Performance of Hierarchical Sparse Detectors for Massive MTC
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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 Machine-type Communications (mMTC) scenarios. In this paper, we take a closer look at the performance of HiHTP 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 which under suitable choice of the signatures equals that of the block correlation detector. A key observation here is that in specific parameter settings non-orthogonal 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. Our extended analysis then shows that, in typical parameter regimes, user detection is independent of the activity level so that essentially HiHTP behaves qualitatively like the classical correlator. We provide mathematically rigorous and easy to handle formulas for numerical evaluations and system design. Finally, we evaluate our findings with numerical examples.
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