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A New Aspect of Transition Between Line and Continuous Spectrum and Its Relation to Seismic in Uence on Structures

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A New Aspect of Transition Between Line and Continuous Spectrum and Its Relation to Seismic in Uence on Structures Scientia Iranica D (2020) 27(3), 1515{1524 Sharif University of Technology Scientia Iranica Transactions D: Computer Science & Engineering and Electrical Engineering http://scientiairanica.sharif.edu A new aspect of transition between line and continuous spectrum and its relation to seismic in uence on structures R. Babica, L. Babicb, and B. Jaksica; a. Department of Electrical and Computing Engineering, Faculty of Technical Sciences, University of Pristina, Knjaza Milosa Street, No. 7, Kosovska Mitrovica, P.O. Box 38220, Serbia. b. Department of Civil Engineering, Faculty of Technical Sciences, University of Pristina, Knjaza Milosa Street, No. 7, Kosovska Mitrovica, P.O. Box 38220, Serbia. Received 4 August 2017; received in revised form 8 November 2017; accepted 23 April 2018 KEYWORDS Abstract. This study considers a new view of the transition from periodic to aperiodic signals in time and spectral domains, thus pointing out the idea of how the concept of Line spectrum; in nity is involved. The results of this paper contribute to a better understanding of the Fourier analysis; nature of both spectral descriptions and conditions of their practical use, particularly in Spectrum invariance; unusual cases. Therefore, this study highlights invariance of convergence of spectrum by Seismic signal; introducing some numerical parameters, which exactly describe such a process. Their Structure dynamics. behavior is numerically examined in detail. In addition, the opposite transition from aperiodic to periodic is considered to clarify the meaning of the spectral line. To suggest the applicability of our analysis, an actual seismic signal is used. By extracting the most prominent waveform part, regarding its in uence on structures, a periodic signal is formed whose line spectrum can clearly show possible resonance with the vibrating tones of structures. © 2020 Sharif University of Technology. All rights reserved. 1. Introduction Although an entity is placed between periodic signal and its function, there are indeed di erent It is well known that the Fourier Series (FS) calculates entities. The rst entity is composed of pure numerical the line spectrum of a periodic function instead of a values, while the other one expresses the values of some periodic signal, since the latter cannot simply exist in physical quantities. As for the FS, it acts only in the strict sense of the word, i.e., to be periodic over the accordance with the given shape ([1], pp. 34{40). There whole t domain, according to the FS de nition. The are those who assign physical meaning to mathematical periodic function is a mathematical representation of shapes and interpret the nature of the entity before and a periodic signal, which is, by itself, a version of the after it is subjected to the FS. \periodic", or real, signal that is periodized to in nity. The term \line" stands for its discrete nature, originating from a suitable graphical representation. *. Corresponding author. Tel.: +38 128425320 This term is of importance to our theoretical consid- Fax: +38 128425322 erations. E-mail addresses: [email protected] (R. Babic); [email protected] (L. Babic);[email protected] Nevertheless, in scienti c research and engineer- (B. Jaksic) ing practice, one often use line spectral representa- tion of \periodic" signals because of its convenience doi: 10.24200/sci.2018.20326 in describing the actual issue. However, always, it 1516 R. Babicet al./Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 1515{1524 should be noted how much a \periodic" signal di ers which covers the in nite ! domain and is readily seen from its aperiodic (theoretical) counterpart in order to if Eq. (1) is rewritten in the following: appropriately explain the actual results. (kZ+1)T Generally, signals that are not strictly periodic 1 S = S(n) = S(n! ) = s(t)ei(n!0)tdt; should be regarded as aperiodic ones, which, depending n 0 T on their duration, are further divided into impulses and kT nite realizations of some processes (random, periodic, etc.). Whatever the actual duration of signal/function n = 0; 1; 2; ; may be, it is always formally extended by zero to in nity to cover the whole t-axis/domain. For theo- k = 0; 1; 2; ; (2) retical purposes, nite random realisations are often where it is observed how the FS maps the whole t generalized to cover t 2 (1; +1) or t 2 (1; 0) [2]. domain to the whole ! domain. Moreover, even a strict periodic signal can be formally Expression (2) gives an implication of t domain treated as an aperiodic one if considered as a whole, in nity such that the same operation is carried out not as a sequence of repeating waveforms ([1], pp. 119{ in each period throughout t domain, i.e., one actually 133). deals with a single period, yet treats it like dealing Herein, a practical rule of thumb is proposed on with all others in the same manner. Therefore, possible how to distinguish impulses from long-lasting aperiodic mistakes and errors are predicted while calculating signals. It is based on the number of changes, i.e., signal's spectrum. If not, the computer program does uctuations, in the signal duration. That is, an not know how many periods there are. aperiodic signal may last long, yet can be considered For a time process (time variations of some physi- as an impulse since it slowly uctuates and comprises cal quantity), described by signal (t), which is of nite several changes. duration: Re ning this rule, i.e., how much changes can di erentiate impulses from the lasting aperiodic sig- 9 (t) 8t 2 (t1; t2) ; (3) nals, calls for a deeper investigation; however, herein, its continuous spectrum is determined by the Fourier this study considers acoustic, or much better seismic, Transform (FT), encompassing the whole t domain: waveforms to nd a clue for such a recommendation. Aperiodic signals are represented in the frequency Z1 Z1 domain by a continuous spectrum. The nature of S (!) = s(t)ei! tdt i:e: []dt: (4) the spectral density is important to better interpret 1 1 research results. The very transition between those two spectral As mentioned before, prior to applying the FT to entities, line and continuous, if properly considered, (t), it must be expressed by function s(t), extending may give a deeper insight and improve one's under- its de nition domain to the whole t-axis as in the standing and even broaden the scope of waveform anal- following: ysis. Seismogram analysts and station operators should 9s (t) 8t 2 (1; +1) ; (5a) possess the capability to \understand each wiggle" in a seismic record [3], especially in unusual cases. The (t) s (t) 8t 2 (t1; t2) ; (5b) same holds for some researchers and engineers that encounter seismic factors in their work. It is implied ensuring that there is no other signal on the rest of t- here that they should have clari ed the notions of all axis. This is not hair-splitting process as it may seem, aspects of the seismogram, including its treatment by yet the need to follow fundamental issues to prevent Fourier tools. The FS calculates the spectrum of a principal mistakes is implied. periodic function (not of a periodic signal) since such Another in nite domain, frequency, is hidden in a signal cannot simply exist in the strict sense of the the FT expression: word, i.e., to be periodic over the whole t domain. Z1 For a periodic time function s(t), its spectral S (!) = s(t)ei! tdt (Fourier) coecients are easily calculated as follows: 1 ZT=2 1 1 < t < 1 1 < ! < 1: (6) S = S(n) = S(n! ) = s(t)ei(n!0)tdt; n 0 T The result of the performance of the FT on s(t) is a T=2 continuous spectrum S(!). In the following, the relationship between these n = 0; 1; 2; ; (1) two spectral entities is examined. R. Babicet al./Scientia Iranica, Transactions D: Computer Science & ... 27 (2020) 1515{1524 1517 2. The essence of the relationship between FS sP (t) ! sA(t): (9) and FT T !1 Considering that spectral analysis, e.g., Yarlagadda [4], Making an aperiodic signal (function) s (t) from its pe- A is well and widely known throughout classical litera- riodic counterpart s (t) is seemingly simple (Figure 1). P ture, let us follow this limit process by formulae. Let us assume: sA(t) = sP (t) 0 t < T: (7) Considering Eqs. (1) and (8), we write: In other words: " Z # X1 1 T +1 in!0x in!0t X sP (t) = sP (x)e dx e s (t) = SP ein!0t T P n n=1 0 n=1 " # X1 Z T +1 1 X = s (x)ein!0xdx ein!0t! = S + 2 SP cos[n! t + (!)] 2 P 0 0 n 0 n=1 0 n=1 0 1 Z1 Zt2 = s (t) 0 < t < T; (8) 1 A ! ei!t @ s(x)ei!tdxA d! T !1 2 where all harmonics outside the interval [0,T ] are cut n! !! 0 1 t1 o . This operation in t domain causes drastic mathe- !0!d! matical disruption since it disintegrates the nature of Z1 1 trigonometric function (Figure 2) by contradicting the = S(!)ei!td! = s (t): (10) very de nition of FS based directly on the space of 2 A trigonometric functions, which are inherently in nite, 1 covering the whole t domain ([1], pp. 87{94). Herein, pulse de nition interval (t1; t2) to (1; +1) To nd a mathematically proper way to relate pe- is simply extended. Now, the formal analogy in these riodic and aperiodic functions/signals, the limit process two expressions is highlighted as follows: is addressed: X1 X1 P in!0t P i[n!0t+n] sP (t)= Sn e = Sn e ; (11) n=1 n=1 and: Z1 1 i!t Figure 1.
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