# Noise Temperature

## Contents

The noise temperature of an amplifier refers to the noise that would be added at the amplifier’s input in order to account for the added noise observed following amplification. This noise is generally the result of the random motions of electrons at a certain temperature, so it isn’t so surprising that the noise generated by them can generally be described as a temperature. Note that, following this definition, the noise temperature does not include the gain of the amplifier.

As discussed in the context of Johnson-Nyquist noise in a resistor, the noise power a resistor injects into a downstream circuit (with a load impedance that matches the source impedance of the noise generator) is given by:

$\frac{P}{B}=k_ B T, \,\!$

where P is the noise power, B is bandwidth, kB is Boltzmann’s constant, and T is the noise temperature. For resistors, we showed that this quantity was independent of the impedance of the resistor, and in fact, the concept can be naturally extended to relate to the input impedance of amplifier circuits. For resistors, the noise temperature T equal to the thermal temperature of the resistor. For amplifiers, this is not always necessarily the case, although it is a good general rule, and cooling amplifiers definitely lowers their noise temperature.

## Noise Figure

Since it would be far too easy to simply report the noise temperature of a receiver, instead, a noise factor F is sometimes used:

$F\equiv \frac{T_0+T}{T_0} \,\!$

with T0 customarily taken to be 290K. To further obfuscate matters, instead of reporting the noise factor F, it is even more common to report a noise figure NF:

$NF\equiv 10 \log _{10}(F), \,\!$

which is to say, the noise figure, NF, is the noise factor F expressed in decibels.

## Amplifier Chains

In general, temperatures add, so the noise temperature that results from chaining multiple noise sources is simply the sum of their respective noise temperatures. Simple enough.

However, if a noise source is an amplifier with gain, G, then downstream noise sources will have their noise temperatures divided by G. For a pair of amplifiers with gains G1 and G2, and noise temperatures T1 and T2, respectively, then the total noise temperature will be given by:

$T=T_1+\frac{T_2}{G_1}+\frac{T_{rest}}{G_1G_2}. \,\!$

This is pretty straightforward to understand: if you calibrate your final output back to the input signal level, you’ll have to divide by the total gain of all the amplifiers. Hence, noise that is added in after a gain stage, when the signal was comparatively stronger relative to a fixed noise level, has less effect on the signal, and, in the units of the input signal, has a noise temperature that is reduced by a factor of the gain.

## Measuring Noise Temperature

The Y-Factor method for computing the noise temperature of an amplifier.

In order to measure the noise temperature of an amplifier (a.k.a. the receiver temperature), you need to solve for two parameters: the gain of the amplifier, G, and its noise temperature, T. A straightforward method for doing this, illustrated above, is called the Y-Factor method. First, you drive the input of the amplifier with the noise off of a resistor that matches the input impedance of your amplifier. You measure the temperature of the resistor (or better, you fix the resistor temperature to a known quantity) so that you know the noise power injected into the amplifier, and measure the output power from the amplifier, which is given by:

$\frac{P_{out,1}}{B}=G(k_ BT_1 + k_ B T), \,\!$

where T1 is the temperature of the resistor. Note that without knowing G very well, you cannot solve for T. We need more information, so next, you heat or cool the resistor to a different temperature T2 and measure the power output:

$\frac{P_{out,2}}{B}=G(k_ BT_2 + k_ B T), \,\!$

Now, with two measurements and two unknowns, you can solve for both G and T. If you are curious, the Y factor is defined to be:

$Y\equiv \frac{P_{out,1}}{P_{out,2}} = \frac{T+T_1}{T+T_2}, \,\!$

so that

$T=\frac{T_1-YT_2}{Y-1}. \,\!$