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Periodic signals PowerPoint PPT Presentation


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S. A. . C. Wrong  : bad   , small A. S. C. Phase. 0. 1. Periodic signals. To search a time series of data for a sinusoidal oscillation of unknown frequency  : “Fold” data on trial period P  Fit a function of the form:. Programming hint: Use phi=atan2(–S,C)

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Periodic signals

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Periodic signals l.jpg

S

A

C

Wrong : bad , small A

S

C

Phase

0

1

Periodic signals

  • To search a time series of data for a sinusoidal oscillation of unknown frequency :

  • “Fold” data on trial period P

  • Fit a function of the form:

Programming hint:

Use phi=atan2(–S,C)

if you care about which

quadrant  ends up in!

Correct : good , large A

S

Phase

0

1

C


Periodograms l.jpg

S

S

C

C

Periodograms

  • Repeat for a large number of  values

  • Plot A() vs  to get a periodogram:

A()


Fitting a sinusoid to data l.jpg

Fitting a sinusoid to data

  • Data: ti, xi ± i, i=1,...N

  • Model:

  • Parameters: X0, C, S, 

  • Model is linear in X0, C, S and nonlinear in 

  • Use an iterative  fit to linear parameters at a sequence of fixed trial .


Slide4 l.jpg

  • Iterate to convergence:

  • Error bars:


Periodogram of a finite data train l.jpg

Periodogram of a finite data train

  • Purely sinusoidal time variation sampled at N regularly spaced time intervals t:

  • The periodogram looks like this:

    • Note sidelobes and finite width of peak.

    • Why don’t we get a delta function?


Spectral leakage l.jpg

Spectral leakage

  • A pure sinusoid at frequency  “leaks” into adjacent frequencies due to finite duration of data train.

  • For the special case of evenly spaced data at times ti = it, i=1,..N with equal error bars:

  • Hence define Nyquist frequency fN = 1/(2Nt)

A()

Note evenly spaced zeroes

at frequency step

 = 2f = 2/Nt = 2fN/(N/2)

x


Two different frequencies l.jpg

Two different frequencies

  • Sum of two sinusoidswith different frequencies, amplitudes, phases:

  • Periodogram of this data train shows two superposed peaks:

  • (This is how Marcy et al separated out the signals from the 3 planets in the upsilon And system)


Closely spaced frequencies l.jpg

Closely spaced frequencies

  • Wave trains drift in and out of phase.

  • Constructive and destructive interference produces “beating” in the light curve.

  • Beat frequency B = |1 - 2|

  • Peaks overlap in periodogram.


Prewhitening l.jpg

Prewhitening

  • Can separate closely-spaced frequencies using pre-whitening :

  • Solution yields X0, 1 , 2 , A1 , A2 , 12


Data gaps and aliasing l.jpg

Data gaps and aliasing

Gap of length Tgap

  • How many cycles elapsed between two segments of data?

    • Cycle-count ambiguity

  • Periodogram has sidelobes spaced by

  • Sidelobes appear within a broader envelope determined by how well the period is defined by the fit to individual continuous segments.


Non sinusoidal waveforms l.jpg

Non-sinusoidal waveforms

  • Harmonics at  = k 0, k = 1, ... modify waveform.

  • Fit by including amplitudes for :

    • sin 2t, cos 2t

    • sin 3t, cos 3t

    • etc

  • The different sinusoids are orthogonal.

  • Can fit any periodic function this way.


Slide12 l.jpg

Sawtooth

Square wave


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