Understanding Noise – Part 4: Noise Factor, Noise Figure & Noise Temperature

# Understanding Noise – Part 4: Noise Factor, Noise Figure & Noise Temperature

Previous posts in this series:

Digital communication systems require received signals to be filtered and amplified before they can be demodulated and passed to the analog to digital converter.  Similarly, transmitted signals must also be passed to an analog amplifier and filter before being transmitted.  These components insert additional noise into the transmitted/received signals and can have a negative impact on the performance/reliability of a communications system.

Several key metrics allow us to characterize the noise performance of these components including Noise Factor, Noise Figure and Noise Temperature.

# Noise Factor

Noise Factor is the ratio between the measured noise power at the output of a system and the noise power at the input.

$F_n = \frac{P_{out}}{P_{in}}$

This is equivalent to looking at the ratio of  the SNR of the signal entering a component/system and the SNR of the signal output:

$F_n = \frac{SNR_{in}}{SNR_{out}}$

If we are looking at a system that amplifies the signal with a certain gain G, then we know that the amplifier will amplify the input noise as well as add additional noise.  This is modeled in the diagram below:

An ideal amplifier that adds no additional noise will have a noise factor of 1, and will only amplify the input noise by the gain factor G:

$P_{no} = P_{ni}.G + P_{na}$

$F_n = \frac{P_{no}}{P_{ni}.G}$

If we assume that the input noise Pni is purely thermal noise, then we define Pni to be:

$P_{ni} = kT_oB$

where To is the standard operating temperature, 290° Kelvin.  It is important to note, when calculating Noise Factor, we always use To = 290° K for Pni. The calculation for Noise Factor of a system is thus:

$F_n = \frac{P_{no}}{k.T_o.B.G}$

# Noise Figure

Noise Figure is simply the logarithmic scale equivalent of Noise Factor, expressed in decibels (dB).

$N_F = 10Log(F_n)$

We can also relate Noise Figure to values of SNR:

$N_F = 10Log(\frac{SNR_{in}}{SNR_{out}})$

$N_{F (dB)} = SNR_{in (dB)} - SNR_{out (dB)}$

# Noise Temperature

Noise Temperature gives us another way to describe how much noise a system adds to a signal.  In this case, we look at the total noise performance of the system, and calculate an equivalent temperature (Te) that would yield the same noise power at the output via additional thermal noise.  It is important to realize that the noise temperature of a component describes the additional noise that the component inserts onto a signal before it is amplified, as shown in the figure below.

## Why Noise Temperature?

This is actually a good question.  The reason we use Noise Temperature is because it allows us an easy way to combine the effects of an antenna and a receiver together.  Antennas are responsible for receiving signals and passing them to a radio receiver where they can be amplified and demodulated.   Antennas also receive noise from the environment they are in as part of the received signal.  Antenna Temperature defines the mount of noise that can be measured at an antenna’s terminals.  Antenna Temperature is not a physical property of the antenna itself, but rather, a function of the antenna’s design and the environment it is installed in.  I will cover Antenna Temperature in some more detail in a future post!

## A Simple Example

Assume that we measure the Noise Density at the output terminals of an arbitrary amplifier with a gain factor of 100 and establish it to be 7×10-19 Watts/Hz. What is the Noise Temperature (Te) of this system?

The Noise Density at the output is given by:

$N_o = GkT_{out}$

Where:

$T_{out} = T_o+T_e$
$k = 1.38064852 \times 10^{-23}$
$T_o = 290^o$

Te is therefore given by:

$T_e = \frac{N_o}{Gk}-T_o$

$T_e = 217^o$ Kelvin

## Converting To Noise Factor

We can also calculate the Noise Factor from the Noise Temperature relatively easily.  Recall that:

$F_n = \frac{P_{no}}{k.T_o.B.G}$

$F_n = \frac{k.T_{out}.B.G}{k.T_o.B.G}$

$F_n = \frac{T_{out}}{T_o}$

$F_n = 1 + \frac{T_{e}}{T_o}$

We can also convert from Noise Factor to Noise Temperature by making Te the subject of the above formula:

$T_e = (F_n - 1)T_o$