We have an asymptotic analysis problem for the eigenvalue performance of the following random matrix:
 $H=\{h\_{ij}\}\_{N\_r\times N\_t}$,


where each entry $h\_{ij}$ is with a probability $p$ to obey the Gaussian distribution $N(0,σ^2)$, and with a probability $1-p$ to be zero.


Then we have following questions


a)Can we use a specific distribution (pdf) to describe the entries of this $N\_r \times N\_t$ matrix as $h\_{ij}\sim (1-p)\delta(0)+p\frac{1}{\sigma\sqrt{2\pi}}e^{-j\frac{x^2}{2\sigma^2}}\delta(x)$?


b)When $N\_r,N\_t\rightarrow \infty$, and $N\_r/N\_t\rightarrow \beta$ (determined value), does the eigenvalues’ PDF go to a deterministic distribution? If the answer is Yes, what is the explicit expression of this deterministic distribution? 


For question b), one potentially useful tool maybe the Mar˘cenko-Pastur law in the random matrix theory, but I do not know if it can be used indeed. 



Asymptotic eigenvalue analysis for a sparse random matrix
I don't really understand your question (a) - but if you look up "mixtures", it will be answered. As for question (b), the magic words are "Girko's circular law", and the magic reference is the [Wikipedia.](http://en.wikipedia.org/wiki/Circular_law)


