Design of random modulation preintegration systemsbased on the restricted-isometry propertymay be suboptimalwhenthe energy of the signals to be acquired is not evenly distributed,i.e., when they are both sparse and localized. To counter this, weintroduce an additional design criterion, that we call rakeness, accountingfor the amount of energy that the measurements capturefrom the signal to be acquired. Hence, for localized signals aproper system tuning increases the rakeness as well as the averageSNR of the samples used in its reconstruction. Yet, maximizing averageSNR may go against the need of capturing all the componentsthat are potentially nonzero in a sparse signal, i.e., againstthe restricted isometry requirement ensuring reconstructability.What we propose is to administer the trade-off between rakenessand restricted isometry in a statistical way by laying down an optimizationproblem. The solution of such an optimization problemis the statistic of the process generating the random waveformsonto which the signal is projected to obtain the measurements. Theformal definition of such a problems is given as well as its solutionfor signals that are either localized in frequency or in moregeneric domain. Sample applications, to ECG signals and smallimages of printed letters and numbers, show that rakeness-baseddesign leads to nonnegligible improvements in both cases.
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