Abstract: The traditional sampling method records the level of signal at the pre-determined time. Therefore the levels sampled are often uniformed. There is another sampling model-implicit sampling, it records the timing of level-crossings which are determined before sampling. Under this situation, the signal determines the times but not level of sampling. In the condition that one scalar factor exists, Logan theorem provides enough conditions for signal reconstruction from zero crossings. Since there are some noises in the measurement of timing, the reconstruction usually is not robust. So this paper introduces additional condition: there are some sparse bases in the signal so that it makes reconstruction robust. Under this assumption, it can regard the reconstruction as minimization of sparsity inducing cost function and provide an algorithm. Although the problem is not convex, the experiments show in the classic cases the algorithm is converged and gives a correct solution with high probability.

Key words: level crossing, signal reconstruction, sampling