Converting spectrometer counts to kelvins

Terminology:

• Bold: variables are arrays with frequency dependence.
  
• Spc: spectrometer counts
• K:   kelvins
  
• Avg: the average value  of the spectra (maybe throwing out some  channels at the edges).
• <>:average over the spectra (throwing out points at the edges if you want).
  
• gCal(f):- frequency dependence of the tabulated cal values. Normalize this to have an average value of unity (maybe throwing out some channels at the edges).
  
• gIF(f)  - frequency dependence of the if/lo bandpass (i'm ignoring the frequency dependence prior to the cal injection). Normalize this to have an average value of unity (maybe throwing out some channels at the edges).
  
• One(f) - an array of 1's
  
• calInK(Avg): Scaler cal value averaged over the band.
  
• calOffSpc(Avg): cal off value averaged over the band (in spectrometer counts)
  
• calDif_cnts(f)=(calOn_cnts(f)-calOff_cnts(f)). This is the cal difference spectrum in counts of the spectrometer. It can also be written:
• calDif_cnts(f)=calDif_cnts(Avg)* gCal*gIF.
  
• calOffSpc_cnts(f)=calOff_cnts(Avg)*gIF
• <gIF(f)> and <gCal(f)> = one by definition.
  

Computing the conversion factor:

• cntsToK(f)=calInK(f) / calDif_cnts(f)
•        
  

		calInK(Avg)*gCal		calInK(Avg)
cntsToK(f)	=	-----------------------------	=	----------------------
		calDif_cnts(Avg)*gIf*gCal		calDif_cnts(Avg)*gIF

• The calInK frequency dependence has cancelled out. It still contains the IF bandpass frequency dependence.


• cntsToK(avg) = <cntsToK(f)> = calInK(Avg) / (calDif_cnts(Avg) * <gIF>)
• but <gIF> = one by definition. so
• cntsToK(avg) =  calInK(Avg) / (calDif_cnts(Avg))

• b_K(f)=b_cnts(f) * cntsToK(avg)
• b_K(f)=

              b_cnts(Avg)*gIF * calInK(Avg)
              ------------------------------
              calDif_cnts(Avg)

• This still contains the gIF Freq response.
• To remove the frequency response of the  IF you need a bandpass correction that includes gIF. eg normalized  calOff:
• b_K(f)/ (calOff_cnts(f)/<calOff_cnts(f)>)=

         =b_cnts(Avg)*gIF * calInK(Avg)    (calOff_cnts(Avg)*<gIF>)
          ----------------------------- * ------------------
          (calDif_cnts(Avg))                calOff_cnts(Avg)*gIF

• This has removed all of the bandpass dependencies
  

• C = <C> * gIF.
  
• the value of <C> will depend on the channels used for the average and the type of average (mean,median).
• You can not cancel gIF 's if they have been computed using different channels or types of average.
  

An example using this:

• Variables:
• BcalOn_cnts cal on spectra
• BcalOff_cnts cal off spectra
• B_cnts spectra to convert to Kelvins
• TcalAvg_K: average cal value for spectra
• Converting a spectrum to Kelvins:
  
• cntsToK(avg)= TcalAvg_K/<BcalOn_cnts - BcalOff_cnts>
• B_k = B_cnts * cntsToK(avg) /   bCalOff_cnts/<bCalOff_cnts>
• Converting spectrum to Kelvins then computing average Tsys:
  
• Before dividing by the normalized calOff spectrum we had:

• Averaging over the spectra gives:

• Converting on/off position switch data to Kelvins
• Bpon_cnts(f)=(Tsrc_cnts(f) + Tsys_cnts(f))*gIF  on source
  
• Bpoff_cnts(f)=(Tsys_cnts(f))*gIF  off Position
  
• Bsrc_tsys(f)=(Bpon_cnts/Bpoff_cnts -1)=Tsrc_cnts(f)/Tsys_cnts(f)
  
• This is now in units of Tsys. The gIF has cancelled out.
• Bsrc_K(f)= (Tsrc_cnts(f)/Tsys_cnts(f)-1)  *X * (calInK(avg)/(calDif_cnts(avg)).
• We Need X to convert from Tsys units back to cnts since cntsToK(avg) converts from _cnts to K.
• X=mean(Bpoff_cnts(f)) = <Bpoff_cnts(avg)*<gIF> > = Bpoff_cnts(avg)
• Bsrc_K(f)=(Bpon_cnts(f)/BpOff_cnts(f) - 1) * (<BpOff_cnts(f)>) * CalInK(avg)/caldif_cnts(avg)
• All of the averages should use the same spectral channels (in case you throw away some channels because of rfi).
  

Computing the conversion from Spc to Kelvins by frequency channel.

• cntsToK(f) =   

• b_K(f)          =
  

• The frequency dependence from the bandpass/cal  has cancelled out.

Summary:

• To compute the Tsys averaged over the band:
• cntsToK(avg) * <b_cnts> 
  
• To compute Tsys vs frequency using the scalar cntstoK(avg) you need to divide the b_K(f) by a normalized bandpass correction (say the off in position switching). This will increase the noise if the divisor does not have as long an integration as the on.
  
• b_K(f)  = b_cnts(avg)*gIF * cntsToK(avg) / (bpc()/<bpc()>)

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