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

Abstract

AbstractA 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.