Deterministic coding theorems for blind sensing: optimal measurement rate and fractal dimension

Completely blind sensing is the problem of recovering bandlimited signals from measurements, without any spectral information beside an upper bound on the measure of the whole support set in the frequency domain. Determining the number of measurements necessary and sufficient for reconstruction has been an open problem, and usually partially blind sensing is performed, assuming to have some partial spectral information available a priori. In this paper, the minimum number of measurements that guarantees perfect recovery in the absence of measurement error, and robust recovery in the presence of measurement error, is determined in a completely blind setting. Results show that a factor of two in the measurement rate is the price pay for blindness, compared to reconstruction with full spectral knowledge. The minimum number of measurements is also related to the fractal (Minkowski-Bouligand) dimension of a discrete approximating set, defined in terms of the Kolmogorov $\epsilon$-entropy. These results are analogous to a deterministic coding theorem, where an operational quantity defined in terms of minimum measurement rate is shown to be equal to an information-theoretic one. A comparison with parallel results in compressed sensing is illustrated, where the relevant dimensionality notion in a stochastic setting is the information (R\'{e}nyi) dimension, defined in terms of the Shannon entropy.