Confusion regarding pdf of circularly symmetric complex gaussian rv

1. What is circularly symmetric complex Gaussian?
2. How do you generate a complex Gaussian random variable in Matlab?
3. What is a gaussian vector?
4. What is gaussian random process?

What is circularly symmetric complex Gaussian?

A complex circularly-symmetric gaussian random variable has the property that eiθz has the same probability density function for all θ. Generalizing, a complex, jointly gaussian random vector z = x + iy is circularly symmetric when the vector eiθz has the same multivariate probability density function for all θ.

How do you generate a complex Gaussian random variable in Matlab?

We have as matlab function randn generates Gaussion randome variables . x = 1/sqrt(2)*(randn(N, 1) + 1i*randn(N,1)); It can be shown that: Therefore the factor of 1/sqrt(2) is correct if one wants to generate 0-mean complex Gaussian variable with variance of 1 (std is also 1 for 0-mean randome variable).

What is a gaussian vector?

A gaussian vector is a vector such that every linear combination of its coefficients follows a gaussian distribution. If the coefficients of a vector follow a multivariate distribution, the vector should be gaussian.

What is gaussian random process?

In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed.