Relations between the possibility of restoration of bandpass‐type band‐limited waves by interpolation and arrangement of sampling points

Abstract

A study has been conducted on band‐limited waves f(t) of bandpass type, the Fourier spectrum F(ω) of which is identically zero outside an occupied band ωτ < ∥ω∥ < ωτ + ωb. Based on this study, we have obtained the following results: first, when an interpolation of a wave is attempted using a set of even‐numbered samples periodically repeated on both sides of the time axis at a mean distance between samples π/ωb (which in general are arranged ununiformly), the necessary and sufficient condition for said samples to restore a desired band‐limited wave having the same occupied bandwidth is demonstrated; second, a concrete formula of interpolation using a set of the prescribed type that satisfies the necessary and sufficient condition above, is introduced. Then a proof that demonstrates the impossibility that a formula which permits a set of samples of an odd number is periodically repeated at a mean interval of π/ωb to restore any desired band‐limited wave having the prescribed occupied bandwidth on the basis of interpolation unless ωτ or ωb satisfies a special condition. Finally, relations between the possibility of interpolation and combinations of ωτ and ωb for sample positions restricted to a certain extent if the mean interval between sampling points is shorter than π/ωb, is discussed.

Kernel-induced sampling theorem for bandpass signals with uniform sampling

“…as mentioned in [1,2] for instance. Although nonuniform sampling scheme has been making great progress in the theory of bandpass sampling [3,2], uniform sampling is still important from a practical point of view. Uniform sampling rate for a given band position achieving perfect reconstruction was thoroughly investigated in [2], and the same result was obtained by a more convincing way in [1].…”