Friis formulas for noise

The terms the Friis formula and Friis's formula (sometimes Friis' formula), named after Harald T. Friis, can refer to either of two formulas used in telecommunications engineering. One relates to noise figure, and the other relates to noise temperature.

The Friis formula for noise figure

Friis's formula is used to calculate the total noise figure of a cascade of stages, each with its own noise factor and gain. It is given as

:$F\_\{total\}\; =\; F\_1\; +\; frac\{F\_2-1\}\{G\_1\}\; +\; frac\{F\_3-1\}\{G\_1\; G\_2\}\; +\; frac\{F\_4-1\}\{G\_1\; G\_2\; G\_3\}\; +\; ...$

where $F\_n$ and $G\_n$ are the noise factor and available power gain, respectively, of the n-th stage. Note that both magnitudes are expressed as ratios, not in decibels.

As an example, consider the cascaded system to be a receiver with the first stage being a low-noise amplifier (LNA). The overall receiver noise figure is then

:$F\_\{receiver\}\; =\; F\_\{LNA\}\; +\; frac\{(F\_\{rest\}-1)\}\{G\_\{LNA$

where $F\_\{rest\}$ is the overall noise factor of the subsequent stages. According to the equation, the overall noise figure, $F\_\{receiver\}$, is dominated by the noise figure of the LNA, $F\_\{LNA\}$, if the gain is sufficiently high.

The Friis formula for noise temperature

Friis's formula can be equivalently expressed in terms of noise temperature:

:$T\_\{total\}\; =\; T\_1\; +\; frac\{T\_2\}\{G\_1\}\; +\; frac\{T\_3\}\{G\_1\; G\_2\}\; +\; ...$

Printed references

*J.D.Kraus, Radio Astronomy, McGraw-Hill, 1966.

Online references

*RF Cafe [http://www.rfcafe.com/references/electrical/noise_figure.htm]
*Microwave Encyclopedia [http://www.microwaves101.com/encyclopedia/cascadeanalysis.cfm]
*Friis biography at IEEE [http://www.ieee.org/web/aboutus/history_center/biography/friis.html]