Abstract
We investigate power spectra of a randomly sampled stationary stochastic signal, e.g., a spatial component of a turbulent velocity. We extend the methods of previous authors that basically assumed point or delta function sampling by including features characteristic of real measurement systems. We consider both the effect on the measured spectrum of a finite sampling time, i.e., a finite time during which the signal is acquired, and a finite dead time, that is a time in which the signal processor is busy evaluating a data point and therefore unable to measure a subsequent data point arriving within the dead time delay.
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Notes
Reduction of sample rate: Let the original sample rate with no dead time be ν. Then the number of samples in time t is n = νt. The probability of n samples in time t with mean number of samples \(\overline{n}\) (Poisson):
$$P(n) = \frac{e^{-n} \overline{n}^n}{n!}$$or with \(\overline{n} = \nu t\): \(P(n) = \frac{e^{-\nu t} (\nu t)^n}{n!}\). The probability that no event occurs in time \(\Updelta t_d\) is then:
$$P(0) = e^{-\nu \Updelta t_{\text{d}}}$$But P(0) is also the probability that the next sample will occur after \(\Updelta t_{\text{d}}\). Thus, the rate of samples occurring after \(\Updelta t_{\text{d}}\), the reduced sample rate ν0, is \(\nu_0 = \nu e^{- \nu \Updelta t_d}\).
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Buchhave, P., Velte, C.M. & George, W.K. The effect of dead time on randomly sampled power spectral estimates. Exp Fluids 55, 1680 (2014). https://doi.org/10.1007/s00348-014-1680-1
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DOI: https://doi.org/10.1007/s00348-014-1680-1