HiHTP: A custom-tailored hierarchical sparse detector for massive MTC

Recently, the Hierarchical Hard Thresholding Pursuit (HiHTP) algorithm was introduced to optimally exploit the hierarchical sparsity structure in joint user activity and channel detection problems, occurring e.g. in 5G massive Massive Machine-type Communications (mMTC) scenarios. In this paper, we take a closer look at the performance of HiHTP for user activity detection under noise and relate its performance to the classical block correlation detector with orthogonal signatures. More specifically, we derive a lower bound for the diversity order of HiHTP and provide explicit and easy to handle formulas for numerical evaluations and (5G) system designs. Furthermore, we surprisingly find that in specific parameter settings nonorthogonal pilots, i.e. pilots of which shifted versions actually interfere with each other, outperform the block correlation detector, which is optimal in the non-sparse situation. Finally, we evaluate our findings with numerical examples.