Antenna and Electronic Delay Calibration

TIP Last Update: JeffMangum - 23 Feb 2006

Specification

  • Antenna Delay
    • Systematic: 7 fs
    • Fluctuating: 50 fs
  • Electronic Delay
    • Systematic: 7 fs
    • Fluctuating: 30 fs
  • LO and Timing Signal Errors
    • Absolute time to be established within 10 nanoseconds

Antenna and Electronic Delay Calibration Memos (ALMA, EVLA, etc.)


Contents


Incorrect Antenna Non-Repeatable Residual Delay Specification

Quite some time ago (2004/06!) Richard Hills pointed out that the current wording for the "non-repeatable residual delay" antenna specification seems to allow for a "double-differencing" of the slowly-varying contributions of path errors in the antenna structure. Richard's note contains all of the details.

After reading Richard's document I noted the following:

  1. This directional-dependent delay error appears to be a problem only for baseline measurements. Can we assume that baseline measurements will require something like one hour?
  2. Note that the AEG measurements of the total path length errors for the VertexRSI prototype included a measurement over 30 minute time scales, measuring a total path variation of about 15 micron or less. Furthermore, Albert Greve was purposely conservative in his analysis of these measurements as he derived this number by RSS of the corresponding daily maximum path length variations.
  3. Note also that for the AEC antenna the similar measurement and calculation yielded about 25 micron or less.

Dick Sramek has written a nice summary document describing this residual delay error.

-- JeffMangum - 21 Oct 2005

Timing Specification Issues

Question from the BE IPT (delivered by DarrelEmerson): What is the needed precision of the timing signal at the antenna, which is derived from the signal sent out along the fiber to each antenna?

On the fiber, the timing signal consists of a 48 ms period pulse, a 25 MHz sine wave and a 2 GHz sine wave. Larry outlines in his documentation how the phase of the 25 MHz is to be referred to the phase of the 2 GHz signal, and then the timing of the 48 ms is slaved to the zero crossing of the 25 MHz. So, you should end up with a 48 ms pulse that has the precision of the zero-crossing waveform of the 2 GHz signal. All these signals are within the line length corrector loop, so at a given antenna should be extremely stable. The derived 48 ms pulse then has to be corrected, in software, for the fixed propagation delay along the fiber to each individual antenna pad. The derived 48 ms pulse then has to be corrected, in software, for the fixed propagation delay along the fiber to each individual antenna pad. The details of the implementation have evolved since Larry's description, but the principle remains the same."

The 48 ms pulse is used for various things at the antenna. It's used for Walsh function switching, which gives a precision requirement of about 0.1 microseconds. That's no problem. It's also used to generate offset frequencies for both the 1st and 2nd LOs. A timing error here will result in a phase error. If the frequency is never changed, then provided the timing offset is constant, it doesn't matter. It will get calibrated out when we measure phase on an astronomical cal source.

However, if the frequency is changed, either to track Doppler shift or to move, within one band, to a different frequency, then the phase will change.

SUB-QUESTION: How big a frequency change of ANY of the local oscillators will we ever want, where we want to refer the interferometric phase of the new frequency to that of the original frequency? An equivalent question is how big a frequency change will we ever want, without having to recalibrate the phase?

If we KNOW what the timing error is, then of course we can always calculate a phase correction in software. However, in some of the hardware designs being considered (even tested), there might be unpredictable jumps in timing, where (say) the 48 ms slips a cycle of the 125 MHz. Do we care? I think that, yes, we do care.

If there were a 1 nanosecond unpredictable timing error at the antenna, then a 1 GHz frequency change would make the phase jump by 360 degrees, so even a 1 MHz change would introduce a phase jump of 0.36 degrees. If we say this tuning shouldn't change the phase by, say, more than 0.1 degrees for a 1 MHz change, then the needed timing precision would be 1 microsec * 0.1/360 or about 0.3 ns. The actual timing change could be greater than that, provided the real time computer knows the timing offset to this precision, and that the timing can be calibrated to this precision. This may need careful measurement of cable lengths at the antenna.

What rate of change can we expect for Doppler tracking? For a source at Declination 0, the maximum rate of change of Doppler, which comes from the earth's rotation, is of order 0.1 km/s per hour. At 950 GHz, this becomes a rate of change of about 320 kHz/hour. If we want to go 20 minutes between phase calibration, where the frequency change will be about 100 kHz, and if we say the phase should not change by (say) 0.1 degrees due to this cause, then the real time computer needs to allow for timing delays with a precision of about 10 microsec *0.1/360 or about 2.8 ns. A reasonable compromise would be to specify 0.3 degrees precision at 950 GHz, which would be 10 ns.

I can see that we will certainly want of order of 1 MHz changes to track Doppler, but will we ever want, say, to move frequency a few GHz, perhaps to move to a maser phase calibration source, and want to refer that phase back to the phase of the original frequency a few GHz away, without any intermediate calibration? This is the most stringent requirement on timing at the antenna that I can think of. (NB none of this has anything to do with the RF phase of the photonic LO signal, but just to offset frequencies added to that LO signal.)

I think that we are concerned about in-band frequency changes, as any band switching would necessarily come with a phase calibrator measurement which would correct for this phase jump. So, if this assumption is correct, here is a list of possible observing modes where we might need to make in-band frequency changes:

  • Doppler tracking:
    • Frequency changes of order 1 MHz.
    • Current design will track in post-processing, but want to leave option for online Doppler tracking.
    • Always used.
  • Maser phase referencing:
    • Jump to a nearby in-band maser line.
    • Need to allow for switching over entire band for this.
    • Used infrequently?
  • Frequency switching:
    • Single dish only, so not a consideration.

I don't think this requirement is written anywhere, but I could be wrong. I looked through the ALMA System Technical Requirements (http://edm.alma.cl/forums/alma/dispatch.cgi/docapproval/docProfile/100923/d20040415180935/No/t100923.htm), but didn't see anything obviously applicable to this situation. Perhaps it is there and I don't see it.

CONCLUSIONS (resulting from ImCal discussion of 2006/02/22):
  1. If we have any large LO frequency change, say of order a few GHz (i.e. "in-band"), we'll perform a separate phase calibration measurement to measure the phase difference at the before and after frequencies. We'll make this measurement rather than using the known LO delay path, since using the LO path to predict the phase offset would require timing precision measured in hundredths (1/100'ths) of a nanosecond.
  2. This isn't so different, in that the way we'd measure the LO delay path would be by measuring the phase change as a function of frequency. We're really making the same measurement, but calling it a phase calibration measurement rather than an LO path measurement.

Question which arose during ImCal discussion 2006/02/22: What is the requirement for timing uncertainty?

We are now being assured that the timing, and the LO delay path, will be stable to high precision (sub-nanosecond), provided there isn't a loss of power or a system reset. The 8 nanosecond uncertainty comes in here - as we all heard today. We are being assured that, in the absence of perturbations such as power failures, there won't be any 8-nanosecond jumps.

The timing accuracy requirement is currently 100 nanoseconds. It should be reduced to 10 ns. With 100 ns, you get a 360 phase change with 10 MHz frequency change, so a 0.36 degree change for 10 kHz frequency change. If this is reduced to 10 ns we end up with (in my view) more reasonable values that would support Doppler tracking without serious issues.

SUMMARY OF TIMING PRECISION AND UNCERTAINTY DISCUSSION:
  1. It's the agreement to perform a separate pre-calibration for large (GHz+) in-band frequency changes. This has removed the requirement for 0.01 ns timing precision.
  2. BE IPT will investigate the feasibility of reducing the timing uncertainty requirement from 100 to 10 ns.

-- JeffMangum - 23 Feb 2006

Description of the Technique

Instrumental delay calibration is required since we have a finite frequency resolution. For a given frequency resolution $\Delta f$ the delays must be matched so that there is no decorrelation inside the band $\Delta f$. During the observations we track the varying geometrical delay by finite steps; the step size (15ps) has been chosen so that the delay error does not produce loss of sensitivity in the worst case. Ideally the (assumed fixed) instrumental delay should be known to the accuracy of half a step so that its uncertainty does not increase the bandwidth decorrelation.

In the correlated spectra of a given baseband the phase will have an instrumental dependence on sky frequency f:


$\phi (f) = f \delta\tau + \phi_{\rm B} (f) + \phi_0 $


The first term is due to the residual differential delay, resulting from pathlength differences between antennas. The second term is the bandpass function. The third term is a frequency independent phase offset.

The goal of the delay calibration is to obtain the best delay compensation. There is some ambiguity between the delay and the bandpass function; the latter can be a rather complex function of frequency so the actual optimal delay will be a function of bandwidth.

In the next section we examine the different causes of delays in the ALMA system.

Origin of delays in ALMA interferometer

Antennas

  • Simple antenna geometry can cause delays in the antenna optics:

    • Antenna position error. This is discussed in antenna position calibration memo (503).

    • Offset between aximuth axis and elevation axis may vary from one antenna to the other. This causes a cosine elevation term in the delay. This is also discussed in memo 503 as it is measured together with antenna position calibration.

  • Thermal deformations will cause focus drift, which will be compensated by moving the subreflector after dedicated focus measurements. To maintaing phase and delay continuity when the subreflector is adjusted in $Z$ (sometimes during the excution of a project) it is necessary to include the Z coordinate corrections in the delay model (as well as in first LO phase tracking).

Receivers

We have to allow for delay offsets due to small differences in optical paths in the receivers, either before or after first down conversion. For this we have to keep track on one delay for each polarization output (1 or 2), and well as for each sideband output (U or L) for 2SB receiver bands. So we have 4 possible paths (U1, U2, L1, L2) and 4 delay values. Naturally there is one such set of 4 delays for each receiver band.

These delays should be to first order constant with time, and stable when antennas are moved. They will have to be remeasured whenever the receiver assembly is exchanged on one antenna.

Backend electronics

Analogue

The picture here gets a little more complicated; one needs to examine the system block diagram 80.04.01.00-004-H.

  • For a given polarization (1 and 2) there are 2 IF inputs (U and L), and 4 down converters, each feeding a digitizer. Each IF input can be connected to any of the four digitizers, giving 8 possible paths and 8 delay values.
  • We have thus 16 delay values (but only 8 cross polarization delays).

    These are expected to be very constant (cabling length), and probably rather small, as we deal with identical modules produced in series.

Note: It is possible that the dependence on the baseband output (0,1,2,3) is small enough to be neglected, and that this set reduces to 4 values only.

Digital

After digitization the main cause of delay is the fibre length. This has to be the same for all carried signals from a given antenna. This is the main parameter that will change when an antenna is moved to a different station and reconnected. To this are added the time-varying compensating delays (added as timing offsets in the correlator and as small phase offsets in the sampling clock). So the main contribution here is a single delay per antenna corresponding to the fibre length. Question: Can we have a different fibre length for each baseband?

Software

To be complete we should remember that a software delay is added post-FFT in the correlator software, to correct for pathlength fluctuations in the atmosphere as measured with the WVR receivers.

Cross Polarization delay

As the ALMA system is capable om measuring cross-polarization it is necessary to measure the delay between the orthogonal polarization channels 1 and 2. For a giben baseband, once the antenna based delays in 1 and 2 have been measured, there is a single cross-polarization delay to be measured for the whole array. There should be one such number for each( 1, 2) pair of baseband paths through the electronic system.

Phase offsets between basebands

Should we keep track of the phase offsets between basebands?


Phases offsets between basebands will occur as the second oscillators follow different paths before they reach the down converters. As the phase of the local oscillators is 100% reproductible, these phase offsets should be constant with time at a given frequency; one may want to keep track of the propagation delays in the LO paths to be able to predict these phases for any given LO2 frequency. It is not clear to be if this prediction actually works.


Given the broad bandwidth (2GHz) of each baseband, these phase offsets should be easy to measure, once the individual delays are known, in a fraction of time needed to measure the bandpass in each baseband.

Measurement Technique

Single Baseline

We record a spectrum on a strong source. The measured phases are:

$ \phi(\nu) = 2 \pi \tau(t) \nu $

So a non-zero residual delay $\tau$ produces a linear phase slope across the passband. By least-square fitting a straight line through the measured baseline phases the delay can be accurately measured. The uncertainty in the delay measurement is approximately (see Thomson p 306):

$\sigma_\tau = \frac{\sqrt{12} \sigma_\phi}{2 \pi B}$

where $\sigma_\phi$ is the statistical rms on phase measurements.

$2 \pi$ phase ambiguities in the phase slope measurement are not a problem. The range of the delay measurement is set by the inverse of the frequency resolution. So to reach a large range one may be sensitivity limited. For a crude measurement with a totally unknown delay one may start by observing a strong, narrow line (SiO maser), to have enough sensitivity; the next step would be to observe a strong continuum source with a wide bandwidth to get an accurate measurement.

Many Baselines

To measure simultaneously the antenna based delay errors one solves for antenna based delays using a method of least squares. The solution is obtained as:

$\tau_i = \frac{1.}{N} \sum_{j\neq i} \tau_{ij}$

This formula will ensure $\sum_{i}\tau_i = 0$. N is the number of antennas in the subarray which is used.

In the simple case of N identical antennas (ALMA main array) one gets:

$\sigma_\tau = \frac{\sigma_\phi}{2 \pi  B / \sqrt{12 N}}$

$\sigma_\phi = \frac{T_{SYS}}{\eta \sqrt{2 B t}} $

See the sensitivity calculation below.

Note: Accurate phase tracking is not required, atmospheric phase fluctuations only degrade the sensitivity through time decorrelation. WVR can be used to improve this.

How often?

We are mainly dealing here with cable and fibre lengths, so it is not necessary to measure them too often, hopefully.

  • A single global delay determination has to be done for each antenna whenever it has been moved. For this we need use only a few reference antennas, very probably the same that are used for the first (crude) baseline determination.

  • Whenever the whole receiver dewar assembly is exchanged in one antenna: one needs a determination of the individual delays between receiver bands, and polarizations.

  • The relative delays in the IF have to be remeasured when the analog and/or digital racks are exchanged (but not the delays between receiver bands).

  • From time to time the whole set of global antenna delays as well as the cross polarization delays should be remeasured (to avoid loss of accuracy after a number of antenna moves). Experience will tell us how often this is necessary (a few times a year?).

How long?

The following table gives the integration times to measure the delay to half a delay step (7.5 ps) in a 2 GHz bandwidth, using a 2 Jy source, for representative frequencies. The 8 antenna column is representative of a measurement done on a few antennas just moved (prior to measuring antenna positions). The 64 antenna column is more relevant for a general measurement done at regular intervals on all antennas. For the cross polarization measurement, all baselines can be averaged, the times are to be divided by a factor of N (8 or 64).


Frequency Tsys time (8 ant.) time (64 ant.)
(GHz) (K) (s) (s)
8 64
43.0 45.0 0.1 0.0
90.0 70.0 0.1 0.0
230.0 150.0 0.7 0.1
350.0 250.0 2.1 0.3
410.0 400.0 6.0 0.7
690.0 1200.0 70.3 8.8
850.0 1200.0 109.8 13.7


The integration times are not a real concern, even at the highest frequencies, for this relatively rare calibration.

What quantities need to be archived

The dimensions are:
  • Antennas: Nant=64,
  • receiver bands: Nband=10,
  • Polarizations: Np=2,
  • Sidebands: Nsb=2 (1 for some bands),
  • Base bands: Nbb=4.

Quantity Description Number When?
TauF Fibre delay Nant=64 Antenna move.
TauR Receiver delay Nant*Nband*Np*Nsb=2560 Receiver exchange
TauIF IF delay Nant*Nsb*Np*Nbb=1024 Backend exchange; measure at low frequency.
TauXP Cross Polarization delay Nband*Nsb*Nbb=80 At least once, but should be checked from time to time.

What further tests and/or studies are needed:

  • The model for the IF delays could be simplified if actual differences are small enough. This will be checked when the first actual frontends/backends are installed on the antennas.

  • One should use one dedicated band/sideband/polarization (e.g. 90 GHz LSB V) to measure the fibre delays. The Receiver delay for that path can be set to zero by convention. The same policy can be used for the IF delays.

- RobertLucas - 29 Sep 2004
Error during latex2img:
ERROR: can't find latex at /usr/bin/latex
INPUT:
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\begin{math}\displaystyle $\phi (f) = f \delta\tau + \phi_{\rm B} (f) + \phi_0 $\end{math}
}
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{
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\begin{math}\displaystyle $\sigma_\tau = \frac{\sqrt{12} \sigma_\phi}{2 \pi B}$\end{math}
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{
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\begin{math}\displaystyle $ \phi(\nu) = 2 \pi \tau(t) \nu $\end{math}
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{
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\begin{math}\displaystyle $\sigma_\tau = \frac{\sigma_\phi}{2 \pi  B / \sqrt{12 N}}$\end{math}
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\begin{math}\displaystyle $\tau_i = \frac{1.}{N} \sum_{j\neq i} \tau_{ij}$\end{math}
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\begin{math}\displaystyle $\sigma_\phi = \frac{T_{SYS}}{\eta \sqrt{2 B t}} $\end{math}
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\end{document}
STDERR:

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