Definitions

# Noise figure

In telecommunication, noise figure (NF) is a measure of degradation of the signal to noise ratio (SNR), caused by components in the RF signal chain. The noise figure is the ratio of the output noise power of a device to the portion thereof attributable to thermal noise in the input termination at standard noise temperature $T_0$ (usually 290 K). The noise figure is thus the ratio of actual output noise to that which would remain if the device itself did not introduce noise. It is a number by which the performance of a radio receiver can be specified.

## General

In heterodyne systems, output noise power includes spurious contributions from image-frequency transformation, but the portion attributable to thermal noise in the input termination at standard noise temperature includes only that which appears in the output via the principal frequency transformation of the system and excludes that which appears via the image frequency transformation.

Essentially, the noise figure is the difference in decibels (dB) between the noise output of the actual receiver to the noise output of an “ideal” receiver with the same overall gain and bandwidth when the receivers are connected to sources at the standard noise temperature $T_0$ (usually 290 K). The noise power from a simple load is equal to $k T B$, where $k$ is Boltzmann's constant, $T$ is the absolute temperature of the load (for example a resistor), and $B$ is the measurement bandwidth.

This makes the noise figure a useful figure of merit for terrestrial systems where the antenna effective temperature is usually near the standard 290 K. In this case, one receiver with a noise figure say 2 dB better than another, will have an output signal to noise ratio that is about 2 dB better than the other. However, in the case of satellite communications systems, where the antenna is pointed out into cold space, the antenna effective temperature is often colder than 290 K. In these cases a 2 dB improvement in receiver noise figure will result in more than a 2 dB improvement in the output signal to noise ratio. For this reason, the related figure of effective noise temperature is therefore often used instead of the noise figure for characterizing satellite-communication receivers and LNA.

## Mathematics

Noise figure is given by
$mathrm\left\{NF\right\} = mathrm\left\{SNR\right\}_mathrm\left\{in\right\} - mathrm\left\{SNR\right\}_mathrm\left\{out\right\}$
where every variable is a dB figure. The previous formula is only valid when the input termination is at standard noise temperature $T_0$.

Sometimes the noise factor F is specified, which is the numerical ratio form of noise figure. Noise Factor is a straight ratio of SNR ratios. Noise Figure is the decibel equivalent of Noise Factor. The following formula is only valid when the input termination is at standard noise temperature $T_0$.

$F = frac\left\{mathrm\left\{SNR\right\}_mathrm\left\{in\right\}\right\}\left\{mathrm\left\{SNR\right\}_mathrm\left\{out\right\}\right\}$
where everything is a ratio

$F = 10^mathrm\left\{NF/10\right\}, quad mathrm\left\{NF\right\} = 10 log\left(F\right)$

The noise factor of a device is related to its noise temperature via

$F = 1 + frac\left\{T_mathrm\left\{e\right\}\right\}\left\{T_0\right\}$

Devices with no gain (e.g., attenuators) have a noise figure equal to their attenuation L (in dB) when their physical temperature equals $T_0$. More generally, for an attenuator at a physical temperature $T_mathrm\left\{phys\right\}$, the noise temperature is $T_mathrm\left\{e\right\} = \left(L-1\right)T_mathrm\left\{phys\right\}$, thus giving a noise factor of $F = 1 + frac\left\{\left(L-1\right)T_mathrm\left\{phys\right\}\right\}\left\{T_0\right\}$

If several devices are cascaded, the total noise factor can be found with Friis' Formula:

$F = F_1 + frac\left\{F_2 - 1\right\}\left\{G_1\right\} + frac\left\{F_3 - 1\right\}\left\{G_1 G_2\right\} + frac\left\{F_4 - 1\right\}\left\{G_1 G_2 G_3\right\} + cdots + frac\left\{F_n - 1\right\}\left\{G_1 G_2 G_3 cdots G_\left\{n-1\right\}\right\},$
where $F_n$ is the noise factor for the n-th device and $G_n$ is the power gain (numerical, not in dB) of the n-th device.