Generalized Haar condition-based phaseless random sampling for compactly supported functions in shift-invariant spaces
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It is proved that the phase retrieval (PR) in the linear-phase modulated shift-invariant space (SIS) V(e^iα·φ), α≠0, is impossible even though the real-valued φ enjoys the full spark property (so does e^iα·φ). Stated another way, the PR in the complex-generated SISs is essentially different from that in the real-generated ones. Motivated by this, we first establish the condition on the complex-valued ϕ such that the PR of compactly supported and nonseparable (CSN) functions in V(ϕ) can be achieved by random phaseless sampling. The condition is established from the perspective of the Lebesgue measure of the zero set of a related function system, or more precisely from the generalized Haar condition (GHC). Based on the proposed reconstruction approach, it is proved that if the GHC holds, then the PR of CSN functions in the complex-generated SISs can be achieved with probability 1, provided that the phaseless random sampling density (SD) ≥3. For the real-generated case we also prove that, if the GHC holds then the PR of real-valued CSN functions can be achieved with the same probability if the random SD ≥2. Recall that the deterministic SD for PR depends on Haar condition(measured in terms of the cardinality of the corresponding zero set). Compared with deterministic sampling, the proposed random sampling enjoys not only the greater sampling flexibility but the lower SD. For the lower SD, the highly oscillatory signals such as chirps can be efficiently reconstructed. To verify our results, numerical simulations were conducted to reconstruct CSN functions in the chirp-modulated SISs.
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