This is not a homework question though it might be trivial for those who are well versed with multivariate distributions. I am trying to understand a paper that has the following product form. Let $A$ be a row matrix that has only single entry picked from a Gaussian distribution and rest of them are zero. Let $H$ be a Walsh-Hadamard matrix and $B$ be a diagonal signed Bernoulli matrix with parameter $p=1/2$. Let $x$ be a unit vector.
What is the probability distribution of $AHBx$?