RANDOM SAMPLING AND RECONSTRUCTION OF SIGNALS WITH FINITE RATE OF INNOVATION

In this paper, we mainly study the random sampling and reconstruction of signals living in the subspace V-p (Phi, Lambda) of L-p(R-d), which is generated by a family of molecules Phi located on a relatively separated subset Lambda subset of R-d. The space V-p(Phi, Lambda) is used to model signals with finite rate of innovation, such as stream of pulses in GPS applications, cellular radio and ultra wide-band communication. The sampling set is independently and randomly drawn from a general probability distribution over R-d. Under some proper conditions for the generators Phi = {phi(lambda) : lambda is an element of Lambda} and the probability density function rho, we first approximate V-p(Phi, Lambda) by a finite dimensional subspace V-N(p)(Phi, Lambda) on any bounded domains. Then, we prove that the random sampling stability holds with high probability for all signals in V-p(Phi, Lambda) whose energy concentrate on a cube when the sampling size is large enough. Finally, a reconstruction algorithm based on random samples is given for signals in V-N(p)(Phi, Lambda).