  Wireless communication systems must increasingly support a multitude of
machine-type communications (MTC) devices, thus calling for advanced strategies
for active user detection (AUD). Recent literature has delved into AUD
techniques based on compressed sensing, highlighting the critical role of
signal sparsity. This study investigates the relationship between frequency
diversity and signal sparsity in the AUD problem. Single-antenna users transmit
multiple copies of non-orthogonal pilots across multiple frequency channels and
the base station independently performs AUD in each channel using the
orthogonal matching pursuit algorithm. We note that, although frequency
diversity may improve the likelihood of successful reception of the signals, it
may also damage the channel sparsity level, leading to important trade-offs. We
show that a sparser signal significantly benefits AUD, surpassing the
advantages brought by frequency diversity in scenarios with limited temporal
resources and/or high numbers of receive antennas. Conversely, with longer
pilots and fewer receive antennas, investing in frequency diversity becomes
more impactful, resulting in a tenfold AUD performance improvement.
