Signal reconstruction from the undersampled signal samples

It is well-known from the celebrated Shannon sampling theorem for bandlimited signals that if the sampling rate is below the Nyquist rate, aliasing takes place and the original signal cannot be reconstructed back by simply passing the signal samples through an ideal lowpass filter. However, researchers such as Stern and Gori have shown the existence of some classes of signals for which the signals are sampled below the Nyquist rate but perfect signal reconstruction is still possible from the given signal samples. Here, we present a generalized lowpass sampling theorem and show that Stern’s and Gori’s lowpass sampling theorems are special cases of it. A sampling theorem for the bandpass signals in the linear canonical transform domains is also presented and its special cases are discussed. Using a modification of the conventional natural sampling waveform with a specific width of the pulses, it is shown that the sampling rate in our generalized lowpass sampling theorem and hence in the Stern’s and the classical Shannon sampling theorems can be further reduced by a factor of two, while for the bandpass signals, the reduction in the sampling rate by some factor is possible only under some restricted conditions.