Download Periodicity search - folding

Periodicity search:
folding
Giampiero Naletto
University of Padova (Italy)
Autumn Workshop
Principles of Multi-wavelength High Time Resolution Astrophysics
Pula, 11 October 2011
From the source…
After looking at an object like this…
The Crab nebula
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
2
… to the instrument…
… with an instrument like this…
Iqueye @ NTT
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
3
…through raw data…
… and obtaining data like these,…
Photon count rate collected from the Crab nebula pulsar,
“integrated” over 0.1 ms time bins.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
4
… to Crab pulsar (periodic) light curve
…how can we get this ?
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
5
Periodicity search: the “folding” techniques
Several are the possible techniques for finding a periodic feature “hidden” in
time signal dominated by counting statistics:
•
•
•
•
•
FFT
epoch folding
Rayleigh folding
Bayesian method
…
The choice of the best method depends on many different factors, as the signalto-noise ratio, the data length, the evenness or non-eveness of sampling, the
characteristics and the quality of the signal to be analyzed.
Here we are going todescribe two methods:
• epoch folding, which is a well consolidated and widely used technique, that
can be advantageously applied to all the cases of non evenly sampled data,
and/or there are data gaps, and with low SNR
• waterfall PCA, which is a novel method based on the Principal Component
Analysis of a waterfall diagram
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
6
Outline
•
•
•
•
•
•
•
Some mathematical background
Preliminary data analysis (FFT, waterfall diagrams, …)
Basic assumptions
Leahy approach
Larsson approach
Waterfall PCA method
Conclusions
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
7
Some mathematical background
•
Poisson distribution
•
χ2 test and χ2 distribution
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
8
Poisson distribution (I)
Let’s have a series of events occurring in time with a known average rate, and
let’s assume that the following conditions are verified:
- the probability to have an event E in an extremely small time interval dt is
proportional to the length of the time interval itself;
- an event is independent of any other preceding or following event
- the probability that more than one event happens in the time interval dt is
higher order infinitesimal with respect to dt.
Under these hypotheses, if the expected
number of events in a given time interval
(i.e., the expected rate) is m, (m = 1,2,3,…),
then the probability of having a
occurrences in the time interval is given
by the Poisson distribution:
ma e−m
P ( a , m) =
a!
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
9
Poisson distribution (II)
NB The Poisson distribution has been here presented considering “time”
intervals. However, the same distribution can be applied to other types of
phenomena, for example related to space intervals.
The Poisson distribution is actually an approximation of the binomial distribution.
It is commonly used when there are a few events on a large statistical base,
because of it allows easier calculations than the binomial distribution.
The Poisson distribution has the following main properties:
• The mean of the distribution is equal to expected number of occurrences:
µ=m
• The variance is equal to the mean of the distribution:
1 n
(xi − µ )2 = µ = m
σ =
n − 1 i =1
2
•
∑
For sufficiently large values of m (say m >1000), the Poisson distribution is an
excellent approximation of the normal (Gaussian) distribution with mean m
and variance m (standard deviation m ),
PPoisson(a, m) ≈ Pnormal(a, µ = m,σ 2 = m)
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
10
χ2 test: definition
The χ2 test is used for testing what scientists call the null hypothesis, which
states that there is no statistically significant difference between the expected
and observed result.
In a very general definition, given an observed data o and an expected data e,
we can define a statistic (a “statistic” is the measurement of an attribute of a set
of data, as could be for example its mean value) called χ2, by means of the
following equation:
χ =∑
2
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
(o − e)2
e
G. Naletto
Periodicity search: folding
11
χ2 test: applications (I)
Let’s make n measurements xi (i = 1,…,n) of a normally (Gaussian) distributed
variable x, known to have an average value µ, and a standard deviation equal to
σ. In this case χ2 is given by
2
n
χ2 = ∑
(xi − µ )
µ
i =1
As previously seen, for large datasets a Gaussian distribution is well
approximated by a Poisson distribution. This implies that µ = σ2, and that
n
χ =∑
2
i =1
(xi − µ )2
σ2
On the average, one expects that each measurement xi deviates from the mean
µ by an amount equal to the standard deviation σ, so that xi − µ ≈ σ .
Thus, it is reasonable to expect that
σ2
χ ≈∑ 2 =n
i =1 σ
n
2
So, generally, we expect χ2 to approximately equal n, the number of data points.
One purpose of χ2 is to compare observed with expected results and see if the
result is likely (if χ2 is "a lot" bigger than expected, something is wrong).
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
12
χ2 test: applications (II)
A different situation applies when χ2 is used in comparing observed counts of
particular cases to the expected counts (for example, you could be interested in
comparing the height of a sample population at steps of 10 cm with respect to
the total population steps). Given the observed counts of the particular cases, xi,
(in the previous example how many people in the sample population have an
height in the ranges [150-160] cm, [160-170] cm,…) and the expected counts of
the particular cases, µexp,i (in the same example, the percentage of the total
population with height in the ranges [150-160] cm, [160-170] cm,… multiplied by
the number of sample), χ2 is given by
n
χ2 = ∑
i =1
(xi − µexp,i )2
µ exp,i
where n is the number of cases.
Assuming that the Gaussian distribution is well approximated by a Poisson
2
distribution, we obtain that µ exp,i = σ exp,i , where σ exp,i is the standard deviation of
the i-th “global” case. At the end, we get
n
(xi − µexp,i )2
i =1
2
σ exp,
i
χ2 = ∑
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
13
χ2 distribution
If these experiments are repeated an infinite number of times, one could obtain
a sampling distribution for χ2 statistic. It can be found that the χ2 distribution is
given by the following probability density function:
(
)
( )
1
χ2
p χ ,k = k / 2
2 Γ( k / 2 )
2
k / 2 −1 − χ 2 / 2
e
where k = n−1 is the number of
degrees of freedom, and Γ is the
Gamma function: Γ(n) = (n−1)!.
In practice, this distribution gives
the probability of obtaining a
certain value for χ2 statistic, given
the number of degrees of freedom.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
14
χ2 distribution
The χ2 distribution is used, for example, in the common χ2 tests for goodness of
fit of an observed distribution to a theoretical one.
The χ2 distribution has the following properties:
• The mean of the distribution is equal to the number of degrees of freedom:
µχ2 = k (this is the value that is expected, on the average, when a χ2 test is
performed)
• The variance is equal to two times the number of degrees of freedom:
σ 2 = 2k
• When k ≥ 2, the maximum value for p(χ2,k) occurs when χ2 = k −2
• As the degrees of freedom increase, the χ2 curve approaches a normal
distribution
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
15
χ2 distribution
As the great majority of the statistical distributions, the χ2 distribution is
constructed so that the total area under the curve is equal to 1.
Since the distribution has the meaning of a probability density, the area under
the curve between 0 and a particular χ2 value is a cumulative probability
associated with that χ2 value. For example, the shaded area in the figure below
represents a cumulative probability associated with a χ2 statistic equal to A: that
is, it is the probability that the value of a χ2 statistic will fall between 0 and A.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
16
The epoch folding technique: preliminary analysis
The standard approach to epoch folding
consists in taking a data set of total length of
T seconds.
To investigate the possibility of having a
periodic feature hidden in the signal, usually
a FFT analysis is preliminary performed.
T
However, not always there are clear features
in the FFT, and anyway the accuracy of FFT
peaks is limited by the signal time bin.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
17
The epoch folding technique: “trial” search
The following step is to define a reasonable target period P* (well or roughly
known, or unknown), and divide the data modulo a “trial” period P1 “close” to P*.
The trial period P1 is divided into n time bins (often, the phase period is used
instead of time period, so the n phase bins range between 0 and 1) (NB with
present software tools is possible to have n as a non-integer number), and the
data modulo P1 are co-added into the time bins.
The same procedure is repeated for a certain number of equi-spaced trial
periods Pi ∈ [ Pmin, Pmax ] with Pmin < P* < Pmax , producing a number of folded
curves.
“Correct” trial period
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
“Wrong” trial period
G. Naletto
Periodicity search: folding
18
The “waterfall” diagram (I)
A very simple, but often unpractical, tool for finding the optimal period among
all those investigated is the so called “waterfall” diagram.
It consists in dividing the whole integration time T into N sub-intervals ∆T, and to
fold the signal with the just described method separately per each sub-interval.
Then, each folded curve is stacked as a row in a matrix. This matrix can finally
be represented as an image.
This method allows to “see” differences in the initial phase of the pulse.
This method can be applied when the searched periodic signal is high with
respect to the noise (SNR > 5; a good example is the Crab pulsar).
“Wrong” trial period
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
“Correct” trial period
G. Naletto
Periodicity search: folding
19
The “waterfall” diagram (II)
For long observation times, and for good
SNR, the waterfall diagram allows to see
phase drifts occurred during the observation.
Trial period “correct” at the middle of the observation
(Crab pulsar, 6000 s observation time. dP/dt ≈4.1⋅10−13 s/s).
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
20
The “waterfall” diagram (III)
Unfortunately, when the SNR is rather poor (SNR < 1), a condition that is often
verified, it is not possible to “see” any regular feature in the waterfalls.
“Correct” trial period for the PSR B0540-69.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
21
Epoch folding technique: basic assumption
It can be easily verified that in the absence of pulsations, or any secular trend,
(that is under the conditions of null hypothesis) the counts in each bin of the
folded curve at any given trial period are Poisson distributed with mean and
variance best estimated by the mean number of counts per bin.
Since the number of events in each time/phase bin is usually rather large
(>1000), we can say that the numbers of counts xi in the bins are normally
distributed with their mean equal to their variance. Then, we can say that the
statistic
n
S =∑
i =1
(xi − µexp )2
µ exp
is χ-squared with n−1 degrees of freedom. In practice, one expects that, under
the hypothesis of absence of pulsation, the value of this statistics is
S ≈ n −1
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
22
Epoch folding technique: basic assumption
If the calculation of S returns such an
approximate value, we can say that the
initial conditions are satisfied, and that
all the counts are “uniformly” distributed
around the average value.
However, if S>>n−1, the statistics is no
longer χ-squared , showing that a nonuniform feature (periodic, because of the
folding) is present in the collected data.
Epoch folding applied to the Vela pulsar.
Period scan: 100 µs; period step: 100 ns
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
23
Epoch folding technique: Leahy approach
The original approach used for calculating the χ2 value (Leahy, 1983) adopted
the following definitions:
• Total counting rate: R = xtot / T , where xtot is the total number of valid collected
events, and T is the total integration time
• Counting rate at the i-th bin: Ri = xi / Ti , where xi is the total number of counts
in the i-th bin, and Ti is the i-th bin integration time (that can differ bin-to-bin,
because of possible gaps in the collected data)
• Expected count rate (that can be different per each bin, if Ti varies):
µexp = µexp,i = RTi
Then
n
S =∑
(xi − µexp )2 = n (RiTi − RTi )2 =
i =1
where σ i2 =
µ exp
∑
i =1
RTi
n
(
Ti (Ri − R )2
Ri − R )2
∑ R =∑ σ2
i
i =1
i =1
n
R
Ti
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
24
Epoch folding technique: detection threshold
Given the χ2 density probability distribution p(χ2,k), function of the number k of
degrees of freedom, it follows that the probability that a calculated statistic S
exceeds by chance a defined level So is
(
∞
) ∫ p(χ , k )dχ
Q χ o2 = S o , k =
2
2
χ o2 = S o
In order to perform a period search, it has to be established a threshold level So
that has a small probability of being exceeded by chance: so, if an S level is
larger than the threshold, there is a large probability of having detected a
periodicity in the signal.
This threshold value is calculated assuming a
percent confidence level c by the following
equation:
(
)
1 − c / 100 = N p Q χ o2 = S o , n − 1
where Np is the total number of periods searched.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
25
Leahy approach applied to a sinusoidal function
It is possible to use a sinusoidal function to establish the method sensitivity
and to set upper limits. Assuming that the count rate is described by the function
r (t ) = ro [1 + A sin(2πt / P + φ )] ,
where P is the searched period, and suitably averaging to remove possible
effects of binning, it is found (Leahy, 1986) that
Ssin
A2
sin 2 (Tπδ / P')
= S noise +
xtot
f ( n ),
2
2
(Tπδ / P')
where P’ is the folding period and
f ( n) = 1 −
π2
3n
2
+2
π4
45n
4
− ...
The first term, Snoise , is the noise statistical contribution which follows a χ2
distribution with n−1 degrees of freedom with mean equal to n−1. This shows that
a large number n of bins increases the variance of the noise term.
The f(n) term is related to a percentage reduction in sensitivity due to binning: for
example, f(n=10) = 0.968. So, an “intermediate” value of n should be selected.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
26
χ2 statistic: analysis of the results
In case of strong signal, a very simple analysis
of the χ2 values as a function of the trial period
allows to estimate the optimal period.
In case the signal is weak, it is possible to
least-squares fit the measured χ2 values with
the analytical curve to find more accurately
period and amplitude as parameters of the fit.
By a Monte Carlo simulation it has been
shown (Leahy, 1987) that the relative errors for
period and amplitude are given by:
σ P / ∆P = 0.71[Ssin / (n − 1) − 1]−0.63
σ A / A = 0.46[Ssin / (n − 1) − 1]−0.65
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
27
Epoch folding technique: Larsson approach
A different way to determine the best period has been introduced by Larsson
(1996).
It always makes use of the epoch folding, but with a more sophisticated
approach.
This technique:
• uses a slightly different way of calculating the χ2 value;
• then it fits the calculated χ2 as a function of the trial period with a synthetic
sinusoidal one to estimate period and amplitude of the pulse;
• it folds the data with this best fit period, and makes a Fourier decomposition of
the obtained pulse profile;
• finally, suitably filtering the data, the process is iterated a few times to obtain
the final result.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
28
Larsson approach: χ2 calculation
A different way of calculating the χ2 value is done by calculating the statistic per
each bin, in the more general approach:
n
χ2 = ∑
i =1
(xi − µexp,i ) 2 = n (xi − µexp,i ) 2
∑
µ exp,i
i =1
2
σ exp,
i
This approach is generally used when the presence of the periodicity is already
consolidated, and a more detailed estimation of the periodicity parameter has to
be performed.
To obtain this better performance, the expected parameters used in the formula
are not those actually expected, but those statistically calculated. In practice, the
measured mean value µi per each bin will be used instead of the expected
value, and the measured variance σi2 of the data points will be used instead of
the expected one ( σ i = σ exp N i , where σexp is the standard deviation of the
unfolded time series, and Ni is the number of data points in the i-th bin):
n
(xi − µi ) 2
i =1
σ i2
χ =∑
2
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
29
Larsson approach: folding & accuracy determination
Once the χ2 function of the trial periods has been computed, the best period is
searched either looking at the χ2 maximum, or by fitting the data with the
synthetic profile (as previously described).
The period error estimation is done in this case by an analysis of the power
density spectra and least-squares fitted sinusoids. In case of a simple sinusoidal
signal, the analytical expression of the error on the obtained period P is
σ P2 =
2
6σ exp
π 2 NA2T 2
P4
where N is the total number of data points, A is the sinusoidal amplitude and T is
the total time length for the data.
Considering a non-sinusoidal oscillation, so that the time series can be
described as a set of Fourier components with frequencies kν , the previous
equation is generalized as
σ P2
=
2
6σ exp
P4
π 2 NT 2
∑k =1 k 2 Ak2
m
NB To calculate this error, it is necessary to know the Ak Fourier components
amplitudes.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
30
Larsson approach: pulse Fourier filtering
It can be shown that for Gaussian white noise the variance estimates, Vk , at the
Fourier frequencies are independently distributed with a probability distribution
proportional to a two degrees of freedom:
Vk
= χ 22
σ n2
This distribution is exponential, and gives a probability
 Vk
 z 1 −s / 2
p 2 > z  =
e
ds = e − z / 2
σn
 02
∫
for Vk / σ n to be larger than a certain value z.
So, after having transformed the pulse in its Fourier components, it is possible to
retain only those components with amplitudes above some significance level,
using this equation.
The “filtered” pulse is then sampled at the same time points as the original data,
and a new χ2 function is calculated and fitted to the χ2 values for the data.
2
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
31
Larsson approach: example
Input data:
portion of
time series
(simulation).
Left: original
signal; right:
noise added.
Calculated χ2 function over a range of trial periods
around the “true” one (continuous). Best fit (dash)
theoretical χ2 function for the case of a sinusoidal
oscillation.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
32
Larsson approach: example (cont.)
Pulse shape obtained by folding
the data at the best fit period.
Fourier pulse obtained by
fitting and filtering the Fourier
components of pulse shape.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
33
A new folding method: the waterfall PCA
A new method for determining the possible presence of a periodicity in a time
series of data has been recently developed (and not yet published…).
It is based too on the search of the best period over a series of trial ones
(around a peak frequency found usually by means of a FFT of the signal), but
the novelty is in the methodology adopted for finding the “best” period.
In this approach, a time series is divided in smaller contiguous time intervals, the
data are then folded with the trial periods, and are finally collected to form a
waterfall diagram.
In order to find the “vertical” pattern indicating the optimal period, the Principal
Component Analysis (PCA) technique is applied.
We applied this new method to some of the data we obtained with Iqueye from
the pulsars visible from NTT.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
34
The Principal Component Analysis (I)
Principal Components Analysis (PCA) is a way of identifying patterns in data,
and expressing the data in such a way as to highlight their similarities and
differences.
From a “geometrical” point of view, the main objective of PCA is to represent the
data in a “reference system” such to highlight their structure.
Graphical representation of a PCA
transformation in only two dimensions.
The variance of the data in the original
Cartesian space (x, y) is best captured
by the basis vectors v1 and v2 in a
rotated space
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
35
The Principal Component Analysis (II)
PCA is a mathematical procedure that uses an orthogonal transformation to
convert a set of observations of possibly correlated variables into a set of values
of uncorrelated variables called principal components.
This transformation is defined in such a way that the first principal component
has as high a variance as possible (that is, accounts for as much of the
variability in the data as possible), and each succeeding component in turn has
the highest variance possible under the constraint that it be orthogonal to
(uncorrelated with) the preceding components.
PCA is a mathematical tool presently used in many techniques of data analysis
in several research fields: from neuroscience to computer graphics. It is a
relatively simple non parametric method to extract useful information from data
of otherwise difficult interpretation.
Since it is mainly used as an image analysis tool, we thought to apply it to the
waterfall images: the “optimal” waterfall is the one where the eigenvalue
associated to the first principal component (PC) is the highest.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
36
Application to the Crab pulsar waterfall
Waterfall diagram for the Crab dataset: original data
(left) and projection over the first PC’s (right). Top:
correct period; bottom: wrong period.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
37
Application to the Crab pulsar: results
Eigenvalue of the first
PC: period search at
0.1 ns steps
Eigenvalue of the first PC:
period search at 100 ns steps
Obtained light curve: period accuracy
of the order of ≈10 ps (with respect to
Jodrell Bank data). Asiago telescope
observation data.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
38
Application to the Vela pulsar: results
Eigenvalue of the first PC:
period search at 100 ns steps
Eigenvalue of the first
PC: period search at
0.1 ns steps
Obtained light curve (top)
compared with previous
measurements.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
39
Comparison EF vs waterfall PCA: Vela pulsar
In order to compare the results obtained with the two methods, we performed a
Monte Carlo analysis. We synthesized a signal equivalent to the two weakest
pulsar observations, noise included. Then we ran 1000 iterations each case and
applied either standard Larsson epoch folding analysis or waterfall PCA to
estimate the relative errors.
•
•
•
Epoch Folding period (± 2σ error):
0.0893669833 ± 44*10−9 s
Waterfall PCA period (± 2σ error):
0.0893669943 ± 21*10−9 s
∆P = 11 ns
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
40
Comparison EF vs waterfall PCA: B0540-69
•
•
•
Epoch Folding period (± 2σ error):
0.0506499852 ± 81*10 −9 s
Waterfall PCA period(± 2σ error):
0.0506499918 ± 62*10 −9 s
∆P = 4 ns
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
41
Conclusions
We described in detail a standard very common method used for estimating the
presence of a (fast) periodicity in a signal and measuring the period by means of
“epoch folding technique”: since the basic assumptions to the actual algorithms
to implement.
This technique is very effective and has been tested (and used with pulsar data
acquired with Iqueye) for SNR down to 0.01.
Also a new technique has been described, which makes use of the PCA for
estimating the best period, which also returned optimal results.
In order to estimate the relative errors, a Monte Carlo analysis has been
performed for both cases, showing that the obtained results are perfectly
equivalent within the errors, even it the PCA technique seems to have a slightly
smaller estimated error.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
42
Essential bibliography
•
•
•
•
•
•
Larsson, S. (1996). Astronomy and Astrophysics Supplement, v.117, p.197201
Leahy, D.A., et al. (1983). Astrophys. J., 266, 160-170
Leahy, D.A., et al. (1983). Astrophys. J., 272, 256-258
Leahy, D.A. (1987). Astronomy and Astrophysics, 180, 275-277
Gregory, P.C.; Loredo, T.J. (1992). Astrophys. J., 398, 46- 168.
Gregory, P.C.; Loredo, T.J. (1996). Astrophys. J., 473, 1059.
Pula, 11 October 2011
Principles of Multi-Wavelength HTRA
G. Naletto
Periodicity search: folding
43