A Sampling Theorem for Time-Frequency Localized Signals

A sampling theorem for signals that are localized in both time and frequency is presented. For an arbitrary finite-energy signal the localization is performed by a time-frequency localization operator (filter) due to Daubechies. Two filter parameters a > 0, ß > 0 with αß > 1 control time-duration and bandwidth. The range of the filter is described as a reproducing kernel Hilbert space. Regular sampling is applied and a reconstruction formula is derived. The basic ingredient, the so-called sampling function, decays exponentially in time. For some filter output functions no perfect reconstruction is possible. The reconstruction error, however, is shown to vanish exponentially if the size of the sampling interval falls under a certain threshold. This threshold is estimated and related to the classical Nyquist interval. It is concluded that a finite number of sample values will allow a practically perfect reconstruction.