Irregular Sampling of Band-limited Signals

Kim Tuan Vu State University of West Georgia, USA

Let f(t) ∈ L_{2}(R) and f̂(λ) be its Fourier transform. If supp f̂ ∈ [−T, T], f is a band-limited signal with the band-width T.

The fundamental of digital signal processing is the Shannon sampling formula

f(t) = ∑_{n=−∞}^{∞} f (\frac{nπ}{T} ) \frac{sin(T t − nπ)}{T t − nπ},

that allows to recover a band-limited signal with band-width T from its samples at equidistant points spaced \frac{π}{T} apart.

Now suppose that our receiver gets a signal from a unknown source. So no information about the band-width of the signal is known. Moreover, the sampling points are irregularly distributed. What conditions should we put on a set of sampling points so that we can recover any band-limited signal f?

A solution of this problem is based on the inverse spectral problem and Kramer’s theorem. The connection between the two is that given a sequence of points, satisfying certain distribution law, we can construct a singular Sturm-Liouville problem such that these points are precisely its eigenvalues. For this particular Sturm-Liouville problem, Kramer’s theorem then allows us to use the spectral theorem as a sampling formula.