Discrete random sampling theory

This paper proposes a new perspective on the relationship between the sampling and aliasing. Unlike the uniform sampling case, where the aliases are simply periodic replicas of the original spectrum, random sampling theory shows that the randomization of sampling intervals shapes the aliases into a noise floor in the sampled spectrum. New insights into both the Fourier random sampling problem and Compressive Sensing theory can be obtained using the theoretical framework of random sampling. This paper extends the theory of continuous time random sampling to deal with random discrete intervals generated from a clock. A key result is established to relate the discrete probability distribution of the sampling intervals to the power spectrum of the aliasing noise. Based on the proposed theory, a generic discrete random sampling hardware architecture is also proposed for sampling and reconstructing a class of spectrally sparse signals at an average rate significantly below the Nyquist rate of the signal.