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Understanding Time-Frequency Analysis: A Deep Dive into Signal Processing

In this forum, delve into time-frequency analysis with Peep Kalv looking through astrophotographic plate (1964-65). Explore topics like multiperiodic processes, carrier fit splines, and Nyquist barrier implications. Learn about Fourier transforms, spectrum replication, and signal reconstruction, including examples on frequency limits and phase dispersion. Discover how bandlimited functions and derivatives play a crucial role in signal processing. Join the discussion with experts Rudy Schild, Sjur Refsdal, and Michael Berry for insights on advanced computational methods and applications. Don't miss out on this informative session in the realm of information and computer science!

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Understanding Time-Frequency Analysis: A Deep Dive into Signal Processing

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  1. Nyquist barrier - not for all!Jaan PeltTartu Observatory Monday, 7. October 2013 Information and computer science forum

  2. Peep Kalv looking through astrophotographic plate (1964-65).

  3. http://www.aai.ee/~pelt/ Ilkka Tuominen

  4. Gravitational lenses Rudy Schild and Sjur Refsdal in wild Estonia

  5. Four views • Time (AR, ARMA, etc) • Frequency (Power spectrum) • Time-Frequency (Wavelets, Wigner TF etc) • Phase dispersion

  6. Phase-process diagram (folding)

  7. Live demo

  8. Weights G are larger than zero when phases of two points in pair are similar, or: G=0 G=1

  9. How to compute?

  10. Multiperiodic processes

  11. An example ???

  12. Why?

  13. Carrier fit Splines Carrier frequency Function with sparse spectra.

  14. Harry Nyquist

  15. Comb function and its Fourier transform

  16. Fourier transform

  17. Sampling

  18. Spectrum replication

  19. Reconstruction

  20. Aliasing

  21. Simple harmonic, regular sampling

  22. Simple harmonic, irregular sampling

  23. Frequency to the right from Nyquist limit

  24. Here it is !

  25. From “Numerical Recipes”

  26. They tell us…

  27. Many possibilities • Some intervals are shorter (as Press et al). • Mean sampling step is to be computed. • Statistical argument, from N data points you can not get more than N/2 spectrum points. • Every time point set is a subset of some regular grid.

  28. Phases Observed magnitudes Phases s – frequency, P=1/s - period Arbitrary trial period (frequency) Correct period (frequency)

  29. Old story

  30. Typical “string length spectrum”

  31. Horse racing argument For “string length” method maximal return time is N! – number of permutations (N is number of data points). For other methods return time scales as NN. This comes from Poincare return theory.

  32. Noiseless case, simple power spectrum.

  33. 10% noise

  34. 25% noise

  35. Comments

  36. More comments

  37. Left from Nyquist limit Bandlimited process

  38. Michael Berry http://www.phy.bris.ac.uk/people/berry_mv/index.html http://michaelberryphysics.wordpress.com/

  39. Ohhh…, no….

  40. But still? Derivatives of bandlimited functions are also bandlimited! Look at red dots! Zeros are maxima and minima after differentiation.

  41. First hints

  42. Aharonov again

  43. Berry is more explicit

  44. Abstract

  45. Kempf is the best seller!

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