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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

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

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

Sawtooth

Square wave

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