Features of Limited Duration Harmonic Signal Transform by Sampling Theorem
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 the conditions of the theorem validity and the peculiar effects of its violation follow naturally. It can be argued that these interesting effects are hardly covered in literature, and the aim of the work is to fill this gap. The harmonic signal s (t) = a cos(2πft+φ) 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(tk ) is calculated, which is a convolution of s (tk) sequence with Dirichlet kernel (used in the Fourier series theory), which depends on continuous time t, length of digital realization N, and frequency band index G - a parameter generalizing classical statement 0 ≤ f Iow uplow and Fup depending on G. The main theoretical question is, under what restrictions on both signal and kernel parameters this transform 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 fT) is the expansion of the function s'(t) into an infinite series (with summation index I ) of transformed signal copies (branches) s1 (t), shifted in frequency by IF and in phase by -2πFt0 - each in its own frequency band, the boundaries 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 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 (complete) 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) = 2s (t) - the selfimposition of the branch I = 0). There is no frequency overlay for integer values of G. The branch of number I may exist in a pure form (and not only for integer G) if the signal’s offset frequency has satisfied another inequality. Actually the sampling theorem is satisfied only in this case, for zero branch only: s'(t) =s 0(t ) = s ( t). In the rest nodes on parameters’ plane (G, fT ) a very interesting phenomenon takes place - complete disappearance of the signal after its conversion: s’(t) = 0. Sampling theorem study should be continued in direction of deepening the Fourier analysis theory and searching for relevant applications while developing modern electronics devices (signal detecting, interpolation, etc.).