Friis formula

The Friis formula is used in communications engineering to calculate the noise figure of a chain of amplifiers or attenuators , which are composed of stages, of which the individual noise figures F _{1,2, ...} and the gain factors G _{1,2, ... are} known are. The formula is named after Harald Friis , who coined the term in 1944.

definition

Amplifier chain with known amplification factors G _1,2,3 and noise figures F _{1,2,3 of} the individual stages.

The Friis formula for determining the noise figure of an amplifier chain with n elements is given by:

${\displaystyle F_{\mathrm {gesamt} }=1+(F_{1}-1)+{\frac {F_{2}-1}{G_{1}}}+{\frac {F_{3}-1}{G_{1}G_{2}}}+{\frac {F_{4}-1}{G_{1}G_{2}G_{3}}}+\cdots +{\frac {F_{n}-1}{G_{1}G_{2}G_{3}\cdots G_{n-1}}}}$

with F _n and G _n as the noise figure and gain of the nth stage. Attenuators, such as cables , have a gain G that is less than one. The gain factors G are also referred to as gain in this context.

The Friis formula expresses that the noise of the subsequent amplifier stages reduced by the gain G of the preceding amplifier stages is included in the total noise figure of the chain. Thus, with identical noise figures of the amplifier stages, the amplifier that has a higher gain and is positioned at the beginning of the chain as far as possible will achieve better noise properties over the chain.

If you consider a low-noise preamplifier (LNA) as the first stage and use F _{rest to} indicate the noise figure of the following stages, then the following applies to the noise figure of the entire chain:

${\displaystyle F_{\mathrm {gesamt} }=F_{LNA}+{\frac {F_{rest}-1}{G_{LNA}}}}$

According to this formula, a low-noise preamplifier can reduce the noise figure of an amplifier chain, provided the gain is sufficiently high.

Friis formula for the intoxication temperature

The Friis formula can also be expressed in terms of the intoxication temperature :

${\displaystyle T_{\mathrm {gesamt} }=T_{1}+{\frac {T_{2}}{G_{1}}}+{\frac {T_{3}}{G_{1}\cdot G_{2}}}+...}$

literature

• Rudolf Müller: Rauschen (=  semiconductor electronics . Volume 15 ). 2nd, revised and expanded edition. Springer, Berlin a. a. 1990, ISBN 3-540-51145-8 .
• Peter Adam Höher: Basics of digital information transfer . From theory to mobile radio applications, 2nd edition, Springer Fachmedien, Wiesbaden 2013, ISBN 978-3-8348-1784-6 .
• H. Meinke, FW Gundlach: Pocket book of high frequency technology. Volume 3: Systems. 5th edition. Springer Verlag, Berlin / Heidelberg 1992, ISBN 3-540-54716-9 .
• Frank Gustrau: High frequency technology: Basics of mobile communication technology. 2nd Edition. Carl Hanser Verlag, Munich 2013, ISBN 978-3-446-43245-1 .

Individual evidence

1. ^ HT Friis : Noise Figures of Radio Receivers. In: Proceedings of the IRE. Volume 32, 1944, ISSN  0096-8390 , pp. 419-422, fi.uba.ar (PDF; 612 kB)

Web links

• Derivation of the Friis formula for derailleur gears (accessed on October 12, 2017)
• Construction of a low-noise high-frequency amplifier (accessed October 12, 2017)