2.4 Minimum Detectable Temperature.
When observing thermal (or other ‘wideband’) signal sources we often need to know how small a temperature change (or signal power level) a receiver can see over the effects of its internal noise. It should be noted that this is not equal to its noise temperature. The output low pass Post Detection Filter will have a bandwidth, 
, This is equivalent to saying that it smooths out fluctuations with a Time Constant, 
. One of the properties of thermal noise is that a mean power level of P will fluctuate by an amount 
from moment to moment. It is these fluctuations in the power level which limit our ability to see low power levels or detect small changes.
The signal power from a thermal source is ‘white’ — i.e. it has a uniform power spectral density. Doubling the IF bandwidth would therefore double the signal power level. This means that the observed fluctuations would only increase by a factor of 
. Hence we can expect the signal to noise performance to improve by 
if we double the IF bandwidth of the receiver. We can therefore expect the minimum detectable temperature (or detectable change in temperature) 
to be such that
The spectrum of fluctuations will be filtered by the post detection filter. Provided this has a bandwidth, 
we can expect it to affect the size of the observed random fluctuations. The narrower its bandwidth (and hence the larger its time constant value), the more of the fluctuation spectrum it will smooth away. Hence we can expect that
where a is some constant value.
For our purposes the precise value of a isn't important (in fact it is usually about 1-2 and its value depends upon various factors we haven't bothered with). The interesting point is that we can use a receiver whose noise temperature is fixed at 
to detect a smaller input level/change either by increasing the IF bandwidth and/or by increasing the post detection filter's time constant.
Expressing this result in terms of a time constant rather than a post detection bandwidth is convenient because — for reasons we won't consider here — what really matters is the Integration Time or Measurement Time. In effect, a time constant provides us with an result ‘averaged’ over a period equal to one time constant. More generally, we can make measurements over some longer period, take their average, and then substitute the total measurement time in place of t in the above expressions. This is an example of a general result; the S/N ratio of a measurement often increases if we record values over a longer period and average them together.
The value of this result can be illustrated by taking an example where we have an IF filter with a bandwidth of 100 MHz and a post detection time constant of 1 second. Using these filters, a receiver whose noise temperature is 1000 °K would be able to detect a 0·1 °K source with a 1:1 measurement S/N ratio in 1 second. Given 100 seconds this would improve to being able to detect an 0·01 °K source (or a change in temperature of 0·01 °K). This behaviour explains how radioastronomers are able to detect distant galaxies with effective Brightness Temperatures of a fraction of a Kelvin using receivers with noise temperatures of thousands of Kelvins. They ‘simply’ sit pointing the telescope and receiver at the same source for hours on end!
Summary.
You should now know how the performance of a superheterodyne receiver system depends upon its basic properties. That the minimum possible Noise Temperature tends to increase with the input signal frequency. That the ability of a system to detect low level signals (or small changes in level) depend upon the noise temperature, the IF bandwidth, and the post detection Time Constant or measurement time.
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University of St. Andrews, St Andrews, Fife KY16 9SS, Scotland.