Volume 18: Pages 63-80, 2005
Spectral Analysis Resolution and Study of the Uncertainty Relationship
M. T. Bliznetsov
Bazhenov Geophysical Expedition, Scheelite, Zarechny, Ekaterinburg 624250 Russia
An uncertainty relationship between the width of a pulse frequency spectrum and its time duration and extension in space is investigated. Analysis of the uncertainty relationship is carried out for causal pulses with minimum‐phase spectrum. Frequency spectra of elementary pulses are calculated using modified Fourier spectral analysis. Modification of the Fourier analysis is justified by the necessity of solving the paradox of ultralow frequencies in the Fourier spectra of unidirectional (nonoscillatory) pulses. It is established in the paper that the finite time of the measurement of signal spectra is the fundamental restriction for the determination of the frequency of harmonic components whose period is larger than the time of measurement. The occurrence of the physically nonobservable spectral components in Fourier spectra is caused by the varying resolution of Fourier analysis along the frequency axis. The modified Fourier spectral analysis has equal resolution along the frequency axis and excludes physically unobservable values from spectral and time concepts. It is proved that the Heaviside unit step function has an infinitely wide spectrum equal to one along the whole frequency range. The Dirac delta function has an infinitely wide spectrum within the scope of infinitely high frequencies. The difference between the propagations of wave and quasi‐wave forms of energy motion is established from the analysis of the uncertainty relationship. The velocity of propagation of unidirectional pulses, or shock waves, depends on the relative width of the pulse spectra. The oscillating pulse propagation velocity is constant in a given nondispersive medium. The idea of the elementary wavelet is introduced. The hypothesis of the elementary wavelet is tested by the example of elastic and electromagnetic waves.
Keywords: spectral analysis, resolution, unidirectional pulse, zero frequency, elementary causal pulses, uncertainty relationship
Received: April 1, 2004; Published online: December 15, 2008