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Determining SNR and sensitivity
How to measure the noise
After calibrating the (not bandpass-corrected) baseline visibilities for phases determined from the fringe-fitting result, we determine the final bandpass by vector-averaging per channel and polarisation. The data are divided by this bandpass, so that they are unity on average and deviations are only due to residual calibration problems and due to thermal noise. We can now determine the rms deviation from 1 (separately for real and imaginary part) in several ways:
| no | name | description |
|---|---|---|
| 0 | rms | rms deviation from 1 |
| 1 | diff in time | from difference between adjacent time bins |
| 2 | diff in freq | from difference between adjacent channels |
| 3 | symmetric diff in time | from difference to mean of both adjacent time bins |
| 4 | symmetric diff in freq | from difference to mean of both adjacent channels |
The correct scaling factors are applied in all cases. For "diff in freq" the difference between channels 0 and 1 is used for 0 etc, the last one is assigned zero. For "symmetric diff in freq" the first and last are assigned zero.
These procedures have different advantages and disadvantages. The "rms" is very simple, but only accurate if the calibration works perfectly, because otherwise residual calibration errors are misinterpreted as thermal noise. The differences are generally less affected by calibration problems. The more channels we have, the higher is the thermal noise relative to calibration residuals, so that the accuracy increases.
From this we get several versions (real/imag for each of the 5 procedures) of noise-to-signal (N/S). If we multiply with the calibrator flux per channel, we obtain the SEFD for the baseline. To a very good approximation this is a factor of sqrt (2) smaller than the SEFD of individual stations (or the geometric mean of both SEFDs if they are different), as we will see below.
Note that real and imaginary part are affected by different calibration problems: real more by amplitude and imag more by phase
Baseline noise
If the SEFDs of two stations are N1 and N2 and the source flux is S, we can derive the following noise quantities:
| quantity | value |
|---|---|
| noise in 1 | N1+S |
| noise in 2 | N2+S |
| noise in baseline 1-2, real | sqrt { [ (N1+S)*(N2+S) + S*S ] / 2 } |
| noise in baseline 1-2, imag | sqrt { [ (N1+S)*(N2+S) - S*S ] / 2 } |
For PKS 1934-638 we have a flux of roughly S=10Jy, the SEFD is about 500Jy. If we completely neglect the source flux, we have:
| quantity | value |
|---|---|
| noise in 1 | N1 |
| noise in 2 | N2 |
| noise in baseline 1-2, real or imag | sqrt ( N1*N2 / 2 ) |
If we take S into account to first order for the baseline, we find:
| quantity | value |
|---|---|
| noise in baseline 1-2, real or imag | sqrt { [ N1*N2 + S*(N1+N2) ] / 2 } |
Even for high S we can combine the real and imaginary noise (if we can trust both) to find that the sum of squares of both is (N1+S)*(N2+S). This can be separated to get the station-based N+S values, from which we can determine the individual SEFD = (N+S) - S.
Remember that our calibration procedure provides not the noise itself, but the noise divided by the signal.
