Noise figure

Number used to evaluate the performance of an amplifier

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, which is equivalent to 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 logarithm of the noise factor, expressed in units of decibels (dB).[1]

## General

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The noise figure is the difference in decibel (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

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

F = S N R i S N R o {\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 unitless power ratios. Note that this specific definition is only valid for an input signal of which the noise is Ni=kT0B.

The noise figure NF is defined as the noise factor in units of decibels (dB):

N F = 10 log 10 ⁡ ( F ) = 10 log 10 ⁡ ( S N R i S N R o ) = S N R i, dB − S N R o, 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 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 Te:[4]

F = 1 + T e T 0 . {\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

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

## Noise factor of cascaded devices

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Main article: Friis formulas for noise

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

F = F 1 + F 2 − 1 G 1 + F 3 − 1 G 1 G 2 + F 4 − 1 G 1 G 2 G 3 + ⋯ + F n − 1 G 1 G 2 G 3 ⋯ G n − 1 , {\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

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The source outputs a signal of power  S i {\displaystyle S_{i}} and noise of power  N i {\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  N a {\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  N a {\displaystyle N_{a}} and the power gain  G {\displaystyle G} of an amplifier.

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

### Derivation

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From the definition of noise factor[3]

F = S N R i S N R o = S i N i S o N o , {\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  N a {\displaystyle N_{a}} , the amplified signal  S i G {\displaystyle S_{i}G} and the amplified input noise  N i G {\displaystyle N_{i}G} ,

S o N o = S i G N a + 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]

F = S i N i S i G N a + N i G = N a + N i G N i G = 1 + N a N i G {\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  N i {\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

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The above describes noise in electrical systems. The optical noise figure is discussed in multiple sources.[7][8][9][10][11] Electric sources generate noise with a power spectral density, or energy per mode, equal to kT, where k is the Boltzmann constant and T is the absolute temperature. One mode has two quadratures, i.e. the amplitudes of cos ω t {\displaystyle \mathrm {\omega } t} and sin ω t {\displaystyle \mathrm {\omega } t} oscillations of voltages, currents or fields. 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. In an optical receiver which can output one available mode or two available quadratures this corresponds to a noise power spectral density, or energy per mode, of hf where h is the Planck constant and f is the optical frequency. In an optical receiver with only one available quadrature the shot noise has a power spectral density, or energy per mode, of only hf/2.

In the 1990s, an optical noise figure has been defined.[7] This has been called Fpnf 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 SNRpnf,in = n2/n = n. Behind an optical amplifier with power gain G there will be a mean of Gn detectable signal photons. In the limit of large n the variance of photons is Gn(2nsp(G-1)+1) where nsp is the spontaneous emission factor. One obtains SNRpnf,out = G2n2/(Gn(2nsp(G-1)+1)) = n/(2nsp(1-1/G)+1/G). Resulting optical noise factor is Fpnf = SNRpnf,in / SNRpnf,out = 2nsp(1-1/G)+1/G.

Fpnf is in conceptual conflict[9][10] with the electrical noise factor, which is now called Fe:

Photocurrent I is proportional to optical power P. P is proportional to squares of a field amplitude (electric or magnetic). So, the receiver is nonlinear in amplitude. The "Power" needed for SNRpnf calculation is proportional to the 4th power of the signal amplitude. But for Fe in the electrical domain the power is proportional to the square of the signal amplitude.

If SNRpnf is a noise factor then its definition must be independent of measurement apparatus and frequency. Consider the signal "Power" in the sense of SNRpnf definition. Behind an amplifier it is proportional to G2n2. We may replace the photodiode by a thermal power meter, and measured photocurrent I by measured temperature change  Δ θ {\displaystyle \mathrm {\Delta \theta } } . "Power", being proportional to I2 or P2, is also proportional to  ( Δ θ ) {\displaystyle (\mathrm {\Delta \theta } )} 2. Thermal power meters can be built at all frequencies. Hence it is possible to lower the frequency from optical (say 200 THz) to electrical (say 200 MHz). Still there, "Power" must be proportional to  ( Δ θ ) {\displaystyle (\mathrm {\Delta \theta } )} 2 or P2. Electrical power P is proportional to the square U2 of voltage U. But "Power" is proportional to U4.

These implications are in obvious conflict with ~150 years of physics. They are compelling consequence of calling Fpnf a noise factor, or noise figure when expressed in dB.

At any given electrical frequency, noise occurs in both quadratures, i.e. 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 SNRpnf takes mainly the in-phase noise into account whereas quadrature noise can be neglected for high n. Also, the receiver outputs only one baseband signal, corresponding to quadrature. So, one quadrature or degree-of-freedom is lost.

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

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

Another optical noise figure Fase for amplified spontaneous emission has been defined.[8] But the noise factor Fase is not the SNR degradation factor in any optical receiver.

All the above conflicts are resolved by the optical in-phase and quadrature noise factor and figure Fo,IQ.[9][10] 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 Fo,IQ = nsp(1-1/G)+1/G ≥ 1. Quantity nsp(1-1/G) is the input-referred number of added noise photons per mode.

Fo,IQ and Fpnf can easily be converted into each other. For large G it holds Fo,IQ = Fpnf/2 or, when expressed in dB, Fo,IQ is 3 dB less than Fpnf. The ideal Fo,IQ in dB equals 0 dB. This describes the known fact that the sensitivity of an ideal optical I&Q receiver is not improved by an ideal optical preamplifier.

## See also

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

## References

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1. ^ "Noise temperature, Noise Figure and noise factor".
2. ^ Keysight (2019), p. 9.
3. ^ a b Keysight (2019), p. 6, Eqn. (1-1).
4. ^ Keysight (2019), p. 8, Eqn. (1-5), with some rearrangement from Te = T0(F − 1).
5. ^ Keysight (2019), p. 10, Eqn. (2-3).
6. ^ Aspen Core. Derivation of noise figure equations (DOCX), pp. 3–4
7. ^ a b E. Desurvire, Erbium doped fiber amplifiers: Principles and Applications, Wiley, New York, 1994
8. ^ a b c 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. ^ a b c 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. ^ a b c R. Noe, "Noise Figure and Homodyne Noise Figure" Photonic Networks; 24th ITG-Symposium, Leipzig, Germany, 09-10 May 2023, pp. 85-91, https://ieeexplore.ieee.org/abstract/document/10173081, presentation https://www.vde.com/resource/blob/2264664/dc0e3c85c8e0cb386cbfa215fe499c4c/noise-figure-and-homodyne-noise-figure-data.pdf
11. ^ 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 Technologies (2019-10-01), Fundamentals of RF and Microwave Noise Figure Measurements (PDF) (Application Note), 5952-8255E, archived from the original (PDF) on 2022-05-21.

## External links

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

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