Abstract
As it was already shown in Sect. 4.2, not all systems can be described by piecewise bijective functions satisfying Definition 2.1. We therefore extend also our measures of relative information loss to stationary stochastic processes. This endeavour is by no means easy, as there are several possible ways for such an extension, some of which we hint at in Sect. 8.3. The most accessible one will be discussed in the first section of this chapter, and we will apply it to multirate systems commonly used in signal processing in Sect. 8.2.
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Notes
- 1.
The introduction of [TG95] gives another example of the common misconception of energy as information. The authors claim: “The filter bank is designed so that most of the signal’s information is optimally concentrated in the first few channels, by employing an appropriate ‘energy compaction’ criterion” (emphasis added).
- 2.
The Paley-Wiener condition is also important for the PSD of stationary processes. If the PSD \(S_{X}(\mathrm {e}^{\jmath \theta })\) violates the Paley-Wiener condition, according to Martin Schetzen [Sch03, p. 169]: “[The future of the signal violating]
$$ \frac{1}{2\pi }\int _{-\pi }^\pi \log S_{X}(\mathrm {e}^{\jmath \theta }) \mathrm {d}\theta > -\infty $$can be completely determined from its own past with arbitrarily small error [...] If the future of your speech waveform were predictable with arbitrary small error, then all that you will say in the future is predetermined and you would not be able to change it. Thus your free will would be definitely limited.” Schetzen thus concludes that the PSD of our speech waveform cannot be nonzero in any band.
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Geiger, B.C., Kubin, G. (2018). Dimensionality-Reducing Functions. In: Information Loss in Deterministic Signal Processing Systems. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-59533-7_8
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DOI: https://doi.org/10.1007/978-3-319-59533-7_8
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