AWGN, WGN, Autocorrelation and PSD Explained using Matlab - YouTube

In this video, we are going to understand and  visualize at a certain abstraction level that what  

is Gaussian distribution, what is Gaussian  noise, what is white Gaussian noise that is wgn  

and what is additive white Gaussian  noise that is awgn. So we would be using  

matlab simulation the code of this simulation is  given in the description of this video so using  

this simulation we would look into a stem time  series plot. So for this random process we would  

look into a probability density function of white  Gaussian noise. Next, we would move-on towards  

auto correlation function of white noise, later on  we would look into power spectral density of white  

noise and lastly we would look into how additive  white question noise affects a given signal.  

Note that noise is prevalent in most systems,  however, let us restrict ourselves to a  

particular system that is a communication  system. Here we have a broadcast station  

and this broadcast station is transmitting a  signal wirelessly to a user so we have a user  

device that is a cell phone this device has an  antenna so the emt wave which was transmitted  

from the broadcast station would be received at  the received antenna of the user device now within  

this receive antenna there is a render movement  of electrons which are agitated and they dissipate  

and unwanted energy so this is a random process  so the emerging signal often is a Gaussian noise

which has a probability density function referred  over here in this expression and it has a  

plot which is referred to over here. this  mu is the mean value or the average value  

where the sigma is the standard deviation and  the square of it is variance, and variance defines  

or identifies the dispersion in data. note that  the Gaussian distribution is also referred to as  

normal distribution or a bell shaped distribution  we call this bell shaped because the opening of  

this bell is controlled by the variance  sigma square such that if the variance  

has a higher value so we would have a bigger  opening as compared to a lower variance value

now for random nice as we  have considered previously  

the mean value is often set to zero  x follows a normal distribution  

which has a certain mu and that mu is equal to 0  in present case and the variance is sigma squared

now as mu is equal to zero so over here  we have a zero mean random variable  

that is this is a zero mean probability density  function of white question nice now if we want  

to find the probability of having a value between  minus 2 and say minus 0.1 so all we need to do is  

to find the area in this region so we will find  the probability between minus 0.2 and minus 0.1  

do note that integrating the pdf curve from minus  infinity to infinity would yield a value of 1. now  

let us relate this random variable with our random  process and that random process is over here and  

this is a stem time series plot so let us hold  our horses for the white aspect for the time being  

now over here this is an outcome of a random  variable and particularly this is the first  

outcome similarly we can have the second  outcome and third outcome and so on so  

this random process can be expressed as p  of t note that this is a time domain plot

now for such random process we cannot take  the fourier transform directly because in that  

scenario we would have to take the fourier  transform of each realization and then we  

have to take the average value to find the power  spectral density and alternate mean is to find the  

autocorrelation function let us revisit the random  process that we refer to as p of t previously and  

we can find the auto correlation function which  is denoted by r and this is a function of tau  

again we have the random process which is p  of t and then we multiply this random process  

with a shifted version of itself where p of t plus  tau is a shifted version shifted in time domain  

of this random process which is p of t and the  amount of shift is controlled by this variable  

which is top so if this is p of t the second  plot is that of p of t plus tau and we shift it

and in the shift at each increment of this shift  we multiply and take the expectation so in this  

way we will find our autocorrelation function  again that is the correlation of this process  

with the time shifted version of itself and which  is plotted over here an important property of  

white noise is that the autocorrelation function  only exists when the lag or time shift is zero  

that is when you set this tau equal to zero  so you would have p of t multiplied by p of t  

and then we would take the expectation so this  is basically the variance of the distribution  

so hence for a white noise we have an  autocorrelation function and that auto correlation  

function is equivalent to sigma square delta  of tau and for all other legs the value is zero  

you would also come across a terminology  which is often referred to as iid

and this is referred to a distribution which  is independent and identically distributed  

random variable so that is the outcome which  is this one is independent of this outcome  

however both of them follow  an identical distribution  

which is gaussian in our case now  using this autocorrelation function  

we can reach the power spectral density  by simply taking the fourier transform

that is we have a random process p of  t we take the autocorrelation function  

of that so we reach r of tau and finally we  can take the fourier transform of this r of tau  

to get back the power spectral density  which is often denoted by s p of f  

now for the case of autocorrelation function we  refer that for white noise the autocorrelation  

function exists only for a time delay of 0 and it  is 0 elsewhere now for the psd and in the context  

of white noise the power is distributed equally  among all frequencies so this is an important  

consideration for all frequencies the power is  distributed equally and that is basically 10  

log 10 of our variance to get that power in  db scale and that is over here which is 20  

note that sigma in our experimentation we  have set it to 0.01 which is also appearing  

over here that is this is 0.01 delta of  t and in the coding it appears over here  

now in the context of additive white question nice  so consider that we have a signal which is simply  

a sine wave and that sine wave is represented by  this bold line so if we add white question noise  

to it that is additive white gaussian nice so  we would have some fluctuations or attenuations  

and this level of spread would be based on  the distribution that we have considered  

previously that is the gaussian distribution and  the spread is controlled by the variance that is  

sigma square now let us move towards the  matlab plot and in this matlab plot we have  

included a preamble over here and in this  preamble we have run the experiment for 10 seconds  

that is the time t in line 5 would start from 0  and it would terminate at time span which is 10  

seconds with the increment of the sampling time  ts which is equal to 1 over fs and the sampling  

frequency we have set to 10 000 hertz in line  6 this capital l identifies the total number of  

samples that would be taken and they are dependent  on t next we include some statistical parameters  

including the mean value which is mu variance  which we have set 2.01 the standard deviation  

which is the square root of variance and again  the variance in db scale that is 10 log 10 of  

various var so in line 14 we generate the random  process that is capital x is equal to square root  

of the variance times rand n that is random normal  which has l number of rows and one column plus  

the mu which is mean but anyhow we have set  the mean equal to zero so next we plot this  

random variable in figure one by using a stem  plot that is the plot in discrete time events  

and we include some aesthetics in terms of title  and labels moreover we limit the x's specifically  

the x-axis why we have limited because the  total number of samples that we consider are  

hundred thousand this is the length of l  previously mentioned but that would be too much  

data to analyze so we are just going to visualize  zero to 25 samples initially and until keyboard  

the simulation would yield this plot which  is a time series plot of the random process  

of white caution noise so let us understand  further this white gaussian noise  

of 100 000 samples by means of an audio signal so  what would be the sound of a white gaussian noise  

and to play the sound in matlab from line 26 to 28  

we have set a command specifically into  line 27 that is sound of x and that sound is

this one now for the plot of the probability  density function of the gaussian random variable  

we have used figure 2 and we have used  specifically the command that is ks density  

of the random process x right and then that  is plotted over here in figure 2 with some  

labeling while for the autocorrelation function  we have used x correlation and included  

the biased option as an argument of the  correlation function and this correlation function  

would give us the autocorrelation function at  different values of lag which are plotted in  

figure number three so until this keyboard in  line number 50 and let us run the experiment so  

now we have two different plots the first  plot is the pdf of white question noise  

whereas the second plot is the autocorrelation  function so previously in figure 1 we mentioned  

that this is our random process and if we take the  correlation function which is this one and then if  

we take the fourier transform of this function  we would reach the power spectral density but  

applying the fourier transform directly on this  autocorrelation function in the present scenario  

is not giving us the power spectral density  and the reasons for such is that our present  

simulation uh deals with a pseudorandom kind of a  number moreover these lags are finite in duration  

so an alternate route to finding the power  spectral density is adopted from this reference  

it is changed a little bit and in figure 4 we  reshape the random process x into a matrix of  

1000 rows by 100 columns that is we partition x  into 100 random processes and each random process  

has 1000 realizations and we have referred  this matrix as x1 so we take the fast fourier  

transform of this x1 then we normalize it in line  60 we compute the mean power from this fft again  

we normalize the x axis and from line 63 onwards  until 9 69 we perform some display settings of the  

pst plot so until line 69 let us run the  experiment to plot the psd which is over here  

in db per hertz now for the additive white  gaussian nice in figure 5 we generate a new signal  

and this signal is simply a sine wave so we  time scale that sine wave by a factor of 2  

and also multiply by 2 as a constant  coefficient so that this new signal  

is of a certain strength and on this new signal we  include the the white caution noise that we have  

previously created and from 76 onwards until line  number 82 we display the settings and label them  

so let us run the experiment again the dark red is  identifying the original signal whereas the signal  

with awgn is indicated by this green waveform  so if we increase the noise power that is we set  

this variance to say 0.1 and we evaluate this  selection next we again perform this awgn analysis

now in this plot you can observe  that the noise is playing a much  

bigger role as compared to previous  situation fit the noise was simply 0.01