Abstract
The prediction and correction of systematic errors in direct spectral estimation from irregularly sampled data taken from a stochastic process is investigated. Different sampling schemes are investigated, which lead to such an irregular sampling of the observed process. Both kinds of sampling schemes are considered, stochastic sampling with non-equidistant sampling intervals from a continuous distribution and, on the other hand, nominally equidistant sampling with missing individual samples yielding a discrete distribution of sampling intervals. For both distributions of sampling intervals, continuous and discrete, different sampling rules are investigated. On the one hand, purely random and independent sampling times are considered. This is given only in those cases, where the occurrence of one sample at a certain time has no influence on other samples in the sequence. This excludes any preferred delay intervals or external selection processes, which introduce correlations between the sampling instances. On the other hand, sampling schemes with interdependency and thus correlation between the individual sampling instances are investigated. This is given whenever the occurrence of one sample in any way influences further sampling instances, e.g., any recovery times after one instance, any preferences of sampling intervals including, e.g., sampling jitter or any external source with correlation influencing the validity of samples. A bias-free estimation of the spectral content of the observed random process from such irregularly sampled data is the goal of this investigation.
Highlights
Digital signal processing normally implies a time-limited, non-interrupted sequence of equidistant samples taken from a signal-generating process under investigation
Spectral estimation from randomly sampled signals in continuous time has been investigated in the past, mainly in the context of controlled sampling with induced variation of the sampling times, known as digital alias-free signal processing or sampling jitter [4,5,6,7], in the context of astrophysical observations [8,9,10,11] or in the context of laser Doppler data processing [12,13,14,15,16,17,18,19,20,21,22,23,24] including the specific role of processor dead times [25,26,27]
4 Conclusion Random sampling of time series causes a systematic error of the spectral estimation compared to the observed process
Summary
Digital signal processing normally implies a time-limited, non-interrupted sequence of equidistant samples taken from a signal-generating process under investigation. Spectral estimation from randomly sampled signals in continuous time has been investigated in the past, mainly in the context of controlled sampling with induced variation of the sampling times, known as digital alias-free signal processing or sampling jitter [4,5,6,7], in the context of astrophysical observations [8,9,10,11] or in the context of laser Doppler data processing [12,13,14,15,16,17,18,19,20,21,22,23,24] including the specific role of processor dead times [25,26,27] More general in their application are investigations in [28,29,30,31,32]. Correlated data gaps have been investigated only for very specific cases [31, 39,40,41,42], without options for generalization or without satisfactory bias correction
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