Nyquist barrier - not for all!

Jaan Pelt

Tartu Observatory

Monday, 7. October 2013

Information and computer science forum

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

http://www.aai.ee/~pelt/

Ilkka Tuominen

Gravitational lenses

Rudy Schild and Sjur Refsdalin 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 pairare similar, or:

G=0

G=1

How to compute?

Multiperiodic processes

An example

???

Why?

Carrier fitCarrier 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 NN.

This comes from Poincare return theory.

Noiseless case, simple power spectrum.

10% noise

25% noise

Comments

More comments

Left from Nyquist limit

Bandlimited process

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

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

http://michaelberryphysics.wordpress.com/