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Noise Figure of resistor network

Posted By Krishna Sankar On April 13, 2012 @ 3:17 am In Analog | 2 Comments

The post on thermal noise [1] described the noise produced by resistor $R$ ohms over bandwidth $B$ at temperature $T$Kelvin. In this post, let us define the noise voltage at the input and output of a resistor network and further use it to define the Noise Figure of such a network.

From Section 2.3.2 of RF Microelectronics, Behzad Razavi [2], the noise figure is defined as

$NF = \frac{SNR_{in}}{SNR_{out}}$, where

$SNR_{in}$ is the input signal to noise ratio and

$SNR_{out}$ is the output signal to noise ratio.

Note : Some textbooks refer to this term as Noise Factor.

## Noise Figure of a resistor network

Consider a simple resistor circuit with $V_{in}$ as the input voltage, $R_s$ source resistance and $R_p$ as the parallel resistance (shown below) :

Figure : Resistor network (Reference : Figure 2.30(a) of  RF Microelectronics, Behzad Razavi [2])

The noise voltage due to the source resistance $R_s$ is

$V^2_{n,in} =4kTR_s$.

The Signal to Noise Ratio at the input is,

$\begin{array}{lll}SNR_{in} &= & \frac{V^2_{s,in}}{V^2_{n,in}}\\ & = & \frac{V^2_{s,in}}{4kTR_s}\end{array}$.

The signal voltage at the output is,

$V_{s,out} =V_{s,in}\frac{R_p}{(R_s+R_p)}$

The noise voltage at the output is,

Figure : Equivalent resistor noise model for two parallel resistor (Reference : Figure 2.30(b) of  RF Microelectronics, Behzad Razavi [2])

$V^2_{n,out} =4kT\frac{R_sR_p}{(R_s+R_p)}$.

The Signal to Noise Ratio at the output is,

$\begin{array}{lll}SNR_{out} &= & \frac{V^2_{s,out}}{V^2_{n,out}}\\ & = & V_{s,in}^2\frac{R_p^2}{(R_s+R_p)^2}\frac{(R_s+R_p)}{R_sR_p}\frac{1}{4kT}\\&=&V_{s,in}^2\frac{R_p}{R_s$$R_s+R_p$$}\frac{1}{4kT}\end{array}$.

Calculating,

$\begin{array}{lll}\frac{SNR_{in}}{SNR_{out}}&=&\frac{V_{s,in}^2}{4kTR_s}\frac{R_s$$R_s+R_p$$}{R_p}\frac{4kT}{V_{s,in}^2}\\&=&\frac{R_p+R_s}{R_p}\end{array}$.

The Noise Figure is,

$\Large{NF=1+\frac{R_s}{R_p}}$

Observations

a) Per the Maximum Power Transfer theorem [3] the source and load resistance should be equal i.e $R_s=R_p$. However that condition does not coincide with the minimum noise figure. The source and load resistance and equal results in $NF=2$ (3dB in decibels)

b) Noise Figure is minimized by maximizing resistance $R_p$ OR minimizing source resistance $R_s$.

## Reference

RF Microelectronics, Behzad Razavi [2]

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URL to article: http://www.dsplog.com/2012/04/13/noise-figure-resistor/

URLs in this post:

[1] thermal noise: http://www.dsplog.com/2012/03/25/thermal-noise-awgn/

[2] RF Microelectronics, Behzad Razavi: http://www.amazon.com/RF-Microelectronics-Behzad-Razavi/dp/0138875715/?_encoding=UTF8&tag=dl04-20&linkCode=ur2&camp=1789&creative=9325

[3] Maximum Power Transfer theorem: http://en.wikipedia.org/wiki/Maximum_power_transfer_theorem

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