Periodically nonuniform sampling of bandpass signals

It is known that a continuous time signal x(i) with Fourier transform X(ν) band-limited to |ν|<Θ/2 can be reconstructed from its samples x(T₀n) with T₀=2π/Θ. In the case that X(ν) consists of two bands and is band-limited to ν₀<|ν|<ν₀ +Θ/2, successful reconstruction of x(t) from x(T₀n) requires an additional condition on the band positions. When the two bands are not located properly, Kohlenberg showed that we can use two sets of uniform samples, x(2T₀n) and x(2T₀n+d₁), with average sampling period T₀, to recover x(t). Because two sets of uniform samples are employed, this sampling scheme is called Periodically Nonuniform Sampling of second order [PNS(2)]. In this paper, we show that PNS(2) can be generalized and applied to a wider class. Also, Periodically Nonuniform Sampling of Lth-order [PNS(L)] will be developed and used to recover a broader class of band-limited signal. Further generalizations will be made to the two-dimensional case and discrete time case