# Estimation of X in Gaussian noise

Given $\textbf{x}=[x_1 x_2 ... x_n]^T$ where $\textbf{x} \in \{ 0, a_1, a_2, a_3 \}^n, a_i \in \mathbb{C}$ and $\textbf{z} = \left\{z_1,z_2,\dots,z_n \right\}$ where $z_i \sim N(0,\sigma^2)$ is a Complex Gaussian RV with mean $0$ and variance $\sigma^2$. Suppose we observe $\textbf{y}$

$$\textbf{y} = H\textbf{x}+\textbf{z}$$

where $H$ is known and its elements are independent complex Gaussian with mean 0 and variance 1 in $\mathbb{C}$ i.e. complex numbers. How can I estimate $\textbf{x}$ observing $\textbf{y}$ when I only want to know whether $x_i$ is zero or non-zero? i.e. I don't want to distinguish between $a_1, a_2, a_3$ and only want to estimate whether $x_i$ was zero or non_zero? Is there any iterative way of finding this out?

• Are the $z_i$ also independent complex Gaussian random variables? – Dilip Sarwate Nov 4 '11 at 12:14
• This sounds like a Rayleigh-fading wireless communications problem. – cardinal Nov 4 '11 at 13:09
• You are right! Its a Rayleigh Fading wireless communication problem. $z_i$ are also independent complex Gaussian RVs. – Aitezaz Nov 4 '11 at 17:04
• What do you mean by you want to "estimate" whether the $x_i$ are zero or non-zero? Do you mean you want to test the hypothesis $H_{0,i}: x_i = 0$ vs. $H_{1,i}: x_i \neq 0$? Are you interested in individual hypotheses or a joint hypothesis, say, that they are all zero? Are you trying to minimize the probability of error of your decision or control the Type I Error rate (which you might call the "false alarm" probability)? – cardinal Nov 4 '11 at 23:24