Noise figure

Template:Short description 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).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

GeneralEdit

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.Template:Sfnp 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.

DefinitionEdit

The noise factor Template:Math of a system is defined as<ref name="NF-def">Template:Harvp.</ref> Template:Equation box 1{\mathrm{SNR}_\text{o}}</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} where Template:Math and Template:Math are the input and output signal-to-noise ratios respectively. The Template:Math quantities are unitless power ratios. Note that this specific definition is only valid for an input signal of which the noise is N_i=kT₀B.

The noise figure Template:Math is defined as the noise factor in units of decibels (dB): Template:Equation box 1{\mathrm{SNR}_\text{o}}\right) = \mathrm{SNR}_\text{i, dB} - \mathrm{SNR}_\text{o, dB}</math> |cellpadding= 6 |border |border colour = #0073CF |background colour=#F5FFFA}} where Template:Math and Template:Math are in units of (dB). These formulae are only valid when the input termination is at standard noise temperature Template:Math, 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 Template:Math:<ref>Template:Harvp, with some rearrangement from Template:Math.</ref>

Attenuators have a noise factor Template:Math equal to their attenuation ratio Template:Math when their physical temperature equals Template:Math. More generally, for an attenuator at a physical temperature Template:Math, the noise temperature is Template:Math, giving a noise factor

Noise factor of cascaded devicesEdit

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} If several devices are cascaded, the total noise factor can be found with Friis' formula:Template:Sfnp

<math>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}},</math>

where Template:Math is the noise factor for the Template:Math-th device, and Template:Math is the power gain (linear, not in dB) of the Template:Math-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 noiseEdit

File:NoiseFactorDefinition.svg

The source outputs a signal of power <math>S_i</math> and noise of power <math>N_i</math>. 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 <math>N_a</math>. 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 <math>N_a</math> and the power gain <math>G</math> of an amplifier. Template:Equation box 1

DerivationEdit

From the definition of noise factor<ref name="NF-def" />

<math>F = \frac{\mathrm{SNR}_\text{i}}{\mathrm{SNR}_\text{o}}=\frac{\frac{S_i}{N_i}}{\frac{S_o}{N_o}},</math>

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 <math>N_a</math>, the amplified signal <math>S_iG</math> and the amplified input noise <math>N_iG</math>,

Substituting the output SNR to the noise factor definition,<ref>Aspen Core. Derivation of noise figure equations (DOCX), pp. 3–4</ref>

In cascaded systems <math>N_i</math> 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 figureEdit

The above describes noise in electrical systems. The optical noise figure is discussed in multiple sources.<ref name="Desurvire1994">E. Desurvire, Erbium doped fiber amplifiers: Principles and Applications, Wiley, New York, 1994</ref><ref name="Haus1998">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</ref><ref name="Noe2022">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</ref><ref name="NoeNF2023">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</ref><ref name="Haus2000">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</ref> Electric sources generate noise with a power spectral density, or energy per mode, equal to Template:Math, where Template:Math is the Boltzmann constant and Template:Math is the absolute temperature. One mode has two quadratures, i.e. the amplitudes of Template:Math<math>\mathrm{\omega}t</math> and Template:Math<math>\mathrm{\omega}t</math> 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 Template:Math where Template:Math is the Planck constant and Template:Math 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 Template:Math.

In the 1990s, an optical noise figure has been defined.<ref name="Desurvire1994"/> This has been called Template:Math for photon number fluctuations.<ref name="Haus1998" /> 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 Template:Math then the variance is also Template:Math and one obtains Template:Math = Template:Math = Template:Math. Behind an optical amplifier with power gain Template:Math there will be a mean of Template:Math detectable signal photons. In the limit of large Template:Math the variance of photons is Template:Math where Template:Math is the spontaneous emission factor. One obtains Template:Math = Template:Math = Template:Math. Resulting optical noise factor is Template:Math = Template:Math = Template:Math.

Template:Math is in conceptual conflict<ref name="Noe2022" /><ref name="NoeNF2023" /> with the electrical noise factor, which is now called Template:Math:

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

If Template:Math is a noise factor then its definition must be independent of measurement apparatus and frequency. Consider the signal "Power" in the sense of Template:Math definition. Behind an amplifier it is proportional to Template:Math. We may replace the photodiode by a thermal power meter, and measured photocurrent Template:Math by measured temperature change <math>\mathrm{\Delta\theta}</math>. "Power", being proportional to Template:Math or Template:Math, is also proportional to <math>(\mathrm{\Delta\theta})</math>Template:Math. 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 <math>(\mathrm{\Delta\theta})</math>Template:Math or Template:Math. Electrical power Template:Math is proportional to the square Template:Math of voltage Template:Math. But "Power" is proportional to Template:Math.

These implications are in obvious conflict with ~150 years of physics. They are compelling consequence of calling Template:Math 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 Template:Math takes mainly the in-phase noise into account whereas quadrature noise can be neglected for high Template:Math. 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 Template:Math it holds Template:Math ≥ 2 whereas for an electrical amplifier it holds Template:Math ≥ 1.

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

Another optical noise figure Template:Math for amplified spontaneous emission has been defined.<ref name="Haus1998" /> But the noise factor Template:Math 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 Template:Math.<ref name="Noe2022" /><ref name="NoeNF2023" /> 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 Template:Math = Template:Math ≥ 1. Quantity Template:Math is the input-referred number of added noise photons per mode.

Template:Math and Template:Math can easily be converted into each other. For large Template:Math it holds Template:Math = Template:Math or, when expressed in dB, Template:Math is 3 dB less than Template:Math. The ideal Template:Math 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.