Closed-Form Word Error Rate Analysis for Successive Interference Cancellation Decoders

We consider the estimation of an integer vector $\text{hbx}\in {\mathbb{Z}}^n$ from the linear observation \y=\A\hbx+\v, where $\text{A}\in {\mathbb{R}}^{m\times n}$ is a random matrix with independent and identically distributed (i.i.d.) standard Gaussian $\mathcal{N}(0,1)$ entries, and $\check{\in }{\mathbb{R}}^m$ is a noise vector with i.i.d. $\mathcal{N}(0,{\sigma }^2)$ entries with given $\sigma$. In digital communications, $\text{hbx}$ is typically uniformly distributed over an $n$-dimensional box $\mathcal{B}$. For this estimation problem, successive interference cancellation (SIC) decoders are popular due to their low complexity, and a detailed analysis of their word error rates (WERs) is highly useful. In this paper, we derive closed-form WER expressions for two cases: (1) $\text{hbx}\in {\mathbb{Z}}^n$ is fixed and (2) $\text{hbx}$ is uniformly distributed over $\mathcal{B}$. We also investigate some of their properties in detail and show that they agree closely with simulated word error probabilities.