A tutorial overview of ADDITIVE WHITE GUASSIAN NOISE (AWGN)

SAMI UR REHMAN

Defining NOISE in AWGN:

In electrical engineering, pertaining to

communication systems, “Noise” is basically an

unwanted signal which carries no information

and hence can corrupt our information bearing

signal. ”Signal to Noise Ratio” is matric

commonly used to characterize the ratio of

information to irrelevant data in a signal. In the

nutshell noise is an unwanted disturbances

superimposed on a useful signal, which tends to

obscure its information content. High noise

levels can block, distort, change or interfere with

the meaning of a message in human, animal and

electronic communication. Similarly in

electronic circuits noise is unwanted

perturbation in voltage signal.

Thermal noise is device inherent noise as it

occurs in all transmission media and

communication equipment, including passive

devices. It arises from random electron motion

and is characterized by a uniform distribution of

energy/power over the frequency, plus it also

follows Gaussian distribution and hence can

rightly be called Additive White Gaussian Noise

which is the topic of this paper. Every

equipment element and the transmission medium

itself contribute thermal noise to a

communication system if the temperature of that

element or medium is above absolute zero.

Thermal noise is heavily dependant on

temperature as the more heat generated or

applied, the greater the level of thermal noise.

Intermodulation (IM) noise is the result of the

presence of intermodulation products. Such type

of noise occurs if two signals of different

frequencies pass through a nonlinear device or

medium, the result will contain IM products that

are spurious frequency energy components.

These components may be inside or outside the

frequency band of interest for a particular

device. Crosstalk refers to unwanted coupling

between signal paths.

Defining WHITE NOISE in AWGN:

White Noise is a random process with flat power

spectral density. Power spectral density (PSD)

shows the variation of power of a signal or

random process with frequency. It is also called

energy spectral density, spectral density or

simply spectrum.

White noise in vector domain has to follow the

following two properties:

( ) ( ) 0X t E x t

This means mean of the white noise is zero.

1 2 1 2 1 2( ) ( ) ( ) ( / 2) ( )Rxx t t E x t x t No t t

There are two implications that can be extracted

from the second property:

This autocorellation function suggests

the PSD must have a value of No/2.

The value of white noise at any pairs of

times is independent or uncorellated.

An extremely important function to calculate

PSD is the Wiener–Khinchin theorem which

states that PSD of a signal is Fourier transform

of its autocorrelation function (function

calculating similarity of signal/process with

itself).

( ) ( )N NS f Fourier R

PSD of White Noise SN(f) is given below

where No represents the noise variation across

frequency.

0( ) / 2NS f N to

Figure 1: PSD of white noise

Similarly the auto correlation function of White

Noise is given as:

0( ) / 2 ( )NR N

Figure 2: Auto correlation function of white

noise

The autocorellation function resembles the dirac

delta function and the Fourier Transform of such

a function is equal to 1. This also implies that

white noise has infinite power at zero. Now

since the PSD is the Fourier Transform of

autocorellation fuction which in white noise is

dirac delta function, the PSD for white noise is

same at all frequncies. We call this noise white

as an analogy to the frequncy spectrum of white

light.

Defining GUASSIAN in AWGN:

In stochastic theory Guassian Noise is the noise

whose Probability Density Function (PSD)

follows Guassian distribution as shown below.

Let „z‟ represent any random process or say the

Guassian noise, then its distribution is given by

the following formulae:

2

22

1 ( )( )

22

z

zz

zf z e

z represents the varience and z represents the

mean of the random variable.

PSD is a function which shows the likelihood of

a random variable to occur at a given point. In

Guassian distribution this point is the mean of

the function where we have the maximum

likelihood of occurrence of that variable. A

Guassian distribution with mean of zero is called

zero mean Guassian and the one with mean zero

and varience of one is called Normal

distribution. A white noise whose probability

density function (PSD) is Guassian distributed,

is called White Guassian Noise. Since white

noise has a mean of zero it is better to call it zero

mean Guassian. Since white noise and Guassian

noise and two separate concepts but we have

added them together so we sometimes call White

Guassian Noise as Additive White Guassian

Noise.

Excerpt from Wikipedia:

“Gaussian noise is properly defined as the

noise with a Gaussian amplitude distribution.

This says nothing of the correlation of the

noise in time or of the spectral density of the

noise. Labeling Gaussian noise as 'white'

describes the correlation of the noise. It is

necessary to use the term "white Gaussian

noise" to be precise. Gaussian noise is

sometimes misunderstood to be white

Gaussian noise, but this is not the case.”