Searching for Periodic Signals in Time
Series Data
1. Least Squares Sine Fitting
2. Discrete Fourier Transform
3. Lomb-Scargle Periodogram
4. Pre-whitening of Data
5. Other techniques
• Phase Dispersion Minimization
• String Length
• Wavelets
Period Analysis
How do you know if you have a periodic signal in your data?
What is the period?
Try 16.3 minutes:
1. Least-squares Sine fitting
Fit a sine wave of the form:
V(t) = A·sin(t + ) + Constant
Where  = 2/P,  = phase shift
Best fit minimizes the 2:
2 = di –gi)2/N
di = data, gi = fit
Advantages of Least Sqaures sine fitting:
• Good for finding periods in relatively sparse data
Disadvantages of Least Sqaures sine fitting:
• Signal may not always be a sine wave (e.g. eccentric orbits)
• No assessement of false alarm probability (more later)
• Don‘t always trust your results
This is fake data of pure random noise with a  = 30 m/s. Lesson: poorly sampled noise 
almost always can give you a period, but it it not signficant
Any function can be fit as a sum of sine and cosines
FT() =  Xj (t) e–it
N0
j=1
A DFT gives you as a function of frequency the amplitude
(power) of each sine wave that is in the data
Power: Px() = | FTX()|2
1
N0
Px() =
1
N0
N0 = number of points
[( Xj cos tj + Xj sin tj) ( )]
2 2
Recall eit = cos t + i sint
X(t) is the time series
2. The Discrete Fourier Transform
The continous form of the Fourier transform:
F(s) =  f(x) e–ixs dx
f(x) = 1/2  F(s) eixs ds
eixs = cos(xs) + i sin (xs)
This is only done on paper, in the real world (computers) you always use 
a discrete Fourier transform (DFT)
The Fourier transform tells you the amplitude of sine
(cosine) components to a data (time, pixel, x,y, etc) string
Goal: Find what structure (peaks) are real, and what
are artifacts of sampling, or due to the presence of
noise
A pure sine wave is a delta function in Fourier space
t
P
Ao
FT

Ao
1/P
x 
A constant value is a delta function with zero frequency in Fourier space:
Always subtract off „dc“ level.
0
 f(u)(x–u)du = f * 
f(x):
(x):
Useful concept: Convolution
Useful concept: Convolution
(x-u)
a1
a2
g(x)
a3
a2
a3
a1
Convolution is a smoothing function
In Fourier space the convolution is just the product of
the two transforms:
Normal Space Fourier Space
f*g F  G
Convolution
DFT of a pure sine wave:
So why isn‘t it a ‐function?
width
Side lobes
It would be if we measured the blue line out to infinity:
But we measure the red points. Our sampling degrades the delta function and 
introduces sidelobes
In time space In Fourier Space
=
×
t
0
×
*
t
1/P
*
=
1/P
t
Window
Data
16 min period sampled regularly for 3 hours
The longer the data window, the narrower is the width of the sinc function 
window:

3
2 N1/2 T A
 = error of measurement
T = time span of your observations
A = amplitude of your signal
N = number of data points
Error in the period (frequency) of a peak in DFT:
A more realistic window
And data
Alias periods:
Undersampled periods appearing as another period
Alias periods:
P 
–1
false P 
–1
alias P 
–1
true
= +
Common Alias Periods:
P 
–1
false (1 day)
–1
P 
–1
true
= +
P 
–1
false
(29.53 d)
–1
P 
–1
true
= +
P 
–1
false (365.25 d)
–1
P 
–1
true
= +
day
month
year
Nyquist Frequency
If T is your sampling rate which corresponds to a frequency of fs, then 
signals with frequencies up to fs/2 can be unambiguously reconstructed. 
This is the Nyquist frequency, N:
N < fs/2
e.g. Suppose you observe a variable star once per night. Then the 
highest frequency you can determine in your data is 0.5 c/d = 2 
days
When you do a DFT on a sine wave with a period = 10, 
sampling = 1: 0=0.1, 1/t = 1
positive  negative  positive  negative 
Nyquist 
frequency
=0=0 =0
00
0
0
The effects of noise:
2 sine waves ampltudes of 100 and 50 m/s. Noise added at different levels
=10 m/s
=50 m/s
=100 m/s
=200 m/s
Real Peaks
3. Lomb-Scargle Periodograms
Power is a measure of the statistical significance of that
frequency (period):
1
2
Px() =
[Xj sin tj–]
2
j
Xj sin2 tj–
[Xj cos tj–]
2
j
Xj cos2 tj–
j
+
1
2
tan(2) = sin 2tj)/cos 2tj)j j
Scargle, Astrophysical Journal, 263, 835, 1982
Power: DFT versus Scargle
DFT Scargle
Amplitude (m/s)
Scargle Power
DFTs give you the amplitude of a periodic signal in the data. This does not change with more data. 
The Lomb‐Scargle power gives you the statistical significance of a period. The more data you have 
the more significant the detection is, thus the higher power with more data
N=40
N=100
Frequency (c/d)
N=15
False Alarm Probability (FAP)
The FAP is the probability that random noise will produce a peak with 
Lomb‐Scargle Power the same as your observed peak in a certain 
frequency range
FAP ≈ 1 – (1–e–P)N
Unknown period:
Where P = Scargle Power
N = number of independent 
frequencies in the frequency 
range of interest
Known period:
FAP ≈ e–P
In this case you have only one 
independent frequency
Scargle Power (significance) is increased by lower level of noise and/or more data points
False Alarm Probability (FAP)
FAP ≈ e–P
The probability that noise can produce the highest peak over a range ≈ 1 – (1–e–P)N
The probability that noise can produce the this peak exactly at this frequency =  e–P
Depth [%] 1.2
Duration [h] ?
Orbital period [d] 
Semi mayor axis[AU]
Number of detections 1
Target field No. 8
Host star K...(?)
Magnitude(B.E.S.T.) 11.25
Radius[Rsun] 0.65‐0.85 
Radius of planet ? [RJup] 0.71‐0.89?
DSS1/POSSI
Why is the FAP Impotant?
Example: A transit candidate from BEST
To confirm you need radial 
velocity measurements, but 
you do not have a period…
16 one hour observations made with the
16 one-hour observations made with the 2m coude echelle
Least Square sine fit yields of 2.69 days
Published (Dreizler et al. 2003) Radial Velocity Curve of the transiting planet 
OGLE 3
Wrong Phase (by 180 degrees) for a transiting planet!
Discovery of a rapidly oscillating Ap star with 16.3 min period
FAP ≈ 10–5
 CrB
Period = 11.5 min
FAP = 0.015
Period = 16.3 min
FAP = 10–5
Lesson: Do not believe any 
FAP < 0.01
My limit: < 0.001
Better to miss a real period 
than to declare a false one
Small FAP does not always mean  a 
real signal
How do you get the number of independent Frequencies?
First Approximation: Use the number of data points N0
Horne & Baliunas (1986, Astrophysical Journal, 302, 757):
Ni = –6.362 + 1.193 N0 + 0.00098N0
2 = number of independent frequencies
Determining FAP: To use the Scargle formula you need the number of 
independent frequencies.
Use Scargle FAP only as an estimate. A more valid determination of the FAP 
requires Monte Carlo Simulations:
Method 1:
1. Create random noise at the same level as your data
2. Sample the random noise in the same manner as your data
3. Calculate Scargle periodogram of noise and determine highest
peak in frequency range of interest
4. Repeat 1.000-100.000 times = Ntotal
5. Add the number of noise periodograms with power greater than
your data = Nnoise
6. FAP = Nnoise/Ntotal
Assumes Gaussian noise. What if your noise is not 
Gaussian, or has some unknown characteristics?
Use Scargle FAP only as an estimate. A more valid determination of the FAP 
requires Monte Carlo Simulations:
Method 2:
1. Randomly shuffle the measured values (velocity, light, etc)
keeping the times of your observations fixed
2. Calculate Scargle periodogram of random data and determine
highest peak in frequency range of interest
3. Reshuffle your data 1.000-100.000 times = Ntotal
4. Add the number of „random“ periodograms with power greater
than your data = Nnoise
5. FAP = Nnoise/Ntotal
Advantage: Uses the actual noise characteristics of your data
FAP comparisons 
Scargle formula using N as number of data points
Monte Carlo Simulation
Using formula and number of data points as your independent frequencies may 
overestimate FAP, but each case is different.
Number of measurements needed to detect a signal of a certain amplitude. The FAP of 
the detection is 0.001. The noise level is  = 5 m/s. Basically, the larger the 
measurement error the more measurements you need to detect a signal.
Amplitude = level of noise
1086421 Amplitude/error
N = 20
Lomb-Scargle Periodogram of 6 data points of a sine
wave:
Lots of alias periods and false alarm probability
(chance that it is due to noise) is 40%!
For small number of data points do not use Scargle,
sine fitting is best. But be cautious!!
Comparison of the 3 Period finding techniques
Least squares sine fitting: The best
fit period (frequency) has the
lowest 2
Discrete Fourier Transform: Gives
the power of each frequency that is
present in the data. Power is in
(m/s)2 or (m/s) for amplitude
Lomb-Scargle Periodogram: Gives
the power of each frequency that is
present in the data. Power is a
measure of statistical signficance
Amplitude(m/s)
4. Finding Multiple Periods in Data: Pre-whitening
What if you have multiple periods in your data? How do you find these and make 
sure that these are not due to alias effects of your sampling window.
Standard procedure: Pre‐whitening. Sequentially remove periods from the data 
until you reach the level of the noise
Prewhitening flow diagram:
Find highest peak in 
DFT/Scargle (Pi)
Fit sine wave to data 
and subtract fit
Calculate new DFT
Are more peaks 
above the level of 
the noise?
Save Pi
yes no Stop, publish all Pi
Noise level
Alias Peaks
False alarm probability ≈ 10–14
False alarm probability ≈ 0.24
Raw data
After removal of
dominant period
http://www.univie.ac.at/tops/Period04/
Useful program for pre‐whitening of time series data:
• Program picks highest peak, but this may be an 
alias
• Peaks may be due to noise. A FAP analysis will 
tell you this
Phase
Amplitude
5. Other Techniques: Phase Dispersion Minimization
Wrong Period
Correct Period
Choose a period and phase the data. Divide phased data into M bins and compute the 
standard deviation in each bin. If 2 is the variance of the time series data and s2 the total 
variance of the M bin samples, the correct period has a minimum value of 
= s2/2
See Stellingwerf, Astromomical Journal, 224, 953, 1978
PDM is suited to cases in which only a few observations are 
made of a limited period of time, especially if the light curves 
are highly non‐sinsuisoidal
In most cases PDM gives the same answer as DFT, Scargle periodograms. With 
enough data it should give the same answer
PDM Scargle
5. Other Techniques: String Length Method
Phase the data to a test period and minimize the distance between adjacent points
Wrong period longer string length
Lafler & Kinman, Astrophysical Journal Supplement, 11, 216, 1965
Dworetsky, Monthly Notices Astronomical Society, 203, 917, 1983
5. Other Techniques: Wavelet Analysis
This technique is ideal for finding signals that are aperiodic, noisy, 
intermittent, or transient.
Recent interest has been in transit detection in light curves
„You have to be careful that you do not fool 
yourself, and unfortunately, you are the easiest 
person in the world to fool“
Richard Feynmann