# Noise figure

Noise figure (NF) and noise factor (F) are measures of degradation of the signal-to-noise ratio (SNR), caused by components in a radio frequency (RF) signal chain. It is a number by which the performance of an amplifier or a radio receiver can be specified, with lower values indicating better performance.

The noise factor is defined as 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 T0 (usually 290 K). The noise factor is thus the ratio of actual output noise to that which would remain if the device itself did not introduce noise, or the ratio of input SNR to output SNR.

The noise figure is simply the noise factor expressed in decibels (dB).[1]

## General

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 matched sources at the standard noise temperature T0 (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 receiver antenna is pointed out into cold space, the antenna effective temperature is often colder than 290 K.[2] 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 low noise amplifiers.

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.

## Definition

The noise factor F of a system is defined as:[3]

$F = \frac{\mathrm{SNR}_\mathrm{in}}{\mathrm{SNR}_\mathrm{out}}$

where SNRin and SNRout are the input and output signal-to-noise ratios, respectively. The SNR quantities are power ratios. The noise figure NF is defined as:

$\mathrm{NF} = 10 \log(F) = 10 \log\left(\frac{\mathrm{SNR}_\mathrm{in}}{\mathrm{SNR}_\mathrm{out}}\right) = \mathrm{SNR}_\mathrm{in, dB} - \mathrm{SNR}_\mathrm{out, dB}$

where SNRin, dB and SNRout, dB are in decibels (dB). The noise figure is the noise factor, given in dB:

$\mathrm{NF} = 10 \log \left(F\right)$

These formulae are only valid when the input termination is at standard noise temperature T0, although in practice small differences in temperature do not significantly affect the values.

The noise factor of a device is related to its noise temperature Te:[4]

$F = 1 + \frac{T_e}{T_0}$

Attenuators have a noise factor F equal to their attenuation ratio L when their physical temperature equals T0. More generally, for an attenuator at a physical temperature T, the noise temperature is $T_\mathrm{e} = (L-1)T$, giving a noise factor of:

$F = 1 + \frac{(L-1)T}{T_0}$

If several devices are cascaded, the total noise factor can be found with Friis' Formula:[5]

$F = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \frac{F_4 - 1}{G_1 G_2 G_3} + \cdots + \frac{F_n - 1}{G_1 G_2 G_3 \cdots G_{n-1}},$

where Fn is the noise factor for the n-th device and Gn is the power gain (linear, not in dB) of the n-th device. The first amplifier in a chain has the most significant effect on the total noise figure than any other amplifier in the chain. The lower noise figure amplifier should usually go first in a line of amplifiers (assuming all else is equal).