# Estimates of sensitivity and detrimental interference levels:

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The theoretical considerations for the sensitivity of radio astronomy systems can be used to estimate quantitatively the sensitivities and interference levels for radio astronomical observations. The results based on an assumed integration time t of 2000 seconds (see: “ITU-R Handbook on Radio Astronomy, 1995, section 4.2.2) are given in Recommendation ITU-R RA769. The sensitivity, expressed in units of temperature or power spectral density, is the level at the receiver input required to increase the output by an amount equal to the rms noise fluctuations.

The detrimental interference levels given in ITU-R RA769 are expressed as the interference level which introduces into the standard deviation of the power, dP (or of the temperature, dT) a component equal to 10% of the r.m.s. fluctuations due to the system noise, i.e.:

```	dPH = 0.1 dPs df                                       (1)
```

where:

dPH: level of harmful interference;

dPs: noise fluctuation in power spectral density in the sensitivity equation – expressed in system temperature as dT;

df: for continuum observations: the bandwidth of the allocated radio astronomy band;
for spectral line observations: the channel bandwidth representative of a spectral line – the values used for df correspond to a velocity of 3 km/s, which is intermediate between values common for spectral lines of sources within our galaxy and in external galaxies.

Interference can be expressed in terms of the power flux-density incident at the antenna, either in the total bandwidth or as a spectral power flux-density SH per 1 Hz of bandwidth. As discussed in the ITU-R Handbook on Radio Astronomy (1995, section 4.2.1), the values in ITU-R RA 769 are given for an antenna having a gain, in the direction of arrival of the interference, equal to that of an isotropic antenna (which has an effective area of c2/4 π f2, where c is the speed of light and f is the frequency). Values of SHdf, in units of dB(W/m2), are derived from dPH, in dBW, by adding:

```	20 log f - 158.5 dB                                    (2)
```

where f is in Hz. SH is then derived by subtracting 10 log df to allow for the bandwidth. SH can also be expressed as a single equation as follows:

```	     0.4 π k(TA + TR)f2
SH = ----------------                                  (3)
c2(t df)0.5
```

The sensitivity of a radio astronomy receiving system to wideband (continuum) radiation improves when the bandwidth is increased. The reason for this is as follows: the noise power increases with bandwidth, but, since the signal also is broadband noise, so does the signal. The signal-to-noise power ratio in the RF or IF stages before the detector remains constant, independent of the bandwidth. However, as the bandwidth increases, the precision of the determination of the power levels improves as (bandwidth)0.5, and thus the sensitivity is correspondingly improved.

Based on the theoretical considerations one may assume that one may achieve any desired sensitivity by making the bandwidth and/or the observing time large enough. In practice, however, factors other than the statistical ones put a practical limit on the sensitivity of a radio astronomy observation. Examples of such factors are the stability of the receiver and fluctuations in the attenuation and phase path in the Earth’s atmosphere. The sensitivity levels given in Recommendation ITU-R RA769 use values for the bandwidth and integration time for which these other factors usually are insignificant. However, it should be emphasised that these sensitivity levels are not fundamental and that they are routinely exceeded in cases where the data can be integrated over periods of many hours (from: “ITU-R Handbook on Radio Astronomy“, 1995, section 4.3.2).