Towards a general theory of reconstruction of bandlimited signals from sine wave crossings

Abstract

Listen

Translate article

Arbitrary bounded bandlimited signals that are real on the real axis cannot be reconstructed from their zero crossings in general, because they can have complex zeros. However, a simple invertible transformation, which consists of taking the difference of such a signal with a properly chosen sine function, creates a function that has only simple and real zeros. It can be shown that the original signal can be reconstructed from these zeros, or, equivalently, from the sine wave crossings of the signal. In this paper we study the reconstruction of bandlimited signals from their sine wave crossings by sampling series, which is one possible method of reconstruction. The reconstruction functions and the sampling points that are used in the reconstruction process are determined by the signal which shall be reconstructed and therefore are, in a certain sense, adapted to the signal. We address the convergence behavior of the reconstruction process without oversampling. Although we have local uniform convergence, we conjectured in a previous paper that the adaptivity is not sufficient for global uniform convergence. By developing a rigorous convergence theory for different signal spaces, and by showing that the Paley–Wiener space PW π 1 of signals that are the inverse Fourier transform of some absolutely integrable function is the border case, we prove that this conjecture is true. The solution of this problem has far reaching consequences for the approximation of stable linear time-invariant systems by sampling series, because it shows that the adaptivity of the system approximation process is not sufficient for its pointwise convergence. The theory of sine-type functions is an important tool for our analysis, and we give several sufficient conditions for the sine-wave crossings of a signal to be the zero sequence of a sine-type function.