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Grant-free protocols exploiting compressed sensing (CS) multi-user detection(MUD) are appealing for solving the random access problem in massivemachine-type communications (mMTC) with sporadic device activity. Suchprotocols would greatly benefit from a prior deterministic knowledge of thesparsity level, i.e., instantaneous number of simultaneously active devices$K$. Aiming at this, herein we introduce a framework relying on coordinatedpilot transmissions (CPT) over a short phase at the beginning of thetransmission block for detecting $K$ in mMTC scenarios under Rayleigh fading.CPT can be implemented either as: i) U-CPT, which exploits only uplinktransmissions, or A-CPT, which includes also downlink transmissions for channelstate information (CSI) acquisition that resolve fading uncertainty. We discusstwo specific implementations of A-CPT: ii) A-CPT-F, which implements CSI-basedphase corrections while leveraging the same statistical inverse power controlused by U-CPT, and iii) A-CPT-D, which implements a dynamic CSI-based inversepower control, although it requires some active devices to remain in silence iftheir corresponding channels are too faded. We derive a signal sparsity leveldetector for each CPT mechanism by relaxing the original integerdetection/classification problem to an estimation problem in the continuousreal domain followed by a rounding operation. We show that the variance of therelaxed estimator increases with $K^2$ and $K$ when operating with U-CPT andA-CPT mechanisms, respectively. The distribution of the estimators under U-CPT,A-CPT-F and A-CPT-D is found to follow an exponential, Gaussian, and Student's$t-$like distribution, respectively. Analyses evince the superiority ofA-CPT-D, which is also corroborated via numerical results. We reveal severalinteresting trade-offs and highlight potential research directions.

Coordinated Pilot Transmissions for Detecting the Signal Sparsity Level in a Massive IoT Network under Rayleigh Fading