Friis formulas for noise

An important consequence of this formula is that the overall noise figure of a radio receiver is primarily established by the noise figure of its first amplifying stage. Subsequent stages have a diminishing effect on signal-to-noise ratio. For this reason, the first stage amplifier in a receiver is often called the low-noise amplifier (LNA). The overall receiver noise "factor" is then

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

where ${\displaystyle F_{\mathrm {rest} }}$ is the overall noise factor of the subsequent stages. According to the equation, the overall noise factor, ${\displaystyle F_{\mathrm {receiver} }}$, is dominated by the noise factor of the LNA, ${\displaystyle F_{\mathrm {LNA} }}$, if the gain is sufficiently high. The resultant Noise Figure expressed in dB is:

${\displaystyle \mathrm {NF} _{\mathrm {receiver} }=10\log(F_{\mathrm {receiver} })}$

For a derivation of Friis' formula for the case of three cascaded amplifiers (${\displaystyle n=3}$) consider the image below.

A source outputs a signal of power ${\displaystyle S_{i}}$ and noise of power ${\displaystyle N_{i}}$. Therefore the SNR at the input of the receiver chain is ${\displaystyle {\text{SNR}}_{i}=S_{i}/N_{i}}$. The signal of power ${\displaystyle S_{i}}$ gets amplified by all three amplifiers. Thus the signal power at the output of the third amplifier is ${\displaystyle S_{o}=S_{i}\cdot G_{1}G_{2}G_{3}}$. The noise power at the output of the amplifier chain consists of four parts:

• The amplified noise of the source (${\displaystyle N_{i}\cdot G_{1}G_{2}G_{3}}$)
• The output referred noise of the first amplifier ${\displaystyle N_{a1}}$ amplified by the second and third amplifier (${\displaystyle N_{a1}\cdot G_{2}G_{3}}$)
• The output referred noise of the second amplifier ${\displaystyle N_{a2}}$ amplified by the third amplifier (${\displaystyle N_{a2}\cdot G_{3}}$)
• The output referred noise of the third amplifier ${\displaystyle N_{a3}}$

Therefore the total noise power at the output of the amplifier chain equals

${\displaystyle N_{o}=N_{i}G_{1}G_{2}G_{3}+N_{a1}G_{2}G_{3}+N_{a2}G_{3}+N_{a3}}$

and the SNR at the output of the amplifier chain equals

${\displaystyle {\text{SNR}}_{o}={\frac {S_{i}G_{1}G_{2}G_{3}}{N_{i}G_{1}G_{2}G_{3}+N_{a1}G_{2}G_{3}+N_{a2}G_{3}+N_{a3}}}}$.

The total noise factor may now be calculated as quotient of the input and output SNR:

${\displaystyle F_{\text{total}}={\frac {{\text{SNR}}_{i}}{{\text{SNR}}_{o}}}={\frac {\frac {S_{i}}{N_{i}}}{\frac {S_{i}G_{1}G_{2}G_{3}}{N_{i}G_{1}G_{2}G_{3}+N_{a1}G_{2}G_{3}+N_{a2}G_{3}+N_{a3}}}}=1+{\frac {N_{a1}}{N_{i}G_{1}}}+{\frac {N_{a2}}{N_{i}G_{1}G_{2}}}+{\frac {N_{a3}}{N_{i}G_{1}G_{2}G_{3}}}}$

Using the definitions of the noise factors of the amplifiers we get the final result:

${\displaystyle F_{\text{total}}=\underbrace {1+{\frac {N_{a1}}{N_{i}G_{1}}}} _{=F_{1}}+\underbrace {\frac {N_{a2}}{N_{i}G_{1}G_{2}}} _{={\frac {F_{2}-1}{G_{1}}}}+\underbrace {\frac {N_{a3}}{N_{i}G_{1}G_{2}G_{3}}} _{={\frac {F_{3}-1}{G_{1}G_{2}}}}=F_{1}+{\frac {F_{2}-1}{G_{1}}}+{\frac {F_{3}-1}{G_{1}G_{2}}}}$.