Nonlinear and Non-Ideal Sampling : Theory and Methods

We study a sampling setup where a continuous-time signal is mapped by a memoryless, invertible and nonlinear transformation, and then sampled in a non-ideal manner. Such scenarios appear, for example, in acquisition systems where a sensor introduces static nonlinearity, before the signal is sampled by a practical analog-to-digital converter. We develop the theory and a concrete algorithm to perfectly recover a signal within a subspace, from its nonlinear and non-ideal samples. Three alternative formulations of the algorithm are developed that provide different insights into the structure of the solution: A series of oblique projections, approximated projections onto convex sets, and quasi-Newton iterations. Using classical analysis techniques for descent-based methods, and recent results on frame perturbation theory, we prove convergence of our algorithm to the true input signal. We demonstrate our method by simulations, and explain the applicability of our theory to WienerHammerstein analog-to-digital hybrid systems.