  Enabling low power wireless devices to adopt Nyquist sampling at high
carriers is prohibitive. In spectrum sensing, this limit calls for an analog
front-end that can sweep different bands quickly, in order to use the available
spectrum opportunistically. In this paper we propose a new framework that
allows to optimize a sub-Nyquist sampling front-end to combine the benefits of
optimum sequential sensing with those of Compressive Spectrum Sensing (CSS).
The sensing strategy we propose is formulated as a sequential utility
optimization problem. The utility function decreases linearly with the number
of measurements (accrued to make a decision) and increases as a function of the
signal components binary states. The optimization selects the best linear
combinations of sub-bands to mix, in order to accrue maximum utility. The
structure of the utility represents the trade-off between exploration,
exploitation and risk of making an error, that is characteristic of the
spectrum sensing problem. We first present the analog front-end architecture,
and then map the measurement model into the abstract optimization problem
proposed, and analyzed, in the remainder of the paper. We characterize the
optimal policy under constraints on the sensing matrix and derive the
approximation factor of the greedy approach. Numerical simulations showcase the
benefits of combining opportunistic spectrum sensing with sub-Nyquist sampling.
