Periodic signals

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# Periodic signals - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Periodic signals' - jake

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Presentation Transcript

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)

Correct : good , large A

S

Phase

0

1

C

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

Sawtooth

Square wave