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Features of Limited Duration Harmonic Signal Transform by Sampling Theorem
Authors
	Khanyan G.S.
Date of publication
	2016
Abstract
	For more than eight decades the sampling theorem has remained the subject of interest in the field of information theory and its applications. The author’s research in this area is distinguished by a particular approach to the disclosure of mathematical aesthetics of Fourier analysis. The theorem proof method used is direct – a digital realization of harmonic signal is substituted into interpolation formula, and in the process of analyzing the conversion result obtained conditions of theorem applicability and peculiar effects of its violation follow maturally. It can be argued that these effects, surely interesting from the point of view both of pure mathematics and possible scope of applications in microelectronics hardly covered in literature, and the aim of the work is to fill this gap. The harmonic signal s (t) = a cos(2f t+) of limited duration T is sampled with frequency F, so that its digital realization is a sequence of N = FT equidistant readings s (tk) obtained at instants tk = t0+k/F; k =0,1,…, N–1 with an arbitrary start time t0. The integral transform s(t ) is calculated, which is a convolution of s (tk) sequence with Dirichlet kernel (the prototype of Dirac -function serving for representation of partial sums of Fourier series), which depends on continuous time t, length of digital realization N, and frequency band index G – a parameter generalizing classical statement 0  f The main theoretical question is, under what re-strictions on both signal and kernel parameters this trans-form is an identity: s(t ) = s (t). The transform result (computable only for quantized with a step of 1/N values of index G, and for integer or half-integer values of signal dimensionless frequency f T ) is the expansion of the function s(t) into an infinite series (with summation index I ) of transformed signal copies (branches) sI (t ), shifted in frequency by IF and in phase by ¬ 2IFt0 – each in its own frequency band, the bounda-ries of which depend on G in a manner, that no more than two such domains may intersect. In the case of intersection (frequencies overlay), the terms of which are set by double inequality, the transform result is a biharmonic signal (with a certain frequency difference and phase shift of its components). The overlay area size depends on the fractional part of index G. The widest of possible (the full) overlay area is achieved at half-integer values of G (at the critical index G = -1/2 this overlay turns into the signal double gaining: s(t) =2 s ( t) – the self-imposition of the branch I = 0). There is no frequency overlay for integer values of G. A branch of number I may exist in pure form (and not only for integer G) if the signal’s offset frequency satisfies another system of inequalities. Actually the sampling the-orem is satisfied only in this case, for zeroth branch only: s(t) =s0 (t ) = s ( t). In the rest nodes on parameters’ plane (G, f T ) a very interesting phenomenon takes place – the complete disap-pearance of the signal after its conversion: s(t) = 0. Sampling theorem study should be continued in direc-tion of deepening the Fourier analysis theory and search-ing for relevant applications while developing modern electronics devices (signal detecting, interpolation, etc.).
Keywords
	sampling theorem, harmonic signal, frequency band index, logical delta-function, overlapping of branches.
Library reference
	Khanyan G.S. Features of Limited Duration Harmonic Signal Transform by Sampling Theorem // Problems of Perspective Micro- and Nanoelectronic Systems Development - 2016. Proceedings / edited by A. Stempkovsky, Moscow, IPPM RAS, 2016. Part 1. P. 210-217.
URL of paper
	http://www.mes-conference.ru/data/year2016/pdf/D041.pdf