Analysis of frequency spectrum and the Fourier series convergence of the unit impulse train

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Journal of Vibration and Shock ›› 2019, Vol. 38 ›› Issue (4) : 15-19.

DU Feng 1 TANG Lan 2

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Abstract

According to the shift theorem in the time and frequency domain,the unit impulse train has two frequency spectrum functions which were called the periodic spectrum and the series spectrum.The periodic spectrum is not rigorous for the lack of the analysis of the Fourier series convergence.According to the definition of the Dirac function,it proved that the series spectrum is in fact an impulse train in the frequency domain whose energy and period are equal to the angular frequency value.The sampling function characteristic,the limit and integral theory were made full use of.It shows that the Fourier series of the unit impulse train is convergent.The convergence was verified again by the analyses about the difference between the series spectrum and the Fourier series of the periodic spectrum.It failed to verify the miss of the Gibbs phenomenon.

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unit impulse train / frequency spectrum / Fourier series convergence / Gibbs phenomenon

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DU Feng 1 TANG Lan 2. Analysis of frequency spectrum and the Fourier series convergence of the unit impulse train[J]. Journal of Vibration and Shock, 2019, 38(4): 15-19

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