Using the Cal Signals To Find Gain Non-linearity

At this point we desired a better method to diagnose the non-linear response of the IF system. Since it would seem that the noise diodes can excite a non-linear response (see Figure 2) we thought that comparing the two states with the noise diode on and off could provide some insight.

We can define the gain, $G$, of the IF system as follows:

$$G = {P_{\rm out}(\rm counts) \over P_{\rm in}(\rm K)}$$ (2)

where $P_{\rm in}$ is the input power to the system (i.e. total system temperature), and $P_{\rm out}$ is the raw output counts of one of the backends. If the IF system response is linear then $G={\rm constant}$. We can use the first derivative of Equation 2 to check for non-linearities. The first derivative is

$${\partial P_{\rm out}(\rm counts) \over \partial K} = {\part... ...m in}(\rm K) + G {\partial P_{\rm in}(\rm K) \over \partial K}$$ (3)

which can be approximated as

$${\partial P_{\rm out}(\rm counts) \over \partial K} \sim {P_... ...+ { \Delta G  P_{\rm in}^{\rm cal off} \over T_{cal}(\rm K)}$$ (4)

for two power levels that are separated by only by firing the noise diode. $\Delta G$ is the change in the gain between the two power levels. Now by taking the data with the cals firing on two different sources with different input power levels (or on a single source and changing the level of attenuation in the IF system) we can look for non-linearities. Taking the ratio (we will refer to this as the gain ratio hereafter)

$$R_{gain} ={ \left( P^{\rm cal on}_{\rm out} - P^{\rm cal off... ...Delta G  P_{\rm in}^{\rm cal off}\right) \vert _{\rm src 2} }$$ (5)

it can be seen that if the gain is linear then $\Delta G = 0$ and $R=1$. If the system gain is non-linear then $R \neq 1$ and $\Delta G \neq 0$.

Figure 3: Gain ratios plotted for two polarizations. The red lines compare $R_{gain}$ for the sources NGC 7027 and 2202+422. The blue lines compare $R_{gain}$ for blank sky between $20^\circ$ and $10^\circ$ elevation (the off positions for the double position switch observations of NGC 7027 and 2202+422.

In Figure 3 we plot $R_{gain}$ for part of the double position switch observations of NGC 7027 and 2202+422. We considered the gain ratio for the NGC 7027 and 2202+422 on source data and for the NGC 7027 and 2202+422 off source data. The off source data differ in $T_{sys}$ slightly since the off positions are at elevations of $20^\circ$ and $10^\circ$ for these observations.

Figure 3 shows us that the GBT IF system gain is non-linear even for changes of input power of $\bf\sim 5\%$ on blank sky. It should also be noted that the frequency dependence of the gain ratio is approximately the shape of the residual baselines in the results of the double position switching data.

Although most Optical Drivers exhibit symptoms of gain supression there is some indication that some Optical Drivers exhibit anti-compression (i.e. the Gain Ratio is less than one). Further tests are needed to isolate which Optical Drivers exhibit this feature - currently Optical Driver 3 is suspect.