both of these signals (left side of Figure 7). The 1 PPS signal from the maser should be completely invariant in the data, since the entire ALMA system is operated from the maser 10 MHz output. On the other hand, as noted above, the 1 PPS signal from the GPS unit typically varies by tens of nanoseconds due to diurnal ionospheric TEC changes.

Zoom In Zoom Out Reset image size

One complicating factor in the adaptation of the ALMA BL correlator for VLBI is the fact that ALMA uses a 48 ms Timing Event (TE) instead of the VLBI standard 1 PPS signal as a timing epoch standard. For this reason, several redundant measures were designed to insure that the timing is correct. The maser 1 PPS signal, a GPS 1 PPS signal, and the site TE are connected to the Timing Generators in each of the eight PICs. By design, the TE is coincident with the 1 PPS signal at seconds 0, 6, ... 54 and is carefully distributed so that its rising edge is within the same 125 MHz clock cycle everywhere in the correlator. To synchronize the PIC 1 PPS signal, the real-time control computer sends a "synchronize" command to each PIC's Timing Generator just prior to the TE at second 0. The 1 PPS Distributor has the ability to cross-check all of the PIC-generated 1 PPS signals against both the GPS and maser 1 PPS signals.

The ALMA BL correlator contains 64 Correlator Antenna Inputs (CAIs), numbered 0 through 63. To support VLBI, the firmware on the Correlator Interface Cards (CICs) was modified by insertion of a logic switch that toggles the input at CAI #63 from a single ALMA antenna to a phased sum of the array antennas (see Figure 7). This allows the phased sum to be correlated against any individual array antenna, providing a means of validating that the summation is correct.

The sum signal itself is produced using the summer logic in the Correlator Plane assemblies (Figure 7). The analog sum logic (gateware) in the dedicated FPGAs on each Correlator Card was modified to properly scale the outputted digitized samples and produce 2 bit signals for the PICs and CIC feedback. The sum is formed on a per TFB channel basis using a mask-selectable set of ${N}_{{\rm{A}}}$ antennas, where the mask is user-defined and relayed to the system via the correlator control system.

The mask is required for multiple reasons. First, in order to faithfully represent the sum, ${N}_{{\rm{A}}}$ must be odd. The reason is that in the available hardware circuit, only two bits per antenna are available. This allows representation of the signal voltage (part of a Gaussian distribution) as one of four possible values (two positive and two negative states). Consequently, with an even number of antennas, the resulting sum can be exactly zero, which in turn cannot be represented by any of the four available states. The simplest solution is to impose the requirement that the number of antennas be odd. Because a "zero" state is in fact the most probable one for a sum of an even number of antennas, the S/N loss incurred by omitting one antenna from the phased sum is always smaller than would result from having signals end up in non-representable zero states. A second reason for the mask is that it is useful to hold some antennas apart from the sum signal; one or more of these may be used as "comparison antennas" to estimate the efficiency of the phasing system (Section 8.2). Finally, depending on the configuration of ALMA, antennas that are on long baselines (>1 km) may be excluded to help insure good phase stability across the selected array.

In addition to the Timing Generator discussed above, the PICs also include a formatter. The formatter takes 2-bit data streams from 32 of the Correlator Cards in each quadrant and for each polarization, integrates them into a VDIF-compliant data stream, and routes them to a 10 GbE interface. The VDIF data stream is packaged in User Datagram Protocol (UDP) packets onto a 10 Gbps optical fiber which is mated to the PIC using an SFP+ network adapter. The PIC formatter supports bandwidths from 62.5 MHz to 2 GHz in binary multiples of 62.5 MHz and can also produce several types of test patterns for troubleshooting.

The Mark 6 VLBI recorder is the current offering in a long line of VLBI recording systems developed by the Massachusetts Institute of Technology (MIT) Haystack Observatory. It is commercially manufactured by Conduant Corporation. The Mark 6 is based on commercial off-the-shelf technology and open-source software. Data are recorded onto "modules" comprising banks of eight commercial-grade 3.5 inch hard disks inserted into a Mark 6 chassis. Typically the disks are "conditioned" prior to use to eliminate bad sectors and to verify their performance.

Mark 6 recording was initially demonstrated in 2012 (Whitney et al. 2013) and first adopted for astronomical VLBI use at ALMA. Figure 8 shows two of the Mark 6 recorders shortly after installation at the ALMA OSF. In the ALMA configuration, each recorder accepts two VDIF packet streams (one per polarization) on two optical network cables and writes data onto four disk modules at an aggregate rate of 16 Gbps. When all four ALMA basebands are in use, the total recording rate is 64 Gbps (2 GHz of bandwidth from each of four basebands, Nyquist sampled, 2 bits per sample, with dual polarizations).

Zoom In Zoom Out Reset image size

In normal operations, there is some manual effort required to mount and unmount the disk modules and verify that these have survived the stress of international shipping. Thereafter the operation of the recorders is completely automatic at the ALMA Control system via the observing script and the VLBIController software.

The Mark 6 system development was concurrent with the APS development, and there were a number of issues to consider in adapting the recorder for use at ALMA. We needed to ensure that the capacity of the modules was sufficient to eliminate the need for module swaps during overnight VLBI observing shifts. Because the recorders were sited at the ALMA OSF (elevation 2900 m) in a climate-controlled computer room, some environmental issues relevant to other VLBI sites were not an issue. Nevertheless, we undertook extensive testing to verify that ordinary hard drives (rated for 10,000 ft) were indeed usable at ALMA.

In contrast to previous hard-disk VLBI recorders which used proprietary disk formats, we opted for an open, simple, non-RAID, "scatter-gather" plan for distributing data in ≈10 MB blocks across multiple disks. In practice, at ALMA the data from each PIC are thus spread over 16 disks so that a useful recording is still possible even in the event of network issues or disk failures. Finally, since the Mark 6 recorders are conventional Linux systems, it is possible to run DiFX correlation software (see Section 6) on the mini-cluster comprising the four recorders in order to perform local VLBI correlations of data. This was valuable for early testing, although in practice, the ability to routinely perform fringe tests with remote sites in near real time is precluded by the limited network bandwidth between ALMA and the other sites.

An OFLS is required to transfer the antenna sum data generated by the APS from the ALMA AOS to the ALMA OSF where the VLBI recorders are located. The OFLS was designed by the National Astronomical Observatory of Japan and Elecs Industry Co. LTD. and fabricated by Elecs Industry. The system multiplexes the eight digital network streams generated by the PICs in the BL correlator (Section 4.2) onto a single ALMA fiber (out of a bundle of 48 fibers) for transmission. The measured signal loss ranges from ∼6–7 dB over the ∼30 km cable path. At the two ends, identical units multiplex the signals onto the single fiber with dense wavelength division multiplexing.

Each of the eight channels of the OFLS has a maximum data rate of 10 Gbps, allowing it to accommodate the data rate generated by each of the eight PICs (8 Gbps, plus a small overhead for the packet header of <1%). There is also a ninth channel available as a built-in spare, so in total there are 18 different data streams: nine downlink streams and nine uplink streams. The OFLS design is bi-directional since it is built based on 10GbE technologies, although the uplinks are not used during observations.

Figure 9 shows a schematic diagram of the OFLS. The data transfer through the single long-distance fiber is done with a 10GBASE-ZR optical module operating at a wavelength of 1550 nm. The optical channels designated ${\lambda }_{1}$ to ${\lambda }_{18}$ are assigned to the International Telecommunications Union (ITU) grids of 21, 23, 25, ..., 55, corresponding to frequencies of 192.1, 192.3, 192.5, ... 195.5 THz (i.e., 100 GHz spacing). In the OFLS the optical data are converted between 10GBASE-ZR and 10GBASE-SR so that the OFLS can interface the correlator PICs and Mark 6 recorders with 10GBASE-SR optical modules.

Zoom In Zoom Out Reset image size

Software connection to the OFLS and network control can be made with the VLBI Standard Interface for Software (VSI-S) protocol on a physical 10BASE-T/100BASE-T ethernet. For monitoring its status, the OFLS provides information such as power status, the link status of each channel, and temperature status. It has no packet monitoring capability, and thus it is a passive instrument in the VLBI phasing system. The OFLS was built with redundancy in terms of the number of cooling fans and DC power supplies to avoid failures in these parts, which tend to have relatively short mean times between failures (between a few to several years).

A VOM at ALMA is required to support the VLBI back end; this is handled by the VLBIController as indicated in Figure 3. By design, the VOM performs only limited functions: the PICs need to be programmed and the recorders need to be told what to record. For the PICs, programming consists of providing them with the VDIF header, which includes the number of channels to record and, critically, the timestamp for the data. As discussed in Section 4.2, the ALMA system is time-managed around TEs which occur every 48 ms. Every six seconds, one of these is coincident with an integral second, so one of these coincidences is chosen for the moment of programming: a VDIF header is downloaded prior to the TE and at the TE, the PIC starts its packaging and broadcasting of data. This stream is generated continuously until the correlator is reset or the PICs are turned off.

The data are only useful during the parts of the observing sequences when the array is phased-up, as described below. Therefore at the start of the observation, the recorders are provided a schedule from the VEX file appropriate for the current set of observations and the recorders then operate autonomously. Since the VLBIController also has access to the schedule, it can "check in" with the recorders and report the result of a scan check that verifies that the data have been recorded properly.

To exploit the full sensitivity of ALMA for VLBI experiments, it is necessary to operate the ALMA array as a virtual single dish rather than an interferometer. This is accomplished by coherently summing the voltages V received at each of the ${N}_{{\rm{A}}}$ individual dishes:

$$\begin{eqnarray}&&{V}_{\mathrm{sum}}=({V}_{1}+{V}_{2}\,+...+{V}_{{N}_{{\rm{A}}}}).\end{eqnarray} \tag{ 1 }$$

Here, the voltage from a given antenna, i, is the combination of a signal voltage s_i and a noise voltage ${\epsilon }_{i}$, i.e., ${V}_{i}=({s}_{i}+{\epsilon }_{i})$. For the APS, ${N}_{{\rm{A}}}$ must be an odd number because of the 2 bit quantization of the signal (see Section 4.2). The power from the combined signals of all possible baselines ij may be expressed as

$$\begin{eqnarray}&&\langle {V}_{\mathrm{sum}}^{2}\rangle =\displaystyle \sum _{{ij}}[\langle {s}_{i}{s}_{j}\rangle +\langle {s}_{i}{\epsilon }_{j}\rangle +\langle {s}_{j}{\epsilon }_{i}\rangle +\langle {\epsilon }_{i}{\epsilon }_{j}\rangle ]\end{eqnarray} \tag{ 2 }$$

(e.g., Thompson et al. 2017; hereafter TMSIII). In the case where the individual antenna phases are random, their signals will be uncorrelated with those of other antennas. Thus the only non-zero term in Equation (2) will be $\langle {s}_{i}{s}_{j}\rangle$ for the case i = j, and the combined array becomes effectively equivalent in sensitivity to that of only a single element i. In contrast, if the signals are optimally phased before summing them, a coherent addition of the signals results in ${N}_{{\rm{A}}}$ times the unphased array sensitivity (e.g., Dewey 1994), minus efficiency losses caused by atmospheric effects and quantization of the signals before summation (see Section 8). It is for this purpose that specialized software needed to be developed to enable phase-up the ALMA array. The use of these phase solutions is discussed in the next subsection.

5.2.1. Approach

The phasing software for the APS was custom-designed for the existing ALMA BL correlator hardware and software. Figure 10 provides an overview of the flow of information among the components of the APS software.

Zoom In Zoom Out Reset image size

In general, the BL correlator generates interferometric visibility data on a "subscan" basis (where a subscan has a duration of order seconds); these are normally gathered into "scans" (of order minutes) for publication to the ALMA Archive, where the raw visibility data are deposited and stored for eventual scientific analysis. Certain calibrations (performed in TelCal) are also routinely applied on this data stream as requested by the observation software. This is the purview of the PhasingController shown in Figure 3.

The BL correlator hardware is set up by the correlator control computer (CCC), and once programmed, it continuously correlates the input signals from all antennas in the observation. Antennas are managed as an "Array" by the control system (e.g., for the purpose of commanding pointing at targets), and the signals from the defined Array are correlated to provide cross-correlations between all antenna pairs (baselines). This computation also contains the antenna phases relative to one arbitrary reference antenna in the Array (see Section 5.2.2), as needed to coherently sum the antennas on the next trip through the BL correlator.

Figure 11 illustrates the temporal relationship between the 5 timelines that occur during a VLBI scan. Information is processed on two timescales or "loops," which may be deployed individually or in parallel. The arrows in the figure refer to transfers of information in these two loops. The "slow" loop (Section 5.2.2) processes the visibility data and applies corrections on the order of every 10 s based on phasing results from the previous scan. It is a closed loop between the BL correlator, where measurements are made and applied, and TelCal, where corrections are calculated. The "fast loop" (Section 5.2.3) operates on timescales of order one second. It applies corrections based on data from the WVRs associated with each 12 m antenna, which measure additional delays in the signal path due to water vapor above each telescope.

Zoom In Zoom Out Reset image size

Of the 64 CAIs, numbered 0–63, one (#63) is reassigned to hold the phased sum (see Section 4.2). Since the sum antenna is not a real antenna, it cannot be pointed, so it is not part of the Array at the control level. However, once the sum is calculated by the CIC, the BL correlator can cross-correlate it as if it were a real antenna.

The BL correlator spectral processing software makes one delay adjustment for the 192 ns required to perform the summation and route it back to the correlation inputs. The correlations are then dumped from the correlator boards into the long-term accumulator (LTA) cards and ultimately read by the nodes of the Correlator Data Processing (CDP) cluster, which is comprised of four slaves per quadrant and a master, which commands the slaves. This is where the final spectral processing is done prior to delivery of the data to the Archive. The full set of correlations is not present in any one computer until the Archive stage. The four slaves are assigned correlation work based on the CAI numbers of the antennas at the ends of each baseline. One slave handles baselines where both antennas have CAI < 32, another handles baselines where both antennas have CAI > 31, and the other two handle the baselines where one CAI < 32 and the other has CAI > 31.

Since there is a significant processing load on all of these machines, as well as the networks connecting them, the phasing calculations are done instead in TelCal. For each subscan, TelCal performs the necessary calculations to phase the array and inform the VOM, which then relays the commands needed for application of the phasing corrections by the CCC. The WVR data (supplied through the CAIs from the antennas) are provided to the CDP slaves so that online spectral processing can optionally include WVR (fast-loop) corrections. If selected, these latter corrections are applied at a cadence of about once per second and are directly forwarded to the CCC. All of this is managed by the PhasingController through commands to the CCC, TelCal, or the CDP master. A complete record of these activities is provided in an ALMA Science Data Model (ASDM) table that is stored in the Archive along with all the other metadata from the observation. The PhasingController ensures that the number of phased antennas is odd, and the observing script (see Section 5.2.5) generally specifies that at least two antennas are omitted from the phased array, making them available as comparison antennas to evaluate the performance of the phasing system (Section 8.2) and facilitate calibration of the VLBI data. Thus a maximum of 61 antennas may contribute to the phased sum.

Optionally, it is possible to retain the last-applied phases in the TFB phase registers rather than clearing them at the start of a subscan sequence for an observation. This allows a "passive" phasing mode whereby a relatively bright calibrator source located within a few degrees of the fainter target is used to phase up the array prior to slewing to the target and observing it without active phasing. Optimal cycle times are currently being explored, but are expected to be of order one minute (see e.g., Holdaway 1997; Carilli et al. 1999). This mode of operation has been successfully tested during APP CSV (Section 8) for sources and calibrators with angular separations up to several degrees and will enable use of the APS on much fainter science targets than is presently possible. Passive phasing is expected to be offered for science observing in future ALMA Cycles (Section 9).

5.2.2. The "Slow" Phasing Loop

Data are delivered to TelCal in the form of channel averages at the end of each correlator subscan. The number of such channel averages in each of the four 2 GHz basebands is programmable, but APS operations in Cycles 4 and 5 use eight channel averages, each spanning 250 MHz (see Section 5.2.4 for an explanation of this choice.) Sufficient time resolution on these channel averages is provided to allow data from the start of the correlator subscan to be excluded without much loss of accuracy in the phase calculation. As shown in Figure 11, the phasing correction is calculated at the end of one subscan and applied at the beginning of the next. In typical APS operations during ALMA Cycle 4, there was a new subscan every 20 s, of which 12 s could be used to determine the phasing solution. This timing constraint is dictated by the network bandwidth available to transfer the data.

The phasing calculations follow a procedure analogous to the standard self-calibration technique commonly implemented in radio interferometry applications (Pearson & Readhead 1984; Cornwell & Fomalont 1989). For an interferometer of ${N}_{{\rm{A}}}$ elements, a phase ${\phi }_{\mathrm{obs}}(t)$ can be measured as a function of time t on the baselines between any two antennas i and j. The observed baseline phases are the sum of the structure phase of the observed source, ${\phi }_{\mathrm{mod}}$, the instrumental phase, ${\phi }_{\mathrm{ins}}$, the atmospheric phase, ${\phi }_{\mathrm{atm}}$, and a thermal noise term, ${\phi }_{{\rm{n}}}$, specific to each baseline:

$$\begin{eqnarray}&&{\phi }_{\mathrm{obs}}(t)={\phi }_{\mathrm{mod}}(t)+{\phi }_{\mathrm{ins}}(t)+{\phi }_{\mathrm{atm}}(t)+{\phi }_{{\rm{n}}}(t).\end{eqnarray} \tag{ 3 }$$

The structure phases (which have to be known with sufficient accuracy from a model) and the instrumental and atmospheric phases have to be removed from the observed baseline phases, which can subsequently be added coherently. In the initial APS implementation (used during ALMA Cycle 4), the structure phases are assumed to be zero—i.e., the target source is assumed to be point-like on the scales sampled by the ALMA array baselines. This is a good approximation for most VLBI target sources, but the APS design allows the option for inclusion of more complex models in future Cycles.

The instrumental, atmospheric, and thermal noise phases are tied to the antennas and do not need to be separated. ${N}_{{\rm{A}}}$ atmospheric phases then need to be determined from the ${N}_{{\rm{A}}}({N}_{{\rm{A}}}-1)$ complex visibilities between pairs of antennas. For arrays with ${N}_{{\rm{A}}}\gt 2$ the problem is thus overdetermined. By selecting one reference antenna (in general, a reliable 12 m antenna located near the array center) and assigning it zero phase, a least-squares minimization allows estimation of the remaining ${N}_{{\rm{A}}}-1$ antenna phases, ${\psi }_{i}$. Specifically, since the baseline phase measurement ${\phi }_{{ij}}$ between antennas (index i and j) may be expressed in terms of the unknown per-antenna phases as ${\psi }_{i}-{\psi }_{j}$, we can form the ${\chi }^{2}$ statistic as

$$\begin{eqnarray}&&{\chi }^{2}\,=\,\displaystyle \sum _{i\ne j}\displaystyle \frac{{[{\phi }_{{ij}}-({\psi }_{i}-{\psi }_{j})]}^{2}}{{\sigma }_{{ij}}^{2}}.\end{eqnarray} \tag{ 4 }$$

Here, ${\sigma }_{{ij}}^{2}$ is the variance. The minimum is found where $\partial {\chi }^{2}/\partial {\psi }_{i}\,=\,0$ for every i, and this provides a linear system of equations to solve. Such a solver was present in TelCal prior to the initiation of the APP. The antenna-based noise ${\psi }_{{\rm{n}}}$ is reduced by a factor $\sqrt{{N}_{{\rm{A}}}}$ from the baseline-based noise ${\phi }_{{\rm{n}}}(t)$. The resulting antenna phases, ${\psi }_{i}$, are then provided to the PhasingController, and their negatives are taken as the corrections to be applied in the TFB phase registers to achieve the optimal station phases. The APP also implemented an additional algorithm to resolve phase lobe ambiguities that result if a phase change close to 2π occurs (see e.g., Fomalont & Perley 1999). The algorithm was based on code originally developed by J. D. Romney (2012, private communication).

A number of notions for determining the quality of the solution on a per-antenna basis were considered to allow the PhasingController to remove problematic antennas (nominally with human supervision). To this end, the solver reports a normalized "quality" (in the range [0, 1], with "1" representing the highest quality) and also estimates a measure of phasing efficiency by comparing the visibility amplitude of the sum and comparison antennas relative to reference and comparison antennas. The PhasingController also has the capability to automatically remove antennas, although this feature has not yet been needed in practice.

5.2.3. The "Fast" Phasing Loop

Each 12 m ALMA antenna¹⁸ is equipped with a WVR that monitors a spectral line of H₂O. The resulting data may be used to estimate the component of the delay due to water vapor for that antenna (Nikolic et al. 2013). These data are provided and processed approximately once per second so that the CDP spectral processors may make adjustments to the baseline phases due to the variable differences in path lengths to the antennas at the end of each baseline. Alternatively, the data may be saved in the ASDM file so that analogous corrections can be applied after the observation, if desired.

The WVR data are also usable by the phasing system. A software agent was created within the CDP cluster to translate the WVR data into a series of phase adjustments (approximately once per second) which are provided to the PhasingController to pass on to the CCC for application at the TFB registers. This "fast" loop is "open" in the sense that it does not provide feedback directly to the CDP agent generating the phasing corrections (see Section 5.2.2). In normal use, however, the TelCal phasing engine is correcting phases that have been modified by WVR corrections, so in that sense the loop is closed on the slower timescale.

While it is envisioned that the fast loop normally will operate in tandem with the slow loop (Section 5.2.2) at times of high water vapor variability, it can also be operated independently, either for testing, or during passively phased observations (see Section 5.2.1). An example of the improvement in phase stability as a result of use of the fast phasing loop alone (no slow loop corrections) is shown in Figure 12. The data are from a Band 6 observation in conditions with PWV ∼ 1.3 mm. Fast-loop corrections were applied to data from one correlator quadrant, while another quadrant received neither fast nor slow phasing corrections. The absolute mean deviation of the corrected phases is a factor of three lower compared with the uncorrected phases. However, since the fast corrections are applied with some latency (about one second), in rapidly varying conditions they may add phase noise rather than remove delays. And in excellent conditions without much variability, the fast loop is not needed, as little coherence is lost on the timescale of the slow cycle, hence the corrections merely add noise. Because of this, the fast loop was not used during the initial APS operations in ALMA Cycle 4. However, it is expected that it may be used in future Cycles during times of high precipitable water vapor (PWV) and/or if the phasing system is operated at higher frequencies (see Section 9).

Zoom In Zoom Out Reset image size

5.2.4. Delay Correction

5.2.5. VEX2VOM and the Observing Script

CSV for the APP was carried out over a two-year period from 2015 January to 2017 January. CSV activities used on-sky testing to validate the fully integrated components of the APS, including hardware, software, observing modes, recording modes, and correlation procedures. The tests that were carried out were designed to demonstrate that the APS met its formal design requirements, that the APS produces robust, scientifically valid data whose characteristics are well understood and documented, and that ALMA is capable of operating as a fully functional VLBI station. Presently these attributes continue to be checked by periodic regression testing of the APS. Details from the various APP CSV campaigns and supplementary test observations have been described in a series of ALMA Technical Notes (Matthews & Crew 2015a, 2015b, 2015c; Matthews et al. 2017).

As noted in Section 1, a perfectly phased array of ${N}_{{\rm{A}}}$ antennas is equivalent to a monolithic aperture with ${N}_{{\rm{A}}}$ times the effective area of one of the individual elements, ${A}_{\mathrm{eff}}$. However, in practice, any real phased array will suffer from efficiency losses (i.e., loss of effective collecting area) caused by a combination of factors. These losses may be collectively characterized in terms of a "phasing efficiency," ${\eta }_{{\rm{p}}}$, where ${\eta }_{{\rm{p}}}=1$ corresponds to perfect efficiency. Routine characterization of the phasing efficiency allows determining whether the system consistently meets theoretical expectations, and is crucial for enabling accurate absolute calibration of VLBI experiments. In addition, such a measure serves as a useful metric for comparing performance of the system under various conditions (e.g., different weather, array configurations, or observing frequencies).

The gain of an antenna may be defined as ${A}_{\mathrm{eff}}/(2k)$ where k is the Boltzmann constant (TMSIII). If the effective area of a single ALMA antenna is ${A}_{12{\rm{m}}}$, then the gain of a phased array of ${N}_{{\rm{A}}}$ such antennas may be written as: ${N}_{{\rm{A}}}{A}_{12{\rm{m}}}/(2k)$.

If we now correlate the phased sum signal ${V}_{\mathrm{sum}}$ (Equation (1)) with the signal ${V}_{{\rm{c}}}$ from another identical "comparison" antenna in the array that is not part of the phased sum, the resulting product can be written as

$$\begin{eqnarray}&&\langle {V}_{\mathrm{sum}}{V}_{{\rm{c}}}\rangle =\left(\displaystyle \frac{{V}_{0}^{2}}{2}\right)\cos (2\pi \nu t),\end{eqnarray} \tag{ 6 }$$

where ν is the observing frequency, t is the integration period, and $({V}_{0}^{2}/2)$ is the amplitude of the correlated signal (e.g., Condon & Ransom 2016). Because ${V}_{\mathrm{sum}}$ and ${V}_{{\rm{c}}}$ are both proportional to the electric field of the source being observed, multiplied by the gains of the sum antenna and comparison antenna, respectively, this implies that the amplitude term in Equation (6) is in turn proportional to $\sqrt{({A}_{12{\rm{m}}})({N}_{{\rm{A}}}{A}_{12{\rm{m}}})}$. Thus for an ideal phased array, the correlated amplitude is expected to grow as the square root of the number of antennas used to construct the sum.

The phasing efficiency of the array, ${\eta }_{{\rm{p}}}$, may then be parametrized as

$$\begin{eqnarray}&&{\eta }_{{\rm{p}}}=\displaystyle \frac{\langle {V}_{\mathrm{sum}}{V}_{{\rm{c}}}\rangle }{\sqrt{{N}_{{\rm{A}}}}\langle {V}_{i}{V}_{{\rm{c}}}\rangle },\end{eqnarray} \tag{ 7 }$$

where the $\langle {V}_{i}{V}_{{\rm{c}}}\rangle$ term in the denominator denotes the mean of the correlated amplitudes between each of the ${N}_{{\rm{A}}}$ antennas used to form the phased sum and the unphased comparison antenna. (TelCal computes this ratio for a designated comparison antenna and provides the result within the ASDM file). Throughout the APP CSV, a variety of tests were used to gauge the overall efficiency of the APS empirically and to isolate the sources and magnitudes of various losses.

One means to characterize the efficiency of the phasing system is the execution of "step scan" sequences (Primiani et al. 2011) on bright, compact sources. During such an observation, antennas are sequentially added to, or subtracted from the phased array, permitting an evaluation of how correlated amplitude (and hence the phasing efficiency) scales with the number of antennas in the phased array. During a step scan test, one or more array antennas are designated as comparison antennas and are never phase-corrected or included in the phased sum.

Several step scan tests were performed using the APS during CSV campaigns. Results from one example are presented in Figure 14. These results are based on Band 6 observations of the quasar PKS J0522−3627 obtained with the APS on 2015 March 31. The weather conditions were relatively stable with PWV ∼ 1.3 mm. There were a total of 22 functioning 12-meter antennas in the array used for the test, three of which were designated as unphased comparison antennas. Baseline lengths ranged from 11 m to 313 m. During the course of a 10-minute observation, pairs of antennas were randomly added to or removed from the phased sum (hereafter designated "APP001," corresponding to the station identification of the phased sum in the ASDM file),¹⁹ with the maximum number of phased antennas being 19. The test was performed independently in the four basebands, resulting in different values of ${N}_{{\rm{A}}}$ in each quadrant at any given time. For clarity, we show here only the results for a single baseband/polarization combination (polarization XX in baseband 1, where the center frequency was 214.6 GHz) and for only one of the comparison antennas ("DA45"). Results for the other basebands and comparison antennas are comparable.

Zoom In Zoom Out Reset image size

Figure 14 shows the correlated amplitude for the DA45–APP001 baseline as a function of the number of phased antennas for each 16 s subscan during the step scan test (closed dots). For each time interval, data from the 8 subbands within the baseband were averaged to yield a single amplitude value, and data from the first minute of data (during which the array was not fully phased) were excluded.

The solid line plotted on Figure 14 shows the result of a least squares fit to the measured amplitudes of the form $A=C\sqrt{{N}_{{\rm{A}}}}$, where C is an arbitrary scaling factor. The correlated amplitude, A, is seen to follow the expected $\sqrt{{N}_{{\rm{A}}}}$ dependence for $11\leqslant {N}_{{\rm{A}}}\leqslant 17$, but it is slightly lower than expected for ${N}_{{\rm{A}}}=19$. For comparison, the asterisk plotted on Figure 14 shows the correlated amplitude on the DA45-APP001 baseline prior to phase-up, i.e., effectively showing the amplitude for a phased array of one antenna. The dotted curve running through this latter point illustrates the correlated amplitudes that are predicted if this is adopted as a benchmark for behavior of the system. The actual measured values lie ∼15% below this curve. Finally, the dashed curve on Figure 14 plots the correlated amplitude expected for a perfectly phased array (${\eta }_{{\rm{p}}}=1$). We see that the measured amplitudes are ∼61% of those expected based on this idealized prediction. Unlike the values used to produce the dotted curve, those used to produce the dashed curve have not been processed by the summer mechanism. This therefore provides us a means of disentangling efficiency losses due to factors inherent to phase-correcting the antennas versus those inherent to the production of the phased sum signal itself.

Efficiency losses in the APS occur at different stages. First, imperfections in the phasing solutions themselves (as a consequence of tropospheric fluctuations, residual delay errors, source structure, and latency in the application of the phasing corrections) all increase the rms scatter, ${\sigma }_{\phi }$, in the phases of the antennas used to form the phased sum. The decorrelation resulting from these fluctuations in turn produces a reduction in the correlated amplitude by a factor ${\epsilon }_{0}\approx {e}^{\tfrac{-{\sigma }_{\phi }^{2}}{2}}$ (e.g., Carilli et al. 1999). For the data set shown in Figure 14, ${\sigma }_{\phi }=0.37$ rad (21°), hence ${\epsilon }_{0}\sim 0.93$. In general, the atmosphere dominates ${\epsilon }_{0}$, hence efficiency will be worse at higher frequencies and in times of poor atmospheric stability. Nonetheless, in terms of these losses, the APS phasing engine routinely meets or exceeds its goal of 90% phasing efficiency as specified in the original functional requirements (see Section 3).

Second, a significant and unavoidable loss of phasing efficiency in the APS arises from the 2 bit quantization of the summed signal, which leads to an amplitude reduction of 12% (i.e., ${\eta }_{{\rm{q}}}=0.88;$ e.g., Kokkeler et al. 2001; Crew 2012). Third, additional smaller efficiency losses arise from discrete sum scaling errors (up to 5%) and the lack of optimized weighting of the contributions of the individual antenna signals to the sum. Ideally, one would want to weight the antenna contributions to the sum using either equal power or variance weighting (Moran 1989). However, the correlator circuit that receives the antenna data to be summed (provided by one of the so-called VLBI hooks; Baudry et al. 2012) does not have access to any additional data and has logic space only for a simple sum and not for additional weighting factors. Because the ALMA 12 m antennas are generally well matched (ALMA Partnership 2017, private communication), the weights should be approximately equal and this type of efficiency loss is not expected to exceed a few per cent.

Figure 15 shows results from another test of the APS, conducted on 2017 April 10 in Band 6 using an array of 37 phased 12 m antennas. The goal was to assess the phasing quality and stability as a function of time for all basebands and both polarizations. In this case, the number of phased antennas was kept fixed throughout. Four scans were obtained on the quasar PKS J1924−2914, during which the array was actively phased. Based on observations from the SMA,²⁰ the 1 mm source flux density near the date of observation was 3.18 ± 0.16 Jy. The baseline lengths ranged from 14 m to 400 m and the PWV was ∼0.5 mm. The figure shows the rms scatter in the phases as a function of time for baselines between the phasing reference antenna and the antennas used to form the phased sum. At start-up, the array was completely unphased, as evidenced by the large phase scatter. Phase-up is achieved after approximately 20 s, after which the mean rms phase scatter is ∼10° across all of the four basebands, both polarizations, implying a decorrelation loss in amplitude of $\lt 2 \%$.

Zoom In Zoom Out Reset image size

These and other test data taken throughout CSV and subsequent regression testing show that under stable weather conditions, efficiency losses as a result of imperfections in the phasing solutions themselves are generally minimal. Note that during the second and fourth scans in the present example, the fast phasing loop was used in concert with the slow loop. We see that the rms scatter is ∼30% higher during those scans, confirming that these WVR-based corrections add noise when used under very stable weather conditions (see Section 5.2.3).

Importantly, the data shown in Figure 15 have been corrected by the APS phasing software, but they have not been processed by the summer, and are therefore not subject to the degradation in the signal caused by quantization errors and other subsequent losses. To assess these latter effects, we may examine the correlated amplitude on a baseline between the phased sum signal and an unphased comparison antenna, ${a}_{S\otimes {\rm{c}}}$, with respect to the correlated amplitude seen on a baseline between a comparison antenna and the phasing reference antenna ${a}_{{\rm{c}}\otimes R}$. For a perfectly phased array subject to 2-bit quantization losses, the predicted ratio ${a}_{S\otimes {\rm{c}}}/{a}_{{\rm{c}}\otimes R}=0.88\sqrt{{N}_{{\rm{A}}}}$ where ${N}_{{\rm{A}}}$ is the number of antennas in the phased sum. Instead, for the data set shown in Figure 15, we find ${a}_{S\otimes {\rm{c}}}/{a}_{{\rm{c}}\otimes R}\,\approx 0.61\sqrt{{N}_{{\rm{A}}}}$, consistent with the results of the step scan test described above. Similar total efficiency values were also confirmed in subsequent VLBI observations (see Section 8.3).

Some portion of the APS efficiency loss can be attributed to the present treatment of baseband delays (see Section 5.2.4). This effect is expected to be of order a few per cent (Crew 2012). Additional factors contributing to an efficiency loss of ∼20% above the theoretical expectation remain a topic of ongoing investigation.

The combined functionality of the various hardware components installed by the APP to enable VLBI operations at ALMA was successfully demonstrated for the first time on 2015 January 13 when a VLBI fringe on the quasar PKS J0522−3627 was detected at a frequency of 214.0 GHz (∼1.3 mm, i.e., Band 6) on a 2.1 km baseline between ALMA and the APEX telescope (Figure 16). Despite the short baseline, both stations used separate, independent frequency references, hence this constituted a true VLBI experiment. Although the ALMA station was operated as a multi-antenna array during this test, the array was unphased, and therefore the equivalent sensitivity was that of a single 12 m antenna.

Zoom In Zoom Out Reset image size

The S/N expected for detection of a VLBI fringe between ALMA and another station may be expressed as

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}={\eta }_{{\rm{p}}}{\eta }_{{\rm{b}}}S\left[\displaystyle \frac{\sqrt{{n}_{\mathrm{pol}}{\rm{\Delta }}\nu t}}{\sqrt{({\mathrm{SEFD}}_{1}\times {\mathrm{SEFD}}_{2})}}\right],\end{eqnarray} \tag{ 8 }$$

where we assume ${\eta }_{{\rm{p}}}$ is the efficiency of the APS (see Section 8.2), ${\eta }_{{\rm{b}}}$ is a digital loss factor equal to 0.88 for 2 bit correlation of VLBI signals, S is the source flux density in Jy, ${n}_{\mathrm{pol}}$ is the number of polarizations, ${\rm{\Delta }}\nu$ is the effective observing bandwidth, t is the integration time in seconds, and ${\mathrm{SEFD}}_{1}$ and ${\mathrm{SEFD}}_{2}$ are the system equivalent flux densities (in Jy) of the two respective VLBI stations. The fringe shown in Figure 16 is based on 5 s of data, a single polarization (mixed linear–circular, R–X), and a usable bandwidth of 1632 MHz (32 channels of 51 MHz). Assuming a source flux density²¹ of ∼3.6 Jy and an SEFD of 3600 Jy for both the APEX and ALMA stations, the predicted S/N is $\sim 79\,{\eta }_{{\rm{p}}}$, while the observed S/N is 68 (note that these data are uncorrected for the use of mixed polarizations as described in Section 7). This implies ${\eta }_{{\rm{p}}}\approx 0.86$, consistent with a nominal phasing efficiency after accounting for 2 bit quantization of the sum signal. Because the array was not summed in this case, ${\eta }_{{\rm{p}}}$ is not subject to the additional efficiency losses described above for the ${N}_{{\rm{A}}}\gt 1$ case.

Several subsequent VLBI fringe tests were performed with ALMA as a phased array, both in Band 3 and in Band 6 (Matthews & Crew 2015b, 2015c; Matthews et al. 2017). These tests included VLBI sessions of several hours duration during which a full suite of ALMA and VLBI-specific calibration sources were observed in order to permit end-to-end testing of PolConvert (see Section 7). One set of observations (experiment code BM452) was performed on 2016 July 10 in conjunction with the 8 stations from the National Radio Astronomy Observatory's²² VLBA that are equipped with 3 mm receivers.²³ Two stations (North Liberty and Pie Town) were excluded from further analysis owing to poor weather.

An example of a fringe on the 6727 km baseline between a phased array of 11 ALMA antennas and the Fort Davis station of the VLBA is shown in Figure 17. The usable observing bandwidth was 204 MHz (four 51 MHz channels per polarization). The target source was the quasar PKS J0607−0834, which had a 3 mm flux density of ∼1 Jy near the time of the observations (see footnote 20). The peak source flux density is therefore comparable to the faintest VLBI science targets permitted during ALMA Cycles 4 and 5. However, the correlated flux density on the observed baseline is expected to be approximately four times lower (see Kovalev et al. 2005). Assuming an SEFD for the Fort Davis station of 3500 Jy, an SEFD for the phased ALMA array²⁴ of 219 Jy, and an integration time of 238.4 s, the predicted S/N is ∼33 after accounting for known efficiency losses as described above (i.e., ${\eta }_{{\rm{p}}}=0.61$ and ${\eta }_{{\rm{c}}}=0.88$) and applying PolConvert corrections (Section 7). Using an optimal segmentation time of ∼6 s for the data (to minimize decorrelation losses in the correlated amplitude), the measured S/N is 32.5, in excellent agreement with our previous characterizations of the system. This result confirms that the end-to-end phasing efficiency of the APS is routinely ∼61%, to within an uncertainty of a few per cent.

Zoom In Zoom Out Reset image size