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Sampling Theorem in Time Domain for Infinite Duration Signal: Analytical Expression and Geometric Illustration
Authors
	Khanyan G.S.
Date of publication
	2020
DOI
	10.31114/2078-7707-2020-4-151-158
Abstract
	The work is devoted to proving the time domain sampling theorem for an infinite duration process occurring in a limited frequency band. The kernel of interpolation transform of a process’ elementary component – the harmonic signal of amplitude a, frequency f, and initial phase Phi – contains the sampling frequency F and a dimensionless parameter G – the frequency band index which determines, along with dimensionless frequency p = 2f/F, the conversion result: a biharmonic signal, both componential amplitudes of which take one of the values a, a/2, 0. This gives rise to the following mutually exclusive spectral phenomena (effects): validity of the theorem in a frequency band the lower cut-off frequency of which is not inferior to [G ]F/2 and the upper one does not exceed [G +1]F/2; frequency overlay; double attenuation and doubling or one-and-a-half-times signal amplification with or without the frequency and phase offset; complete signal disappearance. A geometric illustration of these phenomena shows that their zones (the set of points on (p,G) plane where a phenomenon takes place), each painted with its own color, consist of squares of unity-length diagonal or their sides and vertices covering the entire plane. When rotating the p and G axes of this plane by Pi/4 by introducing variables Ksi = (G+p+1)/2, Eta = (G–p+1)/2, called canonical, squares of size 1/2¬x1/2 are oriented along the coordinate axes, covering the plane (Ksi,Eta) in a “staggered” order so that the white fields are zones where the signal is purely harmonic, the black ones where it is biharmonic or disappears. A compara-tive analysis of both infinite and finite duration signals transform has shown that both theorem versions – infinite and finite – are special cases of each other. The representation of signal frequency and its variation range index in a form of peer canonical variables, by analogy with the equal status of generalized coordinates and momenta in the Hamilton function of a mechanical system, extends the theoretical significance of sampling theorem beyond its scope of applications for transmission of information, data compression, interpolation, etc., and makes it possible to formulate fundamental physical principles in the language of Fourier analysis statements.
Keywords
	harmonic signal, frequency band index, canonical varia-bles, summation of series.
Library reference
	Khanyan G.S. Sampling Theorem in Time Domain for Infinite Duration Signal: Analytical Expression and Geometric Illustration // Problems of Perspective Micro- and Nanoelectronic Systems Development - 2020. Issue 4. P. 151-158. doi:10.31114/2078-7707-2020-4-151-158
URL of paper
	http://www.mes-conference.ru/data/year2020/pdf/D069.pdf