  In light of the ever-increasing demand for new spectral bands and the
underutilization of those already allocated, the concept of Cognitive Radio
(CR) has emerged. Opportunistic users could exploit temporarily vacant bands
after detecting the absence of activity of their owners. One of the crucial
tasks in the CR cycle is therefore spectrum sensing and detection which has to
be precise and efficient. Yet, CRs typically deal with wideband signals whose
Nyquist rates are very high. In this paper, we propose to reconstruct the power
spectrum of such signals from sub-Nyquist samples, rather than the signal
itself as done in previous work, in order to perform detection. We consider
both sparse and non sparse signals as well as blind and non blind detection in
the sparse case. For each one of those scenarii, we derive the minimal sampling
rate allowing perfect reconstruction of the signal's power spectrum in a
noise-free environment and provide power spectrum recovery techniques that
achieve those rates. The analysis is performed for two different signal models
considered in the literature, which we refer to as the analog and digital
models, and shows that both lead to similar results. Simulations demonstrate
power spectrum recovery at the minimal rate in noise-free settings and show the
impact of several parameters on the detector performance, including
signal-to-noise ratio (SNR), sensing time and sampling rate.
