Friis formulas for noise

Friis formula or Friis's formula (sometimes Friis' formula), named after Danish-American electrical engineer Harald T. Friis, is either of two formulas used in telecommunications engineering to calculate the signal-to-noise ratio of a multistage amplifier. One relates to noise factor while the other relates to noise temperature.
The Friis formula for noise factor.
Friis's formula is used to calculate the total noise factor of a cascade of stages, each with its own noise factor and power gain (assuming that the impedances are matched at each stage). The total noise factor can then be used to calculate the total noise figure. The total noise factor is given as
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where formula_1 and formula_2 are the noise factor and available power gain, respectively, of the "i"-th stage, and "n" is the number of stages. Both magnitudes are expressed as ratios, not in decibels. 
Consequences.
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
where formula_4 is the overall noise factor of the subsequent stages. According to the equation, the overall noise factor, formula_5, is dominated by the noise factor of the LNA, formula_6, if the gain is sufficiently high. The resultant Noise Figure expressed in dB is:
Derivation.
For a derivation of Friis' formula for the case of three cascaded amplifiers (formula_8) consider the image below.
A source outputs a signal of power formula_9 and noise of power formula_10. Therefore the SNR at the input of the receiver chain is formula_11. The signal of power formula_9 gets amplified by all three amplifiers. Thus the signal power at the output of the third amplifier is formula_13. The noise power at the output of the amplifier chain consists of four parts:
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:
The Friis formula for noise temperature.
Friis's formula can be equivalently expressed in terms of noise temperature: