As a sidebar, the correct interpretaion of measurement in QM seems to be disputed, though that discussion is beyond my pay grade. I believe Roger Penrose has a new book coming out that challenges some current orthodoxy. See a recent interview in Discover. Also, I’m wondering how the points on deterministic signals apply to fractals. But that’s a different topic.

Thanks for the enlightment and the blog.

The analogy to quantum may be more apt than you indicate. Mind you, I’m not certain, but here is my interpretation. The uncertainty principle implies that certain properties cannot be measured simultaneously to arbitrary accuracy, not they can’t be measured to arbitrary accuracy individually by separate probes. Right?

I think that as far as signal analysis is concerned, one can measure both time features and frequency features to better accuracy than uncertainty principle. I am not familiar with quantum mechanical methods of measuring position and momentum simultaneously, so find it hard to visualize what aspect limits the resolution of measurement. The limit on signal measurement will be imposed by capabilities of the measurement instrument. Following example comes to mind: Capture a signal at very high sampling-rate using the best A2D converter and then compute a continuous wavelet transform or continuous wigner distribution in the other dimension. However, if we compare the time-width and band-width from this measurement and analysis, we will find that their product cannot be smaller than uncertainty limit. Thus, the uncertainty principle limits the time-bandwidth product of a signal rather than its measurement. One may think of making the above distinction as pedantic but it seems an important conceptual difference to me.

Also, all squared phase-space decompositions like a wavelet scalogram can be shown to be a phase-space averaging of the Wigner-Ville with a smoothing kernel. In support of your interpretation the WVD does, however, satisfy phase and space marginal integrals (integrating wrt phase gives the square of the spatial distribution and vice versa), but the WVD is not strictly positive due to interference terms. In fact, it has been shown that no positive quadratic energy distribution can satisfy the marginal integrals. The required smoothing that reduces phase-space resolution seems to bear some resemblance to the uncertainty principle in QM.

This is a very interesting point, viz., all physically meaningful distributions are related to Wigner distribution with a smoothing kernel. As an example, we find that the phase-space representation of partially coherent imaging systems is a convolution of the specimen’s Wigner distribution with a kernel dependent on the imaging system’s parameters (illumination and imaging pupils). This is the topic of my talk on Monday at FiO. I think of this situation as implying that signal obtained from a physical system has larger or at best the same time-bandwidth product as obtained with Wigner distribution.

The analogy to quantum may be more apt than you indicate. Mind you, I’m not certain, but here is my interpretation. The uncertainty principle implies that certain properties cannot be measured simultaneously to arbitrary accuracy, not they can’t be measured to arbitrary accuracy individually by separate probes. Right?

Wavelets are a type of phase-space distribution, for example. Individual wavelet atoms are generally interpreted as covering a certain panel in the phase-space plane (1-D signals). A wavelet that is localized in space and, thus, provides precise spatial information, will have a broad spectrum and vice versa. Thus, the chosen phase-space probe will in fact be subject to a restriction akin to the uncertainty principle.

Also, all squared phase-space decompositions like a wavelet scalogram can be shown to be a phase-space averaging of the Wigner-Ville with a smoothing kernel. In support of your interpretation the WVD does, however, satisfy phase and space marginal integrals (integrating wrt phase gives the square of the spatial distribution and vice versa), but the WVD is not strictly positive due to interference terms. In fact, it has been shown that no positive quadratic energy distribution can satisfy the marginal integrals. The required smoothing that reduces phase-space resolution seems to bear some resemblance to the uncertainty principle in QM.