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 unitless 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 T<sub>0</sub> (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 figure is the power level of the noise factor, calculated in terms of the logarithm of noise factor and expressed in scale of decibels (dB).
General
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 T<sub>0</sub> (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. 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 of a system is defined as
{\mathrm{SNR}_\text{o</math>
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where and are the input and output signal-to-noise ratios respectively. The quantities are unitless power ratios. Note that this specific definition is only valid for an input signal of which the noise is N<sub>i</sub>=kT<sub>0</sub>B.
The noise figure is defined as the noise factor in units of decibels (dB):
{\mathrm{SNR}_\text{o\right) = \mathrm{SNR}_\text{i, dB} - \mathrm{SNR}_\text{o, dB}</math>
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where and are in units of (dB).
These formulae are only valid when the input termination is at standard noise temperature , 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 :
:<math>F = 1 + \frac{T_\text{e{T_0}.</math>
Attenuators have a noise factor equal to their attenuation ratio when their physical temperature equals . More generally, for an attenuator at a physical temperature , the noise temperature is , giving a noise factor
:<math>F = 1 + \frac{(L - 1)T}{T_0}.</math>
Noise factor of cascaded devices
If several devices are cascaded, the total noise factor can be found with Friis' formula:
:<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 is the noise factor for the -th device, and is the power gain (linear, not in dB) of the -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.<!-- yes, the input might be an attenuator or a mixer, so the second stage becomes critical. -->
Noise factor as a function of additional noise
right|thumb|600px|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.
Derivation
From the definition of noise factor
:<math>F = \frac{\frac{S_i}{N_i{\frac{S_iG}{N_a+N_iG=\frac{N_a+N_iG}{N_iG} = 1 + \frac{N_a}{N_iG}</math>
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 figure
The above describes noise in electrical systems. The optical noise figure is discussed in multiple sources. Electric sources generate noise with a power spectral density, or energy per mode, equal to , where is the Boltzmann constant and is the absolute temperature. One mode has two quadratures, i.e. the amplitudes of <math>\mathrm{\omega}t</math> and <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 where is the Planck constant and 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 .
In the 1990s, an optical noise figure has been defined.