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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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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

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 .

- Iterate to convergence:
- Error bars:

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 = it, i=1,..N with equal error bars:
- Hence define Nyquist frequency fN = 1/(2Nt)

A()

Note evenly spaced zeroes

at frequency step

= 2f = 2/Nt = 2fN/(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 , 12

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 2t, cos 2t
- sin 3t, cos 3t
- etc

- The different sinusoids are orthogonal.
- Can fit any periodic function this way.

Square wave

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