tag:blogger.com,1999:blog-3115107570196929760.post7514730494660787701..comments2018-01-05T23:26:47.984-08:00Comments on Reflections: Phase-space optics 1: IntroductionShalin Mehtanoreply@blogger.comBlogger3125tag:blogger.com,1999:blog-3115107570196929760.post-67067117590312353072009-10-06T19:12:36.000-07:002009-10-06T19:12:36.000-07:00"Straightforward extrapolation of the uncerta..."Straightforward extrapolation of the uncertainty principle from quantum mechanics to signal analysis may suggest that one cannot measure the instantaneous spectrum with high temporal resolution. Though plausible, this is not the correct interpretation – for whatever the definition of instantaneous spectrum, one can compute it to arbitrary accuracy since the signal is deterministic."<br><br>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?<br><br>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.<br><br>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.S. Barwicknoreply@blogger.comtag:blogger.com,1999:blog-3115107570196929760.post-27294422974813784872009-10-07T19:22:48.000-07:002009-10-07T19:22:48.000-07:00I see your point. The standard interpretation of ...I see your point. The standard interpretation of the uncertainty principle seems to be that the inability to measure inverse quantities simultaneously is a property of the system itself, not the measurement device, which is not true for deterministic signals. By contrast, my analogy (not an exact one) pertains to how the measurement is mathematically made with a particular phase-space distribution. The connection is that both ultimately stem from the time-bandwidth product.<br><br>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.<br><br>Thanks for the enlightment and the blog.S. Barwicknoreply@blogger.comtag:blogger.com,1999:blog-3115107570196929760.post-37114105699835701132009-10-08T05:20:58.000-07:002009-10-08T05:20:58.000-07:00Thanks for thought-provoking comments. Without you...Thanks for thought-provoking comments. Without your comments, this issue wouldn't be so clear. I find Leon Cohen's book and papers very readable and insightful for such aspects.Shalin Mehtanoreply@blogger.com