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In the first part of the series papers, we set out to answer the followingquestion: given specific restrictions on a set of samplers, what kind of signalcan be uniquely represented by the corresponding samples attained, as thefoundation of sparse sensing. It is different from compressed sensing, whichexploits the sparse representation of a signal to reduce sample complexity(compressed sampling or acquisition). We use sparse sensing to denote a boardconcept of methods whose main focus is to improve the efficiency and cost ofsampling implementation itself. The "sparse" here is referred to sampling at alow temporal or spatial rate (sparsity constrained sampling or acquisition),which in practice models cheaper hardware such as lower power, less memory andthroughput.  We take frequency and direction of arrival (DoA) estimation as concreteexamples and give the necessary and sufficient requirements of the samplingstrategy. Interestingly, we prove that these problems can be reduced to some(multiple) remainder model. As a straightforward corollary, we supplement andcomplete the theory of co-prime sampling, which receives considerable attentionover last decade.  On the other hand, we advance the understanding of the robust multipleremainder problem, which models the case when sampling with noise. A sharpenedtradeoff between the parameter dynamic range and the error bound is derived. Weprove that, for N-frequency estimation in either complex or real waveforms,once the least common multiple (lcm) of the sampling rates selected issufficiently large, one may approach an error tolerance bound independent of N.

On the Foundation of Sparse Sensing (Part I): Necessary and Sufficient Sampling Theory and Robust Remaindering Problem