# Noise figure

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Noise figure (NF) and noise factor (F) are measures of degradation of the signal-to-noise ratio (SNR), caused by components in a 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.

## Contents

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 kTB, 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 factorF of a system is defined as [3]

${\displaystyle F={\frac {\mathrm {SNR} _{\text{i}}}{\mathrm {SNR} _{\text{o}}}}}$

where SNRi and SNRo are the input and output signal-to-noise ratios respectively. The SNR quantities are power ratios. The noise figure NF is defined as the noise factor in dB:

${\displaystyle \mathrm {NF} =10\log _{10}(F)=10\log _{10}\left({\frac {\mathrm {SNR} _{\text{i}}}{\mathrm {SNR} _{\text{o}}}}\right)=\mathrm {SNR} _{\text{i, dB}}-\mathrm {SNR} _{\text{o, dB}}}$

where SNRi, dB and SNRo, dB are in decibels (dB). These formulae are only valid when the input termination is at standard noise temperature T0 = 290 K, 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]

${\displaystyle F=1+{\frac {T_{\text{e}}}{T_{0}}}.}$

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 Te = (L 1)T, giving a noise factor

${\displaystyle F=1+{\frac {(L-1)T}{T_{0}}}.}$

## Noise factor of cascaded devices

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

${\displaystyle F=F_{1}+{\frac {F_{2}-1}{G_{1}}}+{\frac {F_{3}-1}{G_{1}G_{2}}}+{\frac {F_{4}-1}{G_{1}G_{2}G_{3}}}+\cdots +{\frac {F_{n}-1}{G_{1}G_{2}G_{3}\cdots G_{n-1}}},}$

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 usually has the most significant effect on the total noise figure because the noise figures of the following stages are reduced by stage gains. Consequently, the first amplifier usually has a low noise figure, and the noise figure requirements of subsequent stages is usually more relaxed.

## Noise factor as a function of additional noise

The noise factor may be expressed as a function of the additional output referred noise power ${\displaystyle N_{a}}$ and the power gain ${\displaystyle G}$ of an amplifier.

${\displaystyle F=1+{\frac {N_{a}}{N_{i}G}}}$

### Derivation

From the definition of noise factor [3]

${\displaystyle F={\frac {\mathrm {SNR} _{\text{i}}}{\mathrm {SNR} _{\text{o}}}}={\frac {\frac {S_{i}}{N_{i}}}{\frac {S_{o}}{N_{o}}}},}$

and assuming a system which has a noisy single stage amplifier. The signal to noise ratio of this amplifier would include its own output referred noise ${\displaystyle N_{a}}$, the amplified signal ${\displaystyle S_{i}G}$ and the amplified input noise ${\displaystyle N_{i}G}$,

${\displaystyle {\frac {S_{o}}{N_{o}}}={\frac {S_{i}G}{N_{a}+N_{i}G}}}$

Substituting the output SNR to the noise factor definition, [6]

${\displaystyle F={\frac {\frac {S_{i}}{N_{i}}}{\frac {S_{i}G}{N_{a}+N_{i}G}}}={\frac {N_{a}+N_{i}G}{N_{i}G}}=1+{\frac {N_{a}}{N_{i}G}}}$

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## References

1. Agilent 2010 , p. 7
2. Agilent 2010 , p. 5.
3. Agilent 2010 , p. 7 with some rearrangement from Te = T0(F − 1).
4. Agilent 2010 , p. 8.
5. Aspen Core. Derivation of noise figure equations (DOCX), pp. 3–4
• Agilent (August 5, 2010), Fundamentals of RF and Microwave Noise Figure Measurements (PDF), Application Note, 57-1

This article incorporates  public domain material from the General Services Administration document: "Federal Standard 1037C".(in support of MIL-STD-188)