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Instruments and Systems: Monitoring, Control, and Diagnostics Annotation << Back
About the spectra of finite functions |
N.D. ZAKHAROV
Integral transformations of functions that are used in the frequency techniques for analyzing processes and systems are reduced to the
definition of prototypals with using improper integrals depending on parameter. In addition to possible difficulty of taking integrals there is
required also to take account of that the integrands must be absolutely integrated within the infinite limits. Increase in class of analytically
integrated functions by means of numerical methods which are implemented with up-to-date computer involves the newly arisen constraints
caused by the loss of accuracy due-to discretization and finiteness of integration interval. Functions that are forced determined in the finite
interval are usual named finite one (it would be better to name the finishing functions). Effect of finiteness may be very essential. The reason of this effect which is usual given in the corresponding sections of mathematical literature, namely, effect of single filter with the pulse length equal to integration interval is highly disputable. In this paper it is shown by analytical and numerical ways that for individual sinusoid the function of spectral density is the resonance curve and not delta-function that, as is generally known, takes the infinite value when tending the pulse duration to zero and not to infinity. Studied sinusoid may be considered the sole term of Fourier series. Paper presents the elementary transformations which make it possible to understand the meaning of Fourier series coefficients as well as the functions of
Kotelnikov series (count functions) on the basis of those his series is constructed.
Keywords: spectral analysis, frequency function of relative amplitudes, amplitude density, finite functions, limiting transitions of integral transformations, resonance curve, delta-function, spectrum of single pulse, count functions, Fourier series, Kotelnikov series.
Contacts: E-mail: zakharov@rtc.ciam.ru
Pp. 23-27. |
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