We develop sub-Nyquist sampling systems for analog signals comprised of several, possibly overlapping, finite duration pulses with unknown shapes and time positions. To the best of our knowledge, stable and low-rate sampling strategies for a superposition of unknown pulses without knowledge of the pulse locations have not been derived. We propose a multichannel scheme based on Gabor frames that exploits the sparsity of signals in time and enables sampling multipulse signals at sub-Nyquist rates. Our approach is based on modulating the input signal in each channel with a properly chosen waveform, followed by an integrator. As we show, the resulting scheme is flexible and exhibits good noise robustness.

Abstract-We study a sampling setup where a continuous-time signal is mapped by a memoryless, invertible and nonlinear transformation, and then sampled in a nonideal 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 nonideal samples. Three alternative formulations of the algorithm are described 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 of 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 Wiener-Hammerstein analog-to-digital hybrid systems.

The metaplectic representation describes a class of automorphisms of the Heisenberg group H = H (G), defined for a locally compact abelian group G. For G = R d , H is the usual Heisenberg group. For the case when G is the finite cyclic group Z n , only partial constructions are known. Here we present new results for this case and we obtain an explicit construction of the metaplectic operators on C n . We also include applications to Gabor frames.

Nonstationary Gabor frames were recently introduced in adaptive signal analysis. They represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. In this paper we show a general existence result for this family of frames. We then give a perturbation result for nonstationary Gabor frames and construct nonstationary Gabor frames with non-compactly supported windows from a related painless nonorthogonal expansion. Finally, the theoretical results are illustrated by two examples of practical relevance.

Nonstationary Gabor frames, recently introduced in adaptive signal analysis, represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from coefficients obtained from nonstationary Gabor frames. In this paper it is shown that for nonstationary Gabor frames that are related to some known frames for which dual frames can be computed, good approximate reconstruction can be achieved by resorting to approximately dual frames. In particular, we give constructive examples for so-called almost painless nonstationary frames, that is, frames that are closely related to nonstationary frames with compactly supported windows. The theoretical results are illustrated by concrete computational and numerical examples.