Analysis on User Activity in Compressed Sensing based Random Access

In Compressed Sensing based Random Access CHannel (CS-RACH) protocol, a base station leverages compressed sensing technique to detect the active users in the cell coverage and estimate the channel gain between the users and the base station. In a real communication scenario, activity of a specific user usually varies time to time and thus can be seen as a random variable following ON/OFF distribution. Meanwhile, the performance of compressed sensing technique is dependent on the sparsity of the estimating vector, which is closely related to the user activity in CS-RACH scenario. In this perspective, we analyze the condition of the user activity for the stable operation of the protocol. Particularly, we use the least absolute shrinkage and selection operator (LASSO) approach, which gives a closed form expressions of the sparsity condition for the successful active user detection in an asymptotic manner. As a result, we obtain the condition of the user activity for stable operation of CS-RACH and verify the result with numerical simulations.