We discuss the statistical properties of a single-trajectory power spectral density S ( ω , T ) of an arbitrary one-dimensional real-valued centered Gaussian process X(t), where ω is the angular frequency and T the observation time. We derive a double-sided inequality for its noise-to-signal ratio and obtain the full probability density function of S ( ω , T ) . Our findings imply that the fluctuations of S ( ω , T ) exceed its average value μ ( ω , T ) . This implies that using μ ( ω , T ) to describe the behavior of these processes can be problematic. We finally evaluate the typical behavior of S ( ω , T ) and find that it deviates markedly from the average μ ( ω , T ) in most cases.
Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes / Squarcini, A.; Marinari, E.; Oshanin, G.; Peliti, L.; Rondoni, L.. - In: JOURNAL OF PHYSICS. A, MATHEMATICAL AND THEORETICAL. - ISSN 1751-8113. - STAMPA. - 55:405001(2022), pp. 1-10. [10.1088/1751-8121/ac8cc0]
Noise-to-signal ratio of single-trajectory spectral densities in centered Gaussian processes
Peliti L.;Rondoni L.
2022
Abstract
We discuss the statistical properties of a single-trajectory power spectral density S ( ω , T ) of an arbitrary one-dimensional real-valued centered Gaussian process X(t), where ω is the angular frequency and T the observation time. We derive a double-sided inequality for its noise-to-signal ratio and obtain the full probability density function of S ( ω , T ) . Our findings imply that the fluctuations of S ( ω , T ) exceed its average value μ ( ω , T ) . This implies that using μ ( ω , T ) to describe the behavior of these processes can be problematic. We finally evaluate the typical behavior of S ( ω , T ) and find that it deviates markedly from the average μ ( ω , T ) in most cases.File | Dimensione | Formato | |
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https://hdl.handle.net/11583/2974037