as quotient of the input and output SNR: Using the definitions of the noise factors of the amplifiers we get the final result: . General derivation for a cascade of amplifiers: The total noise figure is given as the relation of the signal-to-noise ratio at the cascade input to the signal-to-noise ratio at the cascade output as . The total input power of the -th amplifier in the cascade (noise and signal) is . It is amplified according to the amplifier's power gain . Additionally, the amplifier adds noise with power . Thus the output power of the -th amplifier is . For the entire cascade, one obtains the total output power The output signal power thus rewrites as whereas the output noise power can be written as Substituting these results into the total noise figure leads to Now, using as the noise figure of the individual -th amplifier, one obtains The Friis formula for noise temperature Friis's formula can be equivalently expressed in terms of noise temperature: Published references J.D. Kraus, Radio Astronomy, McGraw-Hill, 1966. Online references RF Cafe Cascaded noise figure. Microwave Encyclopedia Cascade analysis. Friis biography at IEEE Noise (electronics) Equations