Noise figure

Noise figure (NF) and noise factor (F) are figures of merit that indicate degradation of the signal-to-noise ratio (SNR) that is caused by components in a signal chain. These figures of merit are used to evaluate the performance of an amplifier or a radio receiver, 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 factor and noise figure are related, with the former being a unitless ratio and the latter being the same ratio but expressed in units of 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 the Boltzmann 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]

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

where SNR_i and SNR_o are the input and output signal-to-noise ratios respectively. The SNR quantities are unitless power ratios. The noise figure NF is defined as the noise factor in units of decibels (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 SNR_i, dB and SNR_o, dB are in units of (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 T_e:^[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 T_e = (L − 1)T, giving a noise factor

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

Noise factor of cascaded devices

Main article: Friis formulas for noise

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 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 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 source outputs a signal of power ${\displaystyle S_{i}}$  and noise of power ${\displaystyle N_{i}}$ . Both signal and noise get amplified. However, in addition to the amplified noise from the source, the amplifier adds additional noise to its output denoted ${\displaystyle N_{a}}$ . Therefore, the SNR at the amplifier's output is lower than at its input.

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]

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

In cascaded systems ${\displaystyle N_{i}}$  does not refer to the output noise of the previous component. An input termination at the standard noise temperature is still assumed for the individual component. This means that the additional noise power added by each component is independent of the other components.

Optical noise figure

The above describes noise in electrical systems. Electric sources generate noise with a power spectral density equal to kT, where k is the Boltzmann constant and T is the absolute temperature. However, there is also noise in optical systems. In these, the sources have no fundamental noise. Instead the energy quantization causes notable shot noise in the detector, corresponding to a noise power spectral density of hf where h is the Planck constant and f is the optical frequency.

In the 1990s, an optical noise figure has been defined.^[7] This has been called F_pnf for photon number fluctuations.^[8] The powers needed for SNR and noise factor calculation are the electrical powers caused by the current in a photodiode. SNR is the square of mean photocurrent divided by variance of photocurrent. Monochromatic or sufficiently attenuated light has a Poisson distribution of detected photons. If, during a detection interval the expectation value of detected photons is n then the variance is also n and one obtains SNR_pnf,in = n²/n = n. Behind an optical amplifier with power gain G there will be a mean of Gn photons. In the limit of large n the variance of photons is Gn(2n_sp(G-1)+1) where n_sp is the spontaneous emission factor. One obtains SNR_pnf,out = G²n²/(Gn(2n_sp(G-1)+1)) = n/(2n_sp(1-1/G)+1/G). Resulting optical noise factor is F_pnf = SNR_pnf,in / SNR_pnf,out = 2n_sp(1-1/G)+1/G.

F_pnf is in conceptual conflict compared to the electrical noise factor, which is now called F_e:

Photocurrent is proportional to optical power. Optical power is proportional to squares of a field amplitude (electric or magnetic). So, the receiver is nonlinear in amplitude. The power needed for SNR_pnf calculation is proportional to the 4th power of the signal amplitude. But for F_e in the electrical domain the power is proportional to the square of the signal amplitude.

At a certain electrical frequency, noise occurs in phase (I) and in quadrature (Q) with the signal. Both these quadratures are available behind the electrical amplifier. The same holds in an optical amplifier. But the direct detection photoreceiver needed for measurement of SNR_pnf takes mainly the in-phase noise into account whereas quadrature noise can be neglected for highn. Also, the receiver outputs only one quadrature. So, one quadrature is lost.

For an optical amplifier with large G it holds F_pnf ≥ 2 whereas for an electrical amplifier it holds F_e ≥ 1.

Moreover, today's long-haul optical fiber communication is dominated by coherent optical I&Q receivers but F_pnf does not describe the SNR degradation observed in these.

The above conflicts are resolved by the optical in-phase and quadrature noise figure F_o,IQ.^[9] It can be measured using a coherent optical I&Q receiver. In these, power of the output signal is proportional to the square of an optical field amplitude because they are linear in amplitude. They pass both quadratures. For an optical amplifier it holds F_o,IQ = n_sp(1-1/G)+1/G ≥ 1. Quantity n_sp(1-1/G) is the input-referred number of added noise photons per mode.

F_o,IQ and F_pnf can easily be converted into each other. For large G it holds F_o,IQ = F_pnf/2 or, when expressed in dB, F_o,IQ is 3 dB less than F_pnf.

Unified noise figure

Total noise power spectral density per mode is kT + hf. In the electrical domain hf can be neglected. In the optical domain kT can be neglected. In between, say, in the low THz or thermal domain, both will need to be considered. It is possible to blend between electrical and optical domains such that a universal noise figure is obtained.

This has been attempted by a noise figure F_fas^[10] where the subscript stands for fluctuations of amplitude squares. At optical frequencies F_fas equals F_pnf and involves detection of only 1 quadrature. But the conceptual difference to F_e cannot be overcome: It seems impossible that for increasing frequency (from electrical to thermal to optical) 2 quadratures (in the electrical domain) gradually become 1 quadrature (in optical receivers which determine F_fas or F_pnf). The ideal noise factor would need to go from 1 (electrical) to 2 (optical), which is not intuitive. For unification of F_pnf with F_e, squares of signal amplitudes (powers in the electrical domain) must also gradually become 4th powers of amplitudes (powers in optical direct detection receivers), which seems impossible.

A consistent unification of optical and electrical noise figures is obtained for F_e and F_o,IQ. There are no contradictions because both these are in conceptual match (powers proportional to squares of amplitudes, linear, 2 quadratures, ideal noise factor equal to 1). Thermal noise kT and fundamental quantum noise hf are taken into account. The unified noise figure is F_IQ = (kTF_e + hfF_o,IQ) / (kT + hf) = (kT(T + T_e)) + hf(n_sp(1-1/G)+1/G)) / (kT + hf).^[9]

See also

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

References

1. "Noise temperature, Noise Figure and Noise Factor".
2. Agilent 2010, p. 7 harvnb error: no target: CITEREFAgilent2010 (help)
3. Agilent 2010, p. 5 harvnb error: no target: CITEREFAgilent2010 (help).
4. Agilent 2010, p. 7 harvnb error: no target: CITEREFAgilent2010 (help) with some rearrangement from T_e = T₀(F − 1).
5. Agilent 2010, p. 8 harvnb error: no target: CITEREFAgilent2010 (help).
6. Aspen Core. Derivation of noise figure equations (DOCX), pp. 3–4
7. E. Desurvire, „Erbium doped fiber amplifiers: Principles and Applications“, Wiley, New York, 1994
8. H. A. Haus, "The noise figure of optical amplifiers," in IEEE Photonics Technology Letters, vol. 10, no. 11, pp. 1602-1604, Nov. 1998, doi: 10.1109/68.726763
9. R. Noe, "Consistent Optical and Electrical Noise Figure," in Journal of Lightwave Technology, 2022, doi: 10.1109/JLT.2022.3212936, https://ieeexplore.ieee.org/document/9915356
10. H. A. Haus, "Noise Figure Definition Valid From RF to Optical Frequencies," in IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000, pp. 240-247

• Keysight, Fundamentals of RF and Microwave Noise Figure Measurements (PDF), Application Note, 57-1, Published September 01, 2019., archived (PDF) from the original on 2022-10-09

External links

• Noise Figure Calculator 2- to 30-Stage Cascade
• Mobile phone noise figure

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