Friis formulas for noise

Amplifier chain with known power gain factors G_1,2,3 und noise factors F_1,2,3.

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

{{Equation box 1

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|equation = $F\_\backslash text\{total\}\; =\; F\_1\; +\; \backslash frac\{F\_2-1\}\{G\_1\}\; +\; \backslash frac\{F\_3-1\}\{G\_1\; G\_2\}\; +\; \backslash frac\{F\_4-1\}\{G\_1\; G\_2\; G\_3\}\; +\; \backslash cdots\; +\; \backslash frac\{F\_n\; -\; 1\}\{G\_1\; G\_2\; \backslash cdots\; G\_\{n-1\}\}$

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where $F\_i$ and $G\_i$ 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.

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

:$F\_\backslash mathrm\{receiver\}\; =\; F\_\backslash mathrm\{LNA\}\; +\; \backslash frac\{F\_\backslash mathrm\{rest\}-1\}\{G\_\backslash mathrm\{LNA\}\}$

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

:$\backslash mathrm\{NF\}\_\backslash mathrm\{receiver\}=10\; \backslash log(F\_\backslash mathrm\{receiver\})$

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

Chain of three amplifiers

A source outputs a signal of power $S\_i$ and noise of power $N\_i$. Therefore the SNR at the input of the receiver chain is $\backslash text\{SNR\}\_i\; =\; S\_i/N\_i$. The signal of power $S\_i$ gets amplified by all three amplifiers. Thus the signal power at the output of the third amplifier is $S\_o\; =\; S\_i\; \backslash cdot\; G\_1\; G\_2\; G\_3$. The noise power at the output of the amplifier chain consists of four parts:

• The amplified noise of the source ($N\_i\; \backslash cdot\; G\_1\; G\_2\; G\_3$)
• The output referred noise of the first amplifier $N\_\{a1\}$ amplified by the second and third amplifier ($N\_\{a1\}\; \backslash cdot\; G\_2\; G\_3$)
• The output referred noise of the second amplifier $N\_\{a2\}$ amplified by the third amplifier ($N\_\{a2\}\; \backslash cdot\; G\_3$)
• The output referred noise of the third amplifier $N\_\{a3\}$

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

:$N\_o\; =\; N\_i\; G\_1\; G\_2\; G\_3\; +\; N\_\{a1\}\; G\_2\; G\_3\; +\; N\_\{a2\}\; G\_3\; +\; N\_\{a3\}$

and the SNR at the output of the amplifier chain equals

:$\backslash text\{SNR\}\_o\; =\; \backslash frac\{S\_i\; G\_1\; G\_2\; G\_3\}\{N\_i\; G\_1\; G\_2\; G\_3\; +\; N\_\{a1\}\; G\_2\; G\_3\; +\; N\_\{a2\}\; G\_3\; +\; N\_\{a3\}\}$.

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

:$F\_\backslash text\{total\}\; =\; \backslash frac\{\backslash text\{SNR\}\_i\}\{\backslash text\{SNR\}\_o\}\; =\; \backslash frac\{\backslash frac\{S\_i\}\{N\_i\}\}\{\backslash frac\{S\_i\; G\_1\; G\_2\; G\_3\}\{N\_i\; G\_1\; G\_2\; G\_3\; +\; N\_\{a1\}\; G\_2\; G\_3\; +\; N\_\{a2\}\; G\_3\; +\; N\_\{a3\}\}\}\; =\; 1\; +\; \backslash frac\{N\_\{a1\}\}\{N\_i\; G\_1\}\; +\; \backslash frac\{N\_\{a2\}\}\{N\_i\; G\_1\; G\_2\}\; +\; \backslash frac\{N\_\{a3\}\}\{N\_i\; G\_1\; G\_2\; G\_3\}$

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

:$F\_\backslash text\{total\}\; =\; \backslash underbrace\{1\; +\; \backslash frac\{N\_\{a1\}\}\{N\_i\; G\_1\}\}\_\{=F\_1\}\; +\; \backslash underbrace\{\backslash frac\{N\_\{a2\}\}\{N\_i\; G\_1\; G\_2\}\}\_\{=\backslash frac\{F\_2-1\}\{G\_1\}\}\; +\; \backslash underbrace\{\backslash frac\{N\_\{a3\}\}\{N\_i\; G\_1\; G\_2\; G\_3\}\}\_\{=\backslash frac\{F\_3-1\}\{G\_1\; G\_2\}\}\; =\; F\_1\; +\; \backslash frac\{F\_2-1\}\{G\_1\}\; +\; \backslash frac\{F\_3-1\}\{G\_1\; G\_2\}$.

General derivation for a cascade of $n$ amplifiers:

The total noise figure is given as the relation of the signal-to-noise ratio at the cascade input $\backslash mathrm\{SNR\_i\}=\backslash frac\{S\_\backslash mathrm\{i\}\}\{N\_\backslash mathrm\{i\}\}$ to the signal-to-noise ratio at the cascade output $\backslash mathrm\{SNR\_o\}=\backslash frac\{S\_\backslash mathrm\{o\}\}\{N\_\backslash mathrm\{o\}\}$ as

$F\_\{\backslash mathrm\{total\}\}=\backslash frac\{\backslash mathrm\{SNR\_\{i\}\}\}\{\backslash mathrm\{SNR\_\{o\}\}\}=\backslash frac\{S\_\backslash mathrm\{i\}\}\{S\_\backslash mathrm\{o\}\}\backslash frac\{N\_\backslash mathrm\{o\}\}\{N\_\backslash mathrm\{i\}\}$.

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

$S\_\backslash mathrm\{o\}\; +\; N\_\backslash mathrm\{o\}\; =\; \backslash left(\; \backslash left(\; \backslash left(\; \backslash left(\; S\_\{\backslash mathrm\{i\}\}+N\_\{\backslash mathrm\{i\}\}\; \backslash right)\; G\_1\; +\; N\_\{\backslash mathrm\{a\},1\}\; \backslash right)\; G\_2\; +\; N\_\{\backslash mathrm\{a\},2\}\; \backslash right)\; G\_3\; +\; N\_\{\backslash mathrm\{a\},3\}\; \backslash right)\; G\_4\; +\; \backslash dots$

The output signal power thus rewrites as

$S\_\backslash mathrm\{o\}=S\_\backslash mathrm\{i\}\backslash prod\_\{k=1\}^\{n\}\; G\_k$

whereas the output noise power can be written as

$N\_\backslash mathrm\{o\}\; =\; N\_\backslash mathrm\{i\}\; \backslash prod\_\{k=1\}^\{n\}G\_k+\backslash sum\_\{k=1\}^\{n-1\}N\_\{\backslash mathrm\{a\},k\}\; \backslash prod\_\{l=k+1\}^\{n\}\{G\_l\}\; +\; N\_\{\backslash mathrm\{a\},n\}$

Substituting these results into the total noise figure leads to

$F\_\backslash mathrm\{total\}=\backslash frac\{N\_\backslash mathrm\{i\}\; \backslash prod\_\{k=1\}^\{n\}G\_k+\backslash sum\_\{k=1\}^\{n-1\}N\_\{\backslash mathrm\{a\},k\}\; \backslash prod\_\{l=k+1\}^\{n\}\{G\_l\}\; +\; N\_\{\backslash mathrm\{a\},n\}\}\{N\_\backslash mathrm\{i\}\backslash prod\_\{k=1\}^\{n\}\; G\_k\}\; =\; 1\; +\; \backslash sum\_\{k=1\}^\{n-1\}\; \backslash frac\{N\_\{\backslash mathrm\{a\},k\}\; \backslash prod\_\{l=k+1\}^\{n\}\{G\_l\}\}\{N\_\backslash mathrm\{i\}\backslash prod\_\{m=1\}^\{n\}\; G\_m\}+\; \backslash frac\{N\_\{\backslash mathrm\{a\},n\}\}\{N\_\backslash mathrm\{i\}\backslash prod\_\{k=1\}^\{n\}\; G\_k\}\; =\; 1\; +\; \backslash sum\_\{k=1\}^\{n-1\}\; \backslash frac\{N\_\{\backslash mathrm\{a\},k\}\}\{N\_\backslash mathrm\{i\}\backslash prod\_\{m=1\}^\{k\}\; G\_m\}+\; \backslash frac\{N\_\{\backslash mathrm\{a\},n\}\}\{N\_\backslash mathrm\{i\}\backslash prod\_\{k=1\}^\{n\}\; G\_k\}$

$=\; 1\; +\; \backslash frac\{N\_\{\backslash mathrm\{a\},1\}\}\{N\_\backslash mathrm\{i\}G\_1\}\; +\; \backslash sum\_\{k=2\}^\{n-1\}\; \backslash frac\{N\_\{\backslash mathrm\{a\},k\}\}\{N\_\backslash mathrm\{i\}\backslash prod\_\{m=1\}^\{k\}\; G\_m\}+\; \backslash frac\{N\_\{\backslash mathrm\{a\},n\}\}\{N\_\backslash mathrm\{i\}\backslash prod\_\{k=1\}^\{n\}\; G\_k\}$

Now, using $F\_k=1+\backslash frac\{N\_\{\backslash mathrm\{a\},k\}\}\{N\_\backslash mathrm\{i\}G\_k\}$

as the noise figure of the individual $k$

-th amplifier, one obtains

$F\_\backslash mathrm\{total\}\; =\; F\_1\; +\; \backslash sum\_\{k=2\}^\{n\}\; \backslash frac\{F\_k\; -1\}\{\backslash prod\_\{l=1\}^\{k-1\}\; G\_l\}$

$=\; F\_1\; +\; \backslash frac\{F\_2\; -\; 1\}\{G\_1\}\; +\; \backslash frac\{F\_3\; -\; 1\}\{G\_1\; G\_2\}\; +\; \backslash frac\{F\_4\; -\; 1\}\{G\_1\; G\_2\; G\_3\}\; +\; \backslash dots\; +\; \backslash frac\{F\_n\; -\; 1\}\{G\_1\; G\_2\; \backslash dots\; G\_\{n-1\}\}$

Friis's formula can be equivalently expressed in terms of noise temperature:

{{Equation box 1

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|equation = $T\_\backslash text\{eq\}\; =\; T\_1\; +\; \backslash frac\{T\_2\}\{G\_1\}\; +\; \backslash frac\{T\_3\}\{G\_1\; G\_2\}\; +\; \backslash cdots$

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• J.D. Kraus, Radio Astronomy, McGraw-Hill, 1966.

• RF Cafe [http://www.rfcafe.com/references/electrical/noise-figure.htm] Cascaded noise figure.
• Microwave Encyclopedia [http://www.microwaves101.com/encyclopedia/cascadeanalysis.cfm] {{Webarchive|url=https://web.archive.org/web/20130517182715/http://www.microwaves101.com/encyclopedia/cascadeanalysis.cfm |date=2013-05-17 }} Cascade analysis.
• Friis biography at IEEE [http://www.ieeeghn.org/wiki/index.php/Harald_T._Friis]

Category:Noise (electronics)

Category:Equations