•  14 position switch pairs on a source.
  
• Each position switch was 2 minutes on, 2 minutes off, and then a 10 second calon, 10 second caloff.
• The wapps were used to take data with 25 MHz bw, 2048 channels, and 9 level sampling.
• The center frequency was 1397.93 Mhz.

• Tsky is the power from the sky and any scattered radiation
• Tomt is the contribution from the Omt. G2 is the frequency Dependant gain
• Tcal is the injected cal (if present). It is injected after the OMT.
• Trcv this is the amplifier noise in the dewar.
• g1, g2, g3 are the frequency Dependant portions of the different pieces.
  
• g1 sky and standing wave frequency dependence
• g2 omt dependence. mainly it will be from the resonance.
• g3  everything after the dewar. Mostly filter band passes.

Steps in removing the resonances

1. make a model of the resonance from the Off source data.
2. fit    this model to the on source data
3. subtract the resonance from each
   
4. compute posOn/posoff .
   
5. Do this for the avg spectra of each  onoff pair.

The model:

• Y=A0/((w^2 - A1^2)^2 + (a2*w)^2)   + A3 + A4*w
• w is the frequency
• A1 is the resonance frequency
• A2 is gamma (the loss)
• A3 and A4 are a linear fit to the baseline.
• Fit to offsource/(calOn-calOff)
• CalOn-calOff  = (Tcal + Trcv)*g3
  
• G3 is the frequency dependence of everything downstream from the dewar.
• dividing by calon-caloff flattens the offSource data without affecting the resonance.
• The integration time for the calon-caloff is only 10 seconds. This would normally increase the noise in the spectra. But we divide  the onSource and off source by the same function so the the noise is not part of onSrc/Offsrc at  the end.
  
• The units are now equal to the cal value.
  
• Fit the model to the onsource.
• Compute ymodelOff= A0/((w^2-A1^2)^2 + (A2*w)^2. This ignores the linear baseline fit to the offsource
  
• Fit to onsource: a[0] + a[1]*(ymodelOff) + a[2]*w
• This fits the off model to the on data doing a separate linear fit to the baseline.
• Remove the model fit from the on and off data.
• The resonance should be additive noise so subtract it out.
  
• We have lost the baseline so add back in the median value for onSource/caldif, offsource/caldif.
• Compute onSrcModelRemoved/offSrcModelRemoved - 1.
• The division puts us back to units of Tsys
• Accumulate this for each on/off pair.

Plots showing the results:

• There are 14 pages. 1 for each on/off pair.
• 1st Plot: the 2 minute average band passes for onSrc (black), offSrc (red), and calOn - calOff (green).
• 2nd Plot: Onsrc/calDif, offSrc/calDif and the model fits to each.
• The extra noise if because the cal is only 10 seconds long (but it is the same noise for onSrc and OffSrc).
• 3rd Plot: onSrc/calDif - modelFitOfResonance, offSrc/calDif - modelFitOfResonance.
  
• 4th Plot: onSrc/OffSrc -1 with model correction (black) and without model correction (red).
• the rms went from about .0024 to around .0014. The expected rms is .0011.
  

• Top: the average OnSrc (black) and offSrc (red) for the 14 pairs. The green line is the average off source without any corrections.
• bottom: The average on/off-1 for the 14 scans. The black line is with the correction, the green line is with no correction.
• The rms went from .00197 to .00051 when the correction was applied. The expected rms was .00029.

Conclusions:

• Removing the resonance decreased the rms from .00197 to .00051. The expected rms was .00011.
• Some of the difference between .00051 and .00011 is from structure outside the resonance region (around 1390).
• The average On, Off after correction still show a small bump around the resonance region. This is causing a slight dip in the on/off-1 which is probably not real.
  

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