Delay sampling theorem: A criterion for the recovery of multitone signal

Periodic nonuniform sampling (PNS) is widely used in sub-Nyquist sampling schemes of multitone signals. However, what sampling pattern of PNS enables the signal to recover from the sub-Nyquist samples? Although this problem has been discussed in different schemes, the corresponding answers vary greatly. In this paper, we introduce the recurrent delay times to describe a PNS and provide a mathematical definition of signal recoverability from the perspective of delay. On this basis, we review the well-known Shannon–Nyquist sampling theorem and suggest a delay sampling theorem (DST) that a multitone signal bandlimited to [0,f_max] can be perfectly reconstructed from its periodic nonuniform samples when its minimum combined delay time τ_min is not more than the Nyquist interval 1/2f_max. To explain and demonstrate the DST, subspace theory-based algorithms and active aliasing and de-aliasing algorithm (AADA) are proposed and used to recover the spectrum or waveform of multitone signals. The high consistency between practical and theoretical results evidently verifies the correctness of DST. In terms of sampling and recovery of multitone signal, DST extends the Shannon–Nyquist sampling theorem and naturally unifies several existing sampling strategies from the perspective of delay. DST provides a new guideline for the sub-Nyquist sampling of multitone signals. According to the DST criterion, we can optimize the sampling patterns to reduce cosets and improve the supremum frequency such that the reconstructed spectrum is alias-free. By generalizing the sampling theorem from the view of delay, DST opens many avenues for the development of multitone signal processing.