Gaussian complex distribution over noise vector

Hello,

Hope everyone is well!

I have a channel input-output relation : Y(k)= H(k) X(k) + W(k) with k=1,...,L

W(k)~CN(0,I(nc)) where W(k) is a gaussian vector, CN is a complex gaussian and I(nc) is an identity matrix of dimension nc.

How can I generate multiple random variables for W(k)?

Is it correct to write :

V=(randn(0,I) + i*randn(0,I))

Answer 1

[AR]

Hi @Anthony Koussaifi,

To generate a complex Gaussian random vector, the standard approach is to combine two independent real Gaussian vectors (for the real and imaginary parts) and scale by 1/√2 to maintain unit variance.

If you want to generate L such vectors each of dimension nc × 1, use:

W = (randn(n_c, L) + 1i * randn(n_c, L)) / sqrt(2); 

• Each column of W corresponds to one sample W(k)∼CN(0,I(nc)).
• This ensures that the total variance remains 1, as expected for standard complex Gaussian noise.

Additionally, the function “randn” expects integer dimensions, like “randn(rows, cols)”

You can refer to the following documentation page for more information:

https://www.mathworks.com/help/matlab/ref/double.randn.html

I hope this helps!