Radiometer Equation

Prerequisites

• Central Limit Theorem
• Noise Temperature

Reference Material

• Synthesis Imaging in Radio Astronomy II, ed. Taylor, Carilli, Perley, Ch. 9 (Wrobel & Walker)
• NRAO Radio Antennas and Radiometers

The Radiometer Equation

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:

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

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

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

In the denominator, $2Bt$, is simply the number of independent samples (i.e. $N$ 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 $B$, expressed in Hz, contains $2B$ independent pieces of information each second. Thus, for an measurement made over $t$ seconds, we have averaged $2Bt$ independent samples.

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

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