On the correctness of coherence in the determination of preambles for compressive random access

In this paper, we study a criterion for measurement matrices to understand the performance of sparse signal recovery. In particular, we consider the activity detection in compressive random access and derive an approximate upper-bound on the recovery error probability using pair-wise error probability (PEP) when the maximum likelihood (ML) detection is employed. From the approximate upper-bound, we obtain a performance criterion that can be used to choose a set of pilot or preamble sequences. Based on the derived performance criterion, we can show that the performance of Alltop sequences can be similar to that of Zadoff-Chu sequences.