How to create a complex normal i.i.d.?

Hello ,

i want to create a i.i.d CN(0,σ^2)

I create the variable like that :

x=sqrt(var/2)*(randn+1i*randn);

Is this correct?

[Bjorn Gustavsson]

Does that satisfy your definition of a complex normal distribution? Does it fit with for example the definitions here: Complex_normal_distribution?

[Gn Gnk]

I just want to know how to convert it to matlab code and if this : x=sqrt(var/2)*(randn+1i*randn);

is correct or not.

Thank you.

Answer 1

[John D'Errico]

Is it correct? It is what it is. You are creating a rotationally symmetric complex random variable, where if we define the variance to be

N = 1e7;

V = 3;
X = sqrt(V/2)*(randn(N,1) + i*randn(N,1));

Does it have zero mean? That is trivial, yes. The population mean is clearly zero, and the sample mean will approach zero as N-->inf.

Xbar = mean(X)
Xbar =
    0.0002026 - 0.00027059i
    
Vbar = (X-Xbar)'*(X - Xbar)/(N-1)
ans =
        2.999

As you can see, the sample variance seems to approach 3. I could spend some time to verify the expected value of that result for the population, done in symbolic form. But I can as easily just say that is is "correct".

[Gn Gnk]

Thank you a lot sir!