From Co-prime to the Diophantine Equation Based Sparse Sensing in Complex Waveforms
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For frequency estimation, the co-prime sampling tells that in time domain, by two sub-Nyquist samplers with M and N down sampling rates, respectively, up to O(MN) frequencies can be estimated based on autocorrelation, where M and N are co-prime. Similarly, in space domain for Direction-of-arrival (DOA) estimation, co-prime arrays are made of two uniform linear arrays and up to O(MN) sources can be resolved with O(M+N) sensors. In general, the idea behind co-prime sensing is the well-known Bazout's Theorem. However, still from Bazout's Theorem, in frequency estimation, the time delay is proportional to MN. Also, for DOA estimation, though with enhanced degrees of freedom, the sparsity of sensors is not arbitrary. In this letter, we restrain our focus on complex waveforms and present a framework under multiple samplers/sensors for both frequency and DOA estimation. We prove that there exist sampling schemes which can achieve arbitrary sparsity while the time delay is only proportional to the number of samples and resolution required. In the scenario of DOA estimation, we show there exists an array, of which the sparsity can be arbitrary with sufficiently many sensors.
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