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, negatively affecting the performance/reliability of a communications system.

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

# Noise Factor

Noise Factor provides a way to measure the additional noise added to a signal as it passes through a component.

If we are looking at a component that amplifies the signal by gain G, then we know that the system 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 still amplify the input noise (Pni) by the gain G. The output Noise Power of an ideal component is given by:

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

A realistic component will insert additional noise (Pna) to the system.  We model Pna as entering the component before it is amplified by the gain G.  Therefore the output Noise Power (Pno) of a realistic component is given by:

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

The Noise Factor of the component is defined below:

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

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:

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

## Understanding Noise Factor

You should be able to draw the following conclusions from the above equations:

• The Noise Factor of an ideal system is 1.
• The SNR of the input and output signals of an ideal system are equal.
• The Noise Factor of a realistic system is always greater than 1.
• The output SNR of a real system will always be smaller than the input SNR.

## Input Noise Due to Thermal Noise

If we assume that the input noise Pni is purely due to thermal noise (the minimum possible noise level), 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 of a component, we always use To = 290° K for Pni.  Similarly, when testing the Noise Performance of a component, the test is always conducted at this temperature.

The calculation for Noise Factor of a system is thus:

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

This will have some implications further on when we discuss Noise Temperature.

# 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:

$T_e = (F_n - 1)T_o$