Intro

Monitoring the 12meter total power has shown that it varies. As the temperature increases the total power goes down.

see: run winking cal for for 26 hours (after temperature stabilization)

On 8nov21 a peltier cooler was installed to temperature stabilize the cal diode. If the cal diode output is constant, then it can be used to correct for gain variations in the system (as long as the gain variations are slower than the 25Hz cal).

On 17jan22 2 minutes of data was taken with the winking cal. The winking cal was then used to correct for gain variations (more info)

On 16mar22 15 minutes of data was taken with the winking cal to see how far (in integration time) we could extend the correction. The results are reported below.

Setup:

The analysis used the 25Hz hardware winking cal data taken on 16mar22. The setup was:

7 172 MHz bands were recorded with 512 freq channels/band
the band frequencies were:

8219.00 8363.00 8505.00 8647.00 8789.00 8931.00 9073.00 MHz.

the 25Hz hardware winking cal cycled with 20 milliseconds cal on followed by 20 milliseconds cal off
the spectra were dumped at 2 milliseconds (giving 10 samples per calOn and cal Off.
the spectra around the transition were excluded giving 8*2ms = 16 ms for cal on and 16 ms caloff for each cal cycle
Data was taken for 900 seconds tracking blank sky giving 900/.04=22500 cal cycles in the 900 secs of data.

How to correct for gain variations

The hardware cycles at 40ms (20ms on ,20ms off). It can be used to correct for slow gain variations in the electronics.

Let:

tpOnM= measured total power calOn
tpOffM =measured total power calOff (this is also Tsys).
tpOffC =the tpOff corrected for the gain variations
g(t) be a slowly varying electronic gain variation (slower than the 40ms cal calcyle).
The correct tpon,off would be:

tpOn=g(t)*tpOnM
tpOff=g(t)*tpOffM

tpCalM=tpOnM- tpOffM = g(t)*(tpOn - tpOff)=g(t)*calDiodeOutput

The calDiode output is temperature stabilized with a peltier cooler.

tpOffC=tpOffM/tpCalM= tpOff*gt/(g(t)*calDiodeOutput) = tpOff/calDiodeOutput .. with the g(t) canceled

This works if g(t) is relatively constant during the cal cycle.

What the correction does to the noise statistics

Dividing the tpOff (tsys) by the cal deflection will increase the rms noise level.
Let:

bw be the total power bandwidth used
tau be the integration time. For a single calcycle we have 16 ms calon,16ms calOff (since we dropped 2 2ms sample around the cal transition).
The radiometer eq gives:

deltaTp/Tp = 1/sqrt(bw*tau) = expRms.. expected rms
deltaTp= Tp*expRms

deltaTpOn,deltaTpOff then come from the radiometer equation
when adding or subtracting noisy variables:

d(a+b) = sqrt(da^2 + db^2)

When dividing noisy variables:

x/y = sqrt ((dx/x)^2 + (dy/y)^2)

deltaTpCal=sqrt(deltaTpOn^2 + deltaTpOff^2) = expRms*sqrt(tpOn^2 + tpOff^2)
deltaTpOff/tpCal= sqrt( expRms^2 + expRms^2*(tpOn^2 + tpOff^2)/(tpon - tpOff)^2)
or
deltaTpOff/tpCal= expRms*sqrt(1. + (tpOn^2 +tpOff^2)/(tpon-tpoff)^2)
The ratio of tpCal/tpOff varies from .27 to .39 between the 2 pols and across the 1ghz band.
using .33 for the cal fraction of Tsys

the theoretical rms should increase by about a factor of 5 because of the division by the cal deflection.

To not blow up the errors so much we can:

smooth the cal deflection before dividing

this will decrease the noise, but it may not do as good a job the smoothed value gets close to the g(t) variations.

fit a function (polynomial) to the cal deflection.

this will add no extra noise, but it depends how well we cat fit the g(t) variations in the cal deflection.

Processing the data.

For freq band of 900 seconds of data:

The calOn and caloff spectra were input

they were averaged to 16 milliseconds giving 1 cal on, off spectra per cal cycle (40 ms)

The spectra were bandpass corrected:

The median cal off bandpass was computed and then divided by it's mean value.
Each of the 22500 calon, and calOff spectra were then divided by the normalized average bandpass
this will cause the freq channels be be weighted more evenly when computing the total power across the band.

A mask for bad freq channels was created for each freq band to exclude rfi.

the rms/mean was computed for each freq channel using the 22500 calon and caloff spectra

The resulting rms/mean spectra has been flattened in freq.

a linear fit was done to the calOn,and calOff rms/mean spectra

Any points >3 sigma from the mean were flagged as bad (rfi)
4% of the band on each edge was also flagged (20 extra channels on each edge where the filter falloff was largest).
we ended up with about 155MHz/band after excluding the bad channels (the 2nd freq band (8363MHz ended up with 129MHz)

The total power was then computed for the 22500 calON,calOff points for each pol and freq band.

this will be called the "uncorrected" total power since it includes electronic gain variations with time.

the gain correction was tried with various methods:

divide tpOff by tpCal by 40 ms sample

this should give the best gain variation correction, but it will blowup the rms error

divide tpOff by a smoothed version of the tpCal (25 samples.. or .16*25=.4 seconds)

this will give smaller rms errors, but may not do as good a job correcting for the gain variation.

Fit an nth order polynomial to tpCal and then divide tpOff by the fit

this will keep the rms error unchanged but may not do a perfect job of fitting the gain variations.

A check was then made to see if the noise statistics improved as we increased the integration time.

the tpOn/tpCal was used (no smoothing or fit). It should do the best job of cancelling the gain variations.

The 155 MHz band was also broken up into 25 MHz sections and the previous test was redone

this was to check to see if some low level rfi was causing problems.

Looking at the results:

Uncorrected total power calOff vs time (.ps) (.pdf)

the tpCalon (black) and tpCalOn (red) are over plotted.

they have been normalized to their median value.

Each row is a different freq band
The left column is polA, the right column is polB
There is a variation of up to +/- .5% for the 900 seconds.

Over plot tpCaloff for the 7 freq bands (.ps) (.pdf)

The top Frame is polA, the bottom frame is polB
The tpCalOff from the 7 bands in over plotted in different colors.

Each band has been normalized to their median value.

The largest variation in amplitude is common to all 7 freq bands.

Over plot tpOff with and without corrections (.ps) (.pdf)

TpOff is plotted. each plot has been normalized by it's median value.
The upper two frames are polA,B of a freq band
the lower two frames are polA,B of the next bad. (there are 4 pages)/.
Black: no correction

the rms is about .0011

red : divide by tpCal

The rms is about .0035

green: divide by smoothed tpcal (25 samples=.4 seconds)

The rms is about .0011

The expected rms should be .0063

Divide TpOff by a polynomial fit to tpCal and then compute the measured rms. Compare with no correction.(.ps) (.pdf)

Page 1 rms vs polynomial fit order

The x-axis is the order of the polynomial fit used (1st to 5th order)
The vertical scale is the measured rms/mean
top frame : polA
bottom frame: polB
The colors show the 7 freq bands.
The purple line is the expected rms
For polA the rms continues to improve out to the 5th order
For polB the rms bottoms out at the 4th order fit.

Page 2: tpOff/4th order fit and then try various integration times.

tpOff was divided by a 4th order fit to tpCal
the data was then smoothed by .016 to 62 seconds
after each smoothing, the rms was computed.
The plots show how the rms improved as the smoothing time was increased.
Top frame: PolA
Bottom frame: PolB
the axis are log scale.
the colors show the different freq bands.
the straight lines are the expected rms/mean
the measured rms/mean decreases with increasing integration, but it is many orders of magnitude from the theoretical value.
the spread in the rms curves with frequency is probably because the 4th order fit did a better job with some frequency bands.

Page 3: rmsTpoff/tpoff with no corrections.

the data was smoothed similarly to page 2.
after each smoothing, the rms was co
Top frame: PolA
Bottom frame: PolB
the axis are log scale.
the colors show the different freq bands.
the straight lines are the expected rms/mean
the polA values are similar to tpOff/4thorderFit (since polA did not have much gain variation for these 15 minutes.
the polB values are much worse than the tpOff/4thorderFit (polB had a lot more gain variation).

Using tpOff/tpCal plot the rms/mean for different integration times (.ps) (.pdf)

TpOff was divided by tpCal (each .016 ms sample).

Each page is a different frequency band.
the rms will increase by dividing by tpCal\, but this should do the best job of cancelling any gain variations after the cal injection.
the data was then smoothed by .016 up to 62 seconds.
for each smoothing the rms/mean was computed and plotted
Black is PolA, Red is polB
the straight lines (with the +) are the expected rms/mean
the dashed lines are a linear fit to the log of rms,integration using the first 2 points

this gives an idea where the measured rms/mean deviates from the expected rms.

The rms/mean turns over around 1 to 2 seconds integration time.
PolB does a little better than PolA

PolB min rms/mean: .0005 Tsys at 1-2 seconds integration
PolA min rms/mean: .0006 tsys at 1-2 seconds integration.

the rms/mean starts to deviate from the dashed line at .1 to .4 seconds of integration.

Break the 155MHz band into 25MHz sections to see if any rfi affects things (.ps) (.pdf)

The 155 MHz from the 8219 MHz band was split into 6 25 MHz bands and then reprocessed (as above)
* is the measured rms from the 6 bands
+ is the expected rms
Black in polA, red is polB
There is no large change in processing the 25MHz bands separately

so the rfi is probably not causing a degradation.

Summary:

the gain correction was corrected by dividing by the calDeflection.
The correction was done by:

dividing by the calDeflection (the .016 ms samples0
dividing by a smoothed version of the calDefl
fitting a polynomial to the cal deflection
for the 900 secs we ended up using a 4th order polynomial.

With the 4h order fit we got down to .0002 Tsys (for polB)

the rms continued to decrease with increasing integration time ( but no where near the expected rms/mean)

Correct rcvr gain variations with winking cal.