Calibration Equations

The power spectrum at the input of the correlator when the telescope is pointed at the source of interest, , and at the reference target, , can be written as

where



, , and are assumed not to vary significantly with frequency over the bandpass of the spectrometer. Taking ratios and doing appropriate algebra,


Since the gain is the unknown that one aims to remove by this calibration, it is assumed not vary with power level. One then obtains the difference between the signal spectrum and the reference spectrum:
 
The ratio of the mean power levels can be obtained from a broadband detector or power meter or similar device. In general the spectrum will have poor baselines because the system gain is invariably non-linear to some small degree and the non-linearity is a function of frequency.

The system temperatures for the reference spectrum and the signal spectrum, respectively, are

Equation 1 gives the signal, both line and continuum, by which the source differs from the reference:

Since spectral lines are typically weak, and , so that

and therefore

This allows us to rewrite equation 1 as
 
and suggests the definition of a normalized calibrated spectrum as

so that
 

Noise Analysis

The expected r.m.s. noise in is calculated from equation 2:

Substituting in the radiometer equation for the normalized spectrum

we get
 
which, assuming that the sig and ref integration times are the same, leads to the result
 
Generally, the sig/ref duty cycle is 50%, and the total integration sig+ref integration time, is used in the radiometer equation, in which case
 
A tidy way to avoid having to worry about the constant in the radiometer equation, and also to handle cases where the two spectra don't have the same integration times, is to define the integration time for the calibrated spectrum so that the constant in the radiometer equation remains unity:

so that
 
In the case of a 50% duty cycle leads to .

Combining Spectra

We have two spectra and which we want to combine optimally so that the combined spectrum has the minimum noise. To do this we combine them with weights and so that

The proof below shows that this requires that
 
To apply this result, we note from equation 9 above that
 
Since any constant divides out, we can choose and thus

Using equation 4 to replace with and equation 16 (see below) to replace the denominator, we obtain the equivalent normalized spectrum:

Proof: The noise in the combined spectrum is
 
We want to minimize this noise by an optimum choice of :

which proves equation 10.

Noise Propagation

With equation 10, equation 15 reduces to:

Using equation 9, and cancelling out , we can write this as
 
where and are defined as the effective system temperature and integration time of the combined spectrum. If , then proper scaling of the combined spectrum requires that , and thus that

It then follows from equation 16 that