Estimation of X in Gaussian Noise

[ait.abd]

Given
${X}=[x_1 x_2 ... x_n]^T$ where ${x_i} \in \{ 0, a_1, a_2, a_3 \}^n, a_i \in \mathbb{C}$ and $Z = \{z_1 z_2,...,z_n \}$ where $z_i ~ N(0,\sigma^2)$ is a Complex Gaussian RV with mean 0 and variance $sigma^2$. Suppose we observe $Y$

$Y = HX+Z$

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 $X$ observing $Y$ 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?

Thanks

 

[lippyka]

for your help.One approach would be to use a probabilistic model to estimate the probability of each x_i being zero or non-zero, given the observed Y. This can be done using Bayesian inference, where you use a prior distribution on the value of each x_i, and then update it based on the observed Y. You can then make a decision about which x_i is most likely to be zero or non-zero by choosing the one with the highest probability. You could also try using a machine learning algorithm such as a support vector machine (SVM), which will try to learn the best decision boundary between the two classes of x_i values.