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:

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

General derivation for a cascade of ${\displaystyle n}$ amplifiers:

The total noise figure is given as the relation of the signal-to-noise ratio at the cascade input ${\displaystyle \mathrm {SNR_{i}} ={\frac {S_{\mathrm {i} }}{N_{\mathrm {i} }}}}$ to the signal-to-noise ratio at the cascade output ${\displaystyle \mathrm {SNR_{o}} ={\frac {S_{\mathrm {o} }}{N_{\mathrm {o} }}}}$ as

${\displaystyle F_{\mathrm {total} }={\frac {\mathrm {SNR_{i}} }{\mathrm {SNR_{o}} }}={\frac {S_{\mathrm {i} }}{S_{\mathrm {o} }}}{\frac {N_{\mathrm {o} }}{N_{\mathrm {i} }}}}$.

The total input power of the ${\displaystyle k}$-th amplifier in the cascade (noise and signal) is ${\displaystyle S_{k-1}+N_{k-1}}$. It is amplified according to the amplifier's power gain ${\displaystyle G_{k}}$. Additionally, the amplifier adds noise with power ${\displaystyle N_{\mathrm {a} ,k}}$. Thus the output power of the ${\displaystyle k}$-th amplifier is ${\displaystyle G_{k}\left(S_{k-1}+N_{k-1}\right)+N_{\mathrm {a} ,k}}$. For the entire cascade, one obtains the total output power

${\displaystyle S_{\mathrm {o} }+N_{\mathrm {o} }=\left(\left(\left(\left(S_{\mathrm {i} }+N_{\mathrm {i} }\right)G_{1}+N_{\mathrm {a} ,1}\right)G_{2}+N_{\mathrm {a} ,2}\right)G_{3}+N_{\mathrm {a} ,3}\right)G_{4}+\dots }$

The output signal power thus rewrites as

${\displaystyle S_{\mathrm {o} }=S_{\mathrm {i} }\prod _{k=1}^{n}G_{k}}$

whereas the output noise power can be written as

${\displaystyle N_{\mathrm {o} }=N_{\mathrm {i} }\prod _{k=1}^{n}G_{k}+\sum _{k=1}^{n-1}N_{\mathrm {a} ,k}\prod _{l=k+1}^{n}{G_{l}}+N_{\mathrm {a} ,n}}$

Substituting these results into the total noise figure leads to

${\displaystyle F_{\mathrm {total} }={\frac {N_{\mathrm {i} }\prod _{k=1}^{n}G_{k}+\sum _{k=1}^{n-1}N_{\mathrm {a} ,k}\prod _{l=k+1}^{n}{G_{l}}+N_{\mathrm {a} ,n}}{N_{\mathrm {i} }\prod _{k=1}^{n}G_{k}}}=1+\sum _{k=1}^{n-1}{\frac {N_{\mathrm {a} ,k}\prod _{l=k+1}^{n}{G_{l}}}{N_{\mathrm {i} }\prod _{m=1}^{n}G_{m}}}+{\frac {N_{\mathrm {a} ,n}}{N_{\mathrm {i} }\prod _{k=1}^{n}G_{k}}}=1+\sum _{k=1}^{n-1}{\frac {N_{\mathrm {a} ,k}}{N_{\mathrm {i} }\prod _{m=1}^{k}G_{m}}}+{\frac {N_{\mathrm {a} ,n}}{N_{\mathrm {i} }\prod _{k=1}^{n}G_{k}}}}$

${\displaystyle =1+{\frac {N_{\mathrm {a} ,1}}{N_{\mathrm {i} }G_{1}}}+\sum _{k=2}^{n-1}{\frac {N_{\mathrm {a} ,k}}{N_{\mathrm {i} }\prod _{m=1}^{k}G_{m}}}+{\frac {N_{\mathrm {a} ,n}}{N_{\mathrm {i} }\prod _{k=1}^{n}G_{k}}}}$

Now, using ${\displaystyle F_{k}=1+{\frac {N_{\mathrm {a} ,k}}{N_{\mathrm {i} }G_{k}}}}$ as the noise figure of the individual ${\displaystyle k}$-th amplifier, one obtains

${\displaystyle F_{\mathrm {total} }=F_{1}+\sum _{k=2}^{n-1}{\frac {F_{k}-1}{\prod _{l=1}^{k-1}G_{l}}}+{\frac {F_{n}-1}{\prod _{k=1}^{n-1}G_{k}}}}$

${\displaystyle =F_{1}+{\frac {F_{2}-1}{G_{1}}}+{\frac {F_{3}-1}{G_{1}G_{2}}}+{\frac {F_{4}-1}{G_{1}G_{2}G_{3}}}+\dots +{\frac {F_{n}-1}{G_{1}G_{2}\dots G_{n-1}}}}$