  Design of Random Modulation Pre-Integration systems based on the
restricted-isometry property may be suboptimal when the energy of the signals
to be acquired is not evenly distributed, i.e. when they are both sparse and
localized. To counter this, we introduce an additional design criterion, that
we call rakeness, accounting for the amount of energy that the measurements
capture from the signal to be acquired. Hence, for localized signals a proper
system tuning increases the rakeness as well as the average SNR of the samples
used in its reconstruction. Yet, maximizing average SNR may go against the need
of capturing all the components that are potentially non-zero in a sparse
signal, i.e., against the restricted isometry requirement ensuring
reconstructability. What we propose is to administer the trade-off between
rakeness and restricted isometry in a statistical way by laying down an
optimization problem. The solution of such an optimization problem is the
statistic of the process generating the random waveforms onto which the signal
is projected to obtain the measurements. The formal definition of such a
problems is given as well as its solution for signals that are either localized
in frequency or in more generic domain. Sample applications, to ECG signals and
small images of printed letters and numbers, show that rakeness-based design
leads to non-negligible improvements in both cases.
