1 / 12

Periodic signals

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)

jake
Télécharger la présentation

Periodic signals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. S S C C Periodograms • Repeat for a large number of  values • Plot A() vs  to get a periodogram: A() 

  3. 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 .

  4. Iterate to convergence: • Error bars:

  5. 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?

  6. 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

  7. 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)

  8. 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.

  9. Prewhitening • Can separate closely-spaced frequencies using pre-whitening : • Solution yields X0, 1 , 2 , A1 , A2 , 12

  10. 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.

  11. 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.

  12. Sawtooth Square wave

More Related