Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.atnf.csiro.au/research/pulsar/orange10/pdf/BillColesOrange2010.pdf Дата изменения: Wed Sep 29 13:38:32 2010 Дата индексирования: Sun Feb 13 23:35:42 2011 Кодировка: Поисковые слова: вечный календарь

Spectral Analysis of Pulsar Timing Residuals
Bill Coles, ATNF and UCSD and George Hobbs, ATNF

Wednesday, September 29, 2010


Why estimate spectra of pulsar timing residuals?

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curiosity about the timing noise, searching for orbiting objects, improve accuracy of timing model parameters, need to interpolate or extrapolate pulse phase, searching for evidence of the GWB, given time series x(ti), form Px(fk) = |DFT(x(ti))|2 /Nobs this works fine if x(ti) is regularly sampled, has stationary sampling error, and P(f) is approximately white. alas, none of these apply to most timing residuals.

What's the problem - we do this ever y day?

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Wednesday, September 29, 2010


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Leakage, Spectral Windows, and all of That!
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Wednesday, September 29, 2010


Steep Power-Law Spectra are a Problem
a quadratic has been removed from R(t)
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P(f)



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Wednesday, September 29, 2010


Add White Noise and Remove a Quadratic
01'+2&34-9627:32'

P(f)



f-1 + N

average of 100 simulations theoretical spectrum
01'+2&34-5'*672,-.,8/

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!#$

average of 100 simulations with quadratic removal

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!!

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Wednesday, September 29, 2010


Again with a Steeper Spectrum
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average Spectral Estimate LSQ of 100 simulations

P(f)



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+N

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no quadratic removal theoretical spectrum

Spectral Density (y3)

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prewhitened

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no pre-whitening

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removing a quadratic reduces the leakage, but doesn't eliminate it
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Wednesday, September 29, 2010


A Really Steep Spectrum - Cubic-like Residuals
average of 100 simulations all with quadratic removal LSQ Spectral Estimate
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P(f)



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no prewhitening

Spectral Density (y )

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prewhitened with first difference prewhitened with second difference theoretical spectrum

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second difference works on steep power-law
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Wednesday, September 29, 2010


A Really Steep Spectrum - with White Noise
average of 100 LSQulationsEstimate quadratic removal sim Spectral all with
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no prewhitening
Spectral Density (y3)

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prewhitened with first difference prewhitened with second difference N

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neither first nor second difference works on steep power-law with white noise theoretical spectrum

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Wednesday, September 29, 2010


The Cholesky Prewhitening Transformation
Let R be a column vector of residuals, then T> can be factored in the form Cov(R) = = U-1 < RRT> U-1T Cov(RPW) = < U = U-1 U UT U-1T = I i.e. RPW is white and has unit variance. This transformation is now available in TEMPO2.

Wednesday, September 29, 2010


How do we use this transformation?
Least Squares: R = M P + E where the errors E are not white -1 MP + U-1 E RPW = U = MPW P + EPW where the errors EPW are white Spectrum Analysis: The model is then A sin(2 f t) + B cos(2 f t) + C so P=[A, B, C]T. In this way spectrum analysis can be done for unequally spaced data with variable errors AND a steep power spectrum!
Wednesday, September 29, 2010


Simulated Residuals
Power Spectrum, P(f) = A
regular sampling, equal error bars
x 10 6
Timing Residual (s)
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-5 f

+W

irregular sampling, unequal error bars
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Timing Residual (s)
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4 2 0 -2 -4 0 500 1000 1500 2000 2500 3000 Time (d) 3500 4000 4500 5000

2 1 0 -1 -2 0 500 1000 1500 2000 2500 3000 Time (d) 3500 4000 4500 5000

Whitened Timing Residual

3 2 1 0 -1 -2 500 1000 1500 2000 2500 3000 Time (d) 3500 4000 4500 5000

Whitened Timing Residual

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top panels, original residuals bottom panels, prewhitened resduals
Wednesday, September 29, 2010


Cholesky Power Spectral Estimates
two simulations: P(f) = A f-5 + W and same plus two sine waves spectra are the average of 100 simulations regular sampling, equal error bars
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irregular sampling, unequal error bars
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green = theoretical spectrum; red = weighted least squares; blue = Cholesky least squares suppression of the first point in the red spectrum is caused by quadratic removal, prewhitening corrects this
Wednesday, September 29, 2010

fluctuations in the red and blue traces are not estimation error, they are sidelobes of the spectral window


Search for GWB
Here we used A = 3x10-15 and the J1713+0747 sampling, but reduced the error bars by a factor of 3. one realization
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average of 100 realizations

Timing Residual (s)

Spectral Density (y3)

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Whitened Timing Residual

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Wednesday, September 29, 2010


Covariance Matrices of Spectral Estimates P(fk)
The name of the game here is "degrees of freedom". Prewhitening makes the spectral estimates independent and gives us the DoF we need to detect the GWB

Wednesday, September 29, 2010


Summar y
Spectral leakage is a critical factor in spectrum analysis. It is particularly difficult to control with irregular sampling and variable sampling error. Fitting a timing model is an exercise in leakage control. The Cholesky transformation can eliminate leakage, make the spectral estimates independent, and restore the effects of quadratic removal on P(f=1/Tobs).

Wednesday, September 29, 2010