Derivation
For a derivation of Friis' formula for the case of three cascaded amplifiers () consider the image below.
A source outputs a signal of power and noise of power . Therefore the SNR at the input of the receiver chain is . The signal of power gets amplified by all three amplifiers. Thus the signal power at the output of the third amplifier is . The noise power at the output of the amplifier chain consists of four parts:
- The amplified noise of the source ()
- The output referred noise of the first amplifier amplified by the second and third amplifier ()
- The output referred noise of the second amplifier amplified by the third amplifier ()
- The output referred noise of the third amplifier
Therefore the total noise power at the output of the amplifier chain equals
and the SNR at the output of the amplifier chain equals
- .
The total noise factor may now be calculated 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