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 SNR_in and SNR_out 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:

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

where SNR_in, dB and SNR_out, dB are in decibels (dB). These formulae are only valid when the input termination is at standard noise temperature T₀=290K, 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 T_e:^[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 F_n is the noise factor for the n-th device and G_n 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).

See also

• Noise
• Noise (electronic)
• Noise figure meter
• Noise level
• Thermal noise
• Signal-to-noise ratio
• Y-factor

References

1. http://www.satsig.net/noise.htm
2. Agilent 2010, p. 7
3. Agilent 2010, p. 5
4. Agilent 2010, p. 7 with some rearrangement from T_e=T₀(F-1).
5. Agilent 2010, p. 8

• Agilent (August 5, 2010), Fundamentals of RF and Microwave Noise Figure Measurements (PDF), Application Note, 57-1

External links

• Noise Figure Calculator 2- to 30-Stage Cascade
• Noise Figure and Y Factor Method Basics and Tutorial
• Mobile phone noise figure

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