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

Peep Kalv looking through astrophotographic plate ( ).

Ilkka Tuominen

Gravitational lenses Rudy Schild and Sjur Refsdal in wild Estonia

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

Phase-process diagram (folding)

Live demo

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

How to compute?

Multiperiodic processes

An example ???

Why?

Carrier fit Carrier frequency Splines Function with sparse spectra.

Harry Nyquist

Comb function and its Fourier transform

Fourier transform

Sampling

Spectrum replication

Reconstruction

Aliasing

Simple harmonic, regular sampling

Simple harmonic, irregular sampling

Frequency to the right from Nyquist limit

Here it is !

From “Numerical Recipes”

They tell us…

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.

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

Old story

Typical “string length spectrum”

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.

Noiseless case, simple power spectrum.

10% noise

25% noise

Comments

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Left from Nyquist limit Bandlimited process

Michael Berry

Ohhh…, no….

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

First hints

Aharonov again

Berry is more explicit

Abstract

Kempf is the best seller!

Another example

Spectrum of it, no hint of SO-s

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

Antenna beamforming

But sparse and random array?

Transplanckian frequencies

Superoscillating particles

And finally… Where are the super-oscillations here?

Gateway to superoscillations: PROFESSOR SIR MICHAEL VICTOR BERRY, FRS