11 May 2014

The Carvallo paradox - part II

I mentioned the Carvallo paradox in a previous post. Here, I will give a simpler version and some comments. Consider that, instead of using a spectrometer, we only use the dispersive element (e.g. a prism). After refraction, the signal components (the various colors) are projected onto a perfectly absorbing screen:
From Wikimedia; under CC-SA 1.0 License.

Even though the incoming signal \(s(t)\) is of finite length (and thus energy), each component (of infinite length) \(f_i(t)\), where \(s(t)=\sum_i f_i(t)\), has finite and constant power, meaning that over a period of time larger than the duration of the original signal the screen will receive a higher amount of energy (which can in fact be made arbitrarily large). The Carvallo paradox can then be restated as follows:

1) A finite-length signal is a sum of infinite-length components,
2) which can be separated and manipulated individually.

There is obviously a problem with 1) or 2) (or both!), since accepting them would break both causality and energy conservation. Note that invoking quantum mechanics does not solve the paradox: we can envision a similar setup using for instance sound waves.

Whether 2) is valid or not, the "separation" step is non-trivial, as one can see from the power spectrum of the signal: \(P(\omega) = |\tilde{S}(\omega)|^2 = \left | \sum_i \tilde{F}(\omega)\right |^2\), where the uppercase and tilde combination denote the Fourier transform. On the other hand, the power spectrum after separation (e.g. that absorbed by the screen) is: \(P'(\omega) = \sum_i  \left | \tilde{F}(\omega)\right |^2\). What I called separation thus amounts to decorrelating the various signal components.

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