nyquist barrier - not for all! jaan pelt tartu observatory monday, 7. october 2013 information and...
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Nyquist barrier - not for all!
Jaan Pelt
Tartu Observatory
Monday, 7. October 2013
Information and computer science forum
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Peep Kalv looking through astrophotographic plate (1964-65).
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http://www.aai.ee/~pelt/
Ilkka Tuominen
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Gravitational lenses
Rudy Schild and Sjur Refsdalin wild Estonia
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Four views
• Time (AR, ARMA, etc)
• Frequency (Power spectrum)
• Time-Frequency (Wavelets, Wigner TF etc)
• Phase dispersion
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Phase-process diagram (folding)
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Live demo
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Weights G are larger than zero when phases of two points in pairare similar, or:
G=0
G=1
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How to compute?
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Multiperiodic processes
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An example
???
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Why?
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Carrier fitCarrier frequency
Splines
Function with sparse spectra.
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Harry Nyquist
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Comb function and its Fourier transform
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Fourier transform
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Sampling
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Spectrum replication
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Reconstruction
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Aliasing
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Simple harmonic, regular sampling
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Simple harmonic, irregular sampling
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Frequency to the right from Nyquist limit
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Here it is !
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From “Numerical Recipes”
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They tell us…
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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.
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Phases
Arbitrary trial period (frequency) Correct period (frequency)
Observed magnitudes
Phases
s – frequency, P=1/s - period
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Old story
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Typical “string length spectrum”
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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.
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Noiseless case, simple power spectrum.
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10% noise
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25% noise
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Comments
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More comments
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Left from Nyquist limit
Bandlimited process
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Michael Berry http://www.phy.bris.ac.uk/people/berry_mv/index.htmlhttp://michaelberryphysics.wordpress.com/
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Ohhh…, no….
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But still?
Derivatives of bandlimited functions are also bandlimited! Look at red dots! Zeros are maxima and minima after differentiation.
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First hints
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Aharonov again
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Berry is more explicit
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Abstract
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Kempf is the best seller!
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Another example
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Spectrum of it, no hint of SO-s
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Research programme?
1. Super-resolution using super-oscillations. Already done – using nanohole patterns
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Antenna beamforming
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But sparse and random array?
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Transplanckian frequencies
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Superoscillating particles
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And finally…
Where are the super-oscillations here?
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Gateway to superoscillations:
PROFESSOR SIR MICHAEL VICTOR BERRY, FRS
http://michaelberryphysics.wordpress.com/