Theoretical Formulation/Development of Signal Sampling with an Equal Arc Length Using the Frame Theorem

Non-uniform sampling with equal arc length intervals can be found in shape measurements with contact sensor arrays. In this study, the conditions of non-uniform spatial sampling with an equal arc length interval are derived from two frame theorems. First, for general non-uniform sampling, the condition is that the equal arc length interval of the sensors should be less than 1/4Ω. Second, for strictly increasing sampling, the condition is that the equal arc length interval of the sensors should be less than 1/2Ω. The Ω is the maximum frequency of the detected object. For the latter, if the sampling frequency is more twice than the sampling frequency required, the reconstruction error (RRMSE and MRE) is less than 5%. If the sampling frequency is more than 2.5 times, the reconstruction error is less than 3%. The simulation and the experiment are carried out and the results show that a sensor array with equal arc length interval can reconstruct the detected object with high accuracy.