1. Nyquist barrier - not for all! Jaan Pelt Tartu 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?

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

18. Harry Nyquist

19. Comb function and its Fourier transform

20. Fourier transform

21. Sampling

22. Spectrum replication

23. Reconstruction

24. Aliasing

25. Simple harmonic, regular sampling

26. Simple harmonic, irregular sampling

27. Frequency to the right from Nyquist limit

28. Here it is !

29. From “Numerical Recipes”

30. They tell us…

31. 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.

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

33. Old story

34. Typical “string length spectrum”

35. 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 N N. This comes from Poincare return theory.

36. Noiseless case, simple power spectrum.

37. 10% noise

38. 25% noise

39. Comments

40. More comments

41. Left from Nyquist limit Bandlimited process

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

43. Ohhh…, no….

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

46. First hints

47. Aharonov again

48. Berry is more explicit

49. Abstract

50. Kempf is the best seller!

51. Another example

52. Spectrum of it, no hint of SO-s

53. Research programme? 1. Super-resolution using super-oscillations. Already done – using nanohole patterns

54. Antenna beamforming

55. But sparse and random array?

56. Transplanckian frequencies

57. Superoscillating particles

58. And finally… Where are the super-oscillations here?

59. Gateway to superoscillations: PROFESSOR SIR MICHAEL VICTOR BERRY, FRS http://michaelberryphysics.wordpress.com/