Sampling and reconstruction of non-bandlimited signals using Slepian functions

In this paper, we show that the Whittaker-Shannon (WS) sampling theory can be modified for the reconstruction of non-bandlimited signals. According to the uncertainty principle, non-bandlimited signals have finite time support and thus are more common in practical application. Prolate spheroidal wave functions also called Slepian functions have finite time support and maximum energy concentration within a given bandwidth, so instead of infinite length sinc functions, we consider Slepian functions. We show that by projecting non-bandlimited signals onto the space represented by an orthonormal Slepian basis the minimum sampling rate can be reduced nearly by half, with no aliasing. Moreover, the reconstruction error is much lower than the one obtained by the WS theory. In some cases, depending on the desired reconstruction accuracy, it is possible to lower the rate even further. Simulations show the efficiency of the Slepian functions in the reconstruction of uniformly or non-uniformly sampled bandlimited or non-bandlimited signals.