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### Reference Material

The radiometer equation, at its heart, is a relatively straight-forward application of the Central Limit Theorem. It describes how the uncertainty in measuring a noise temperature decreases as the square-root of the number of samples averaged together:

${\displaystyle \sigma _{T}={\frac {T_{sys}}{\sqrt {Bt}}},\,\!}$

where ${\displaystyle \sigma }$ is the residual (root-mean-square) uncertainty in a noise temperature measurement, ${\displaystyle T_{sys}}$ is the noise temperature of a circuit (or “system"), ${\displaystyle B}$ is the bandwidth over which a single measurement is made (i.e. the integrated bandwidth), and ${\displaystyle t}$ is the time over which a measurement is averaged (i.e. the integration time).

Although the above equation is correct, to fully understand it, it would be better to write it as follows:

${\displaystyle \sigma _{T}={\frac {{\sqrt {2}}T_{sys}}{\sqrt {2Bt}}}\,\!}$

In the denominator, ${\displaystyle 2Bt}$, is simply the number of independent samples (i.e. ${\displaystyle N}$ in the Central Limit Theorem) that were averaged together into a single measurement. The accounting goes as follows: according to the Nyquist Theorem of sampling, it takes two samples per period to uniquely characterize a sine wave. Said another way, a signal with bandwidth ${\displaystyle B}$, expressed in Hz, contains ${\displaystyle 2B}$ independent pieces of information each second. Thus, for an measurement made over ${\displaystyle t}$ seconds, we have averaged ${\displaystyle 2Bt}$ independent samples.

The numerator of the (re-expressed) radiometer equation, ${\displaystyle {\sqrt {2}}T_{sys}}$, has one bit of trickiness in it, and to understand it, we first need to understand what exactly we are measuring, and why there is uncertainty in our measurements. ${\displaystyle T_{sys}}$ is a noise temperature, which means that it characterizes the variance of a noise signal with zero mean. When you go to measure a noise temperature, you are really trying to measure the variance of a random noisy signal.

For any limited number of samples ${\displaystyle x_{1},x_{2},\dots }$ generated by a random process, there is an inherent uncertainty in the variance, ${\displaystyle \sigma ^{2}}$, you compute for that sample, just as there would be an inherent uncertainty in the mean, ${\displaystyle {\bar {x}}}$. However, whereas ${\displaystyle \sigma }$ characterizes the per-sample uncertainty for the purpose of calculating ${\displaystyle {\bar {x}}}$, if you are trying to measure ${\displaystyle \sigma ^{2}}$, the uncertainty is actually ${\displaystyle {\sqrt {2}}\sigma }$ for each sample. This is because, to compute the variance, you have to square each sample (e.g. ${\displaystyle x_{1}^{2},x_{2}^{2},\dots }$), and then average. The math can get messy, but if you just take your computer and calculate the standard deviation of ${\displaystyle x^{2}}$, for ${\displaystyle x}$ drawn from a Gaussian distribution with ${\displaystyle \sigma =1}$, you’ll find that it is ${\displaystyle {\sqrt {2}}}$.

This is all to say that if you are trying to measure the noise temperature, ${\displaystyle T_{sys}}$, which relates to the variance of a distribution, then the uncertainty of each variance measurement is ${\displaystyle {\sqrt {2}}T_{sys}}$. That is the measurement error that we are beating down by ${\displaystyle {\sqrt {N}}}$ according to the Central Limit Theorem (where ${\displaystyle N=2Bt}$), and so that is what goes in the numerator of the Radiometer equation. It’s just a confusing accident that the ${\displaystyle {\sqrt {2}}}$ for the measurement error associated with measuring the variance cancels out the factor of 2 in ${\displaystyle N}$ associated with Nyquist sampling.