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! A  S C S C S C Phase 0 1 Wrong  : bad  , small A Phase 0 1 Correct  : good  , large A

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

3. Fitting a sinusoid to data Data: t i, x i ±  i, i=1,...N Model: Parameters: X 0, C, S,  Model is linear in X 0, 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 t i = i  t, i=1,..N with equal error bars: Hence define Nyquist frequency f N = 1/(2N  t) Note evenly spaced zeroes at frequency step  = 2  f = 2  /N  t = 2  f N /(N/2) x A(  )

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 X 0,  1,  2, A 1, A 2,  1  2

10. Data gaps and aliasing 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. Gap of length T gap

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