Normally, to generate a sequence of noise values that fit a particular probability distribution, one computes the cumulative integral of the pdf and then a uniform random number is calculated and fed into the cumulative integral to yield the noise value of the desired pdf. In MATLAB, use the cumsum function on an array representing the pdf. Then use the rand function to generate an index (with a uniform distribution) to pick values out of the resulting array of the cumsum function. For example:

A = -10 ;%example of lowest value of interest

B = 10; %example of highest value of interest

N = 1000; %example of number data points

X = [A : (B - A +1)/N : B]; %results in N data points between A and B

f = ;

y = cumsum ( f );

index = floor ( rand (1,N) * N + 1); %generate indices based on uniform distribution

NoiseValues = y (index);

I hope this helps.

Hello,

You need mandatory to use and add AWGN : Additive White Gaussian Noise to your signal ! Then random signal is not the correct term. You can use AWGN function in matlalb software.

In the PDF document you don't need it. But calculation in matlab is not difficult to do with the right method ?

Then I continue to explain you how to do with AWGN :

Why ?

1. Because noise signal power density you add have a Gaussian distribution.

2. White : All frequencies will be present in noise.

3. When you use Gaussian distribution you suppose that universe is an isotropic one.

That is a good estimation on propagation channel. If you use other statistic random signal you don't care about this. I think it is not correct.

4. Calculation with CRLB (Cramer-Rao Lower Bound) will be much easier (but remains very difficult to understand or estimate) than other noise definition because CRLB application on AWGN propagation channel.

5. Gaussian variance or error estimation are easy to calculate.

Matlab example :

You define a short impulse - You can use sinus fonction if you use Narrow Band (NB) telecom signal. If NB then think to multiplie it by a windows to limitate the time duration of signal.

% Enveloppe Gaussienne

% for exemple a short Gaussian pulse (ex :pw=10-9 s)

x=A*exp(-2*pi.*((tx-t0)/pw).^2);

% Add AWGN to x telecom signal

% -------------------------------------------------------

% Noise_level is for exemple -10 (db)

Noise_Level = -10;

x_awgn = awgn(x,Noise_Level,'measured');

% then use a Foureir Transform to "see" you spectral dispesion

X_awgn = fft(x_awgn);

% display : that all !

Best regards

Jean-Paul Cipria

Telecom Engineer.

Technologies Teacher.

http://www.nanotechinnov.com

See : (English translation is : CRLB Nanosecond impulsive system precision localization)

Précision de localisation en système impulsif nanoseconde – Cramer Rao Lower Bound – CRLB

In addition, it is important to understand the noise characteristics of your propagation environment. For example, noise at HF frequencies is largely impulsive and can be described by a Bi-Kappa distribution (see my papers on HF noise density function). One cannot blindly assume that AWGN applies to all propagation environments. This is a common mistake that many researchers make. It is true, that environments cluttered with many sources of noise will tend towards a Gaussian noise distribution, however, less cluttered environments tend to have noise sources that are directional and consequently the PDF of noise can be impulsive. The noise distribution is also dependent on frequency, and so at HF, the noise is impulsive, but in the widely used 2.4GHz band the noise will be more Gaussian.

https://www.researchgate.net/search.Search.html?query=James%20giesbrecht%20hf%20noise&type=publication

@James

Sir I appreciate your effort in answer and it is really helpful but the paper which I am following (shared link) is using PDF which involve impulse function which is shown in equation 2

f(x)= c/(b-a) + (1-c)δ(x);

c is the probability of appearance of impulse noise which I have taken 10^(-3) and noise value range is -10 to 10.......

problem is, as I fed this function in your procedure to generate Noise I first of all get all of the values in pdf function 0.00005 and secondly I get many Infinite in the noise value.

So if you clarify this in more detail it would be much more helpful because I am unable to sort out this problem.

@James

this this the procedure I am following

A = -10 ;%example of lowest value of interest

B = 10; %example of highest value of interest

C=10^(-3);

N = 1000; %example of number data points

X = [A : (B - A)/N : B]; %results in N data points between A and B

f = C/(B-A)+(1-C)*dirac(X);

y = cumsum (f);

index = floor(rand (1,N)*N+1); %generate indices based on uniform distribution

NoiseValues = y(index);

The integral of your PDF function must sum to unity (1). Your PDF is everywhere equal to 0.0005 except at X=0 where f(0)=0.0005+0.999=0.9995. The cumsum of your function is 1.4995. This is not equal to 1. I suggest that your PDF is being incorrectly used. You must redefine your PDF for every data point such that the integral of the PDF from -infinity to +infinity is equal 1. The values for A = -10 and B = 10 are examples only and represent the bounds of your PDF. For example, if your PDF is non-zero from -1 to 1, then A = -1 and B= 1.

@James

Sir thanks for the comment.

Sir as you have said I changed my bound values A= -100 to B=100. Now I am getting cumsum of my function is 1.0040 which is nearly equal to 1. Now is it correct or should I have to do something else. And Now I am using codes like this

A = -100 ;

B = 100;

C=10^(-3);

N = 1000;

X = [A : (B - A)/N : B];

for i=1:length(X)

if X(i)==0

f(i)=C/(B-A)+(1-C);

else

f(i) = C/(B-A);

end

end

y = cumsum (f);

index = floor(rand (1,N)*N+1);

NoiseValues = y(index);

waiting for your reply

Depending on your values for A and B you may find that there is no value of X equal to zero, in which case the first part of your IF statement will not execute. You need to decide on how many samples you want. N=1000 is only an example. Then you need to choose A and B such that X=0 at some point. I have run your matlab code and it works fine, provided A and B and N are correctly chosen. As the paper suggests, A and B might be chosen as the maximum and minimum of an ADC. You may also wish to consider adjusting the number of samples and the values of A and B so that the cumsum gets even closer to 1. Hope this helps.

@James

Thank you very much sir for your suggestion. Sir One more thing which I have to ask is from the above mention code which generate noise, is it mixture of Impulse and AWGN or its only Impulse noise??? because nowhere we have generated Gaussian noise in our code. And if It is a mixture of Gaussian and Impulse then noise values is having less then 1.5 but we have mentioned it should be between -100 to 100. Please clarify if possible . Waiting for your reply

It is not a mixture of gaussian and impulsive noise. Your PDF models only impulsive noise as described in the paper. If you want to add in gaussian noise you will have to either change your PDF to model both impulsive and gaussian noise, or, you can add in gaussian noise after generating your impulsive noise.

@James

Thanks for the reply.

 Is there any procedure to follow for addition of  Impulse noise with Gaussian or we can directly add Gaussian noise. This question is raised because probability of occurrence of impulse noise is defined "c", which we have used in generation of impulse noise. 

Yes, you can directly add in Gaussian noise.

yes, you can directly add gaussion noise with impulse

Can we add two pdf of Gaussian and Impulsive noise if they occur simultaneously. I mean is it feasible. ??? 

Yes, you can directly add as many noise sources as you like.  These noise sources can have any number of distributions.  However, you can directly add the probability density functions ONLY after each function is appropriately weighted such that the combined cumulative distribution ranges from 0 to 1.

@James

Don't we have to convolve this two pdf ????

Strictly, the combined PDF depends on the de

Strictly, the combined PDF depends on the dependencies within your system of observations. Probabilities can be added or multiplied. I refer you to any reputable text on probability theory.