A sampling theorem for non-bandlimited signals using generalized Sinc functions

A ladder shaped filter of two real parameters a1,a2∈(−1,1)a1,a2∈(−1,1) is introduced in this note. The impulse response of the corresponding Linear Time Invariant (LTI) system is a generalized Sinc function of two parameters. Consequently a generalized Shannon-type sampling theorem is established for a class of non-bandlimited signals with special spectrum properties associated with a ladder shaped filter of two parameters. Finally, a mathematical characterization for the class of non-bandlimited signals satisfying the generalized sampling theorem is offered. These signals are restrictions to the real line of certain analytic functions in stripped domains symmetric about the real axis in the complex plane. For these signals, their spectra in higher frequency bands are measured by the spectrum of their base bands.