Abstract
In the first paper of this series, we propose a multi-resolution theory of Fourier spectral estimates of finite duration signals. It is shown that multi-resolution capability, achieved without further observation, is obtained by constructing multi-resolution signals from the only observed finite duration signal. Achieved resolutions meet bounds of the uncertainty principle (Heisenberg inequality). In the forthcoming parts of this series, multi-resolution Fourier performances are observed, applied to short signals and extended to time-frequency analysis.
Highlights
IntroductionAnalyzing single realization of noisy short-time signals (short data records), multi-resolution and space-frequency or time-frequency approaches, estimating frequencies of multiple signals in noise, reduction of noise variance, recovery of missing parts of signals and so on, are important topics in many areas of sciences and industries (radar and sonar data processing, communications, geophysical and seismic exploration, biomedical engineering, non destructive testing, and so on)
Analyzing single realization of noisy short-time signals, multi-resolution and space-frequency or time-frequency approaches, estimating frequencies of multiple signals in noise, reduction of noise variance, recovery of missing parts of signals and so on, are important topics in many areas of sciences and industries
It is shown that multi-resolution capability, achieved without further observation, is obtained by constructing multi-resolution signals from the only observed finite duration signal
Summary
Analyzing single realization of noisy short-time signals (short data records), multi-resolution and space-frequency or time-frequency approaches, estimating frequencies of multiple signals in noise, reduction of noise variance, recovery of missing parts of signals and so on, are important topics in many areas of sciences and industries (radar and sonar data processing, communications, geophysical and seismic exploration, biomedical engineering, non destructive testing, and so on). Well known drawbacks of this approach are excessive sensitivity to observation noise (resolution varies as a function of the signal-to-noise ratio), important computation times with respect to FFT analysis, computational complexity [14,15,16], multiplicity of algorithms estimating model parameters and the necessity of subjective judgement in the selection of the order [17]. That devised various methods [23] to overcome drawbacks such as weak robustness to both modeling errors and the presence of a strong background noise add to computational complexity and requires high enough SNRs. An alternative non parametric approach for the resolution of mentioned problems is developed by wavelets. Resolution properties, contraction of spectral leakage, improvement of frequency estimation and recovering of missing parts of short signals are discussed and observed in the forthcoming parts of this series
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