Signal recovery with a limited number of ideal discrete samples
Vladimir Ivanovich Guzhov, Ekaterina E. Trubilina, I. O. Marchenko · Science Bulletin of the Novosibirsk State Technical University · 2020
The article discusses the restoration of continuous signals from a limited set of discrete values.The spectrum of discrete values is a periodically repeating spectrum of a continuous signal.Kotelnikov's theorem determines the conditions under which the spectrum of a single signal can be distinguished without distortion, and then an original signal can be restored from them.However, interpolation using a series implies an infinite number of samples.The article shows that when the Nyquist conditions are fulfilled, signal restoration with a limited number of discrete values using the interpolation series gives significant errors.However, if the signal is periodic, it is possible to completely restore the signal from a limited set of sampling points.In this case, the discretization process can be represented as the action of a limited Dirac lattice.A simple procedure is proposed for reconstructing the original signal which consists of a discrete Fourier transform from the samples, complementing the spectrum with zeros to the range of the selected signal, and the inverse Fourier transform.For a non-periodic signal, complete reconstruction from discrete points does not occur, even if the conditions of the Kotelnikov theorem are satisfied when the initial function is discretized.This is due to the overlapping of single spectra as a result of convolution with the Fourier transform of a rectangular pulse which cuts out a non-periodic signal from a periodic one.Each single spectrum becomes infinite as a result of the convolution with the Fourier transform of the cutting rectangular pulse.It is not possible to precisely single out a single spectrum of a continuous signal.This leads to an error in the restoration of the original signal.The error can be reduced by reducing the sampling interval or by interpolating the edges of the selected single pulse.