Antenna and Electronic Delay Calibration
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
Contents
Incorrect Antenna NonRepeatable Residual Delay Specification
Quite some time ago (2004/06!) Richard Hills pointed out that the current wording for the "nonrepeatable residual delay" antenna specification seems to allow for a "doubledifferencing" of the slowlyvarying 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:
 This directionaldependent delay error appears to be a problem only for baseline measurements. Can we assume that baseline measurements will require something like one hour?
 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.
 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 zerocrossing 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.
SUBQUESTION: 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 inband 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 inband frequency changes:
 Doppler tracking:
 Frequency changes of order 1 MHz.
 Current design will track in postprocessing, but want to leave option for online Doppler tracking.
 Always used.
 Maser phase referencing:
 Jump to a nearby inband 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):
 If we have any large LO frequency change, say of order a few GHz (i.e. "inband"), 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.
 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 (subnanosecond), 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 8nanosecond 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:
 It's the agreement to perform a separate precalibration for large (GHz+) inband frequency changes. This has removed the requirement for 0.01 ns timing precision.
 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:
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
80.04.01.00004H.
 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 timevarying 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
postFFT 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 crosspolarization 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 crosspolarization
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:
So a nonzero residual delay $\tau$ produces a linear phase slope
across the passband. By leastsquare 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):
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:
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:
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:
\documentclass[fleqn,12pt]{article}
\usepackage{amsmath}
\usepackage[normal]{xcolor}
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\definecolor{silver}{gray}{0.75}
\usepackage{latexsym}
\begin{document}
\pagestyle{empty}
\pagecolor{white}
{
\color{black}
\begin{math}\displaystyle $\phi (f) = f \delta\tau + \phi_{\rm B} (f) + \phi_0 $\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle $\sigma_\tau = \frac{\sqrt{12} \sigma_\phi}{2 \pi B}$\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle $ \phi(\nu) = 2 \pi \tau(t) \nu $\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle $\sigma_\tau = \frac{\sigma_\phi}{2 \pi B / \sqrt{12 N}}$\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle $\tau_i = \frac{1.}{N} \sum_{j\neq i} \tau_{ij}$\end{math}
}
\clearpage
{
\color{black}
\begin{math}\displaystyle $\sigma_\phi = \frac{T_{SYS}}{\eta \sqrt{2 B t}} $\end{math}
}
\clearpage
\end{document}
STDERR: