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Search for periodic sources
C. Palomba
• Signal duration much larger than typical observation
time
Possibility to reduce the false alarm probability to
negligible values
• Signal often (but not always) predictable and
• depending on the kind of source
• depending on a (large) number of (poorly known)
parameters
• Different approach in the analysis depending if we know
the source parameters (targeted search) or not (blind
search)
• Blind searches are computationally bound
• Different kinds of periodic sources can be considered:
• isolated NS
• NS in binary systems
• accreting NS (r-modes excitation)
• We will focuse attention on the first type and discuss
what changes in the other cases
Signal characterization - 1
• Doppler frequency modulation, due to the detector
motion and to the source motion
f (t ) : observed frequency
f 0 (t ) : intrinsic frequency


v  nˆ
v
: detector  source velocity
f (t )  f 0 (t )  f 0 (t )
c
nˆ : source direction
• Spin-down or spin-up
• Intrinsic frequency modulation, due to a companion, an
accretion disk or a wobble
• Amplitude modulation, due to the detector radiation
pattern and possibly to intrinsic effects (e.g. wobble)
• Glitches
Signal at the detector
h(t )  F (t ; ) h  (t )  F (t ; ) h (t )
F(t,ψ )
 detector beam pattern function
F(t,ψ ) 
h   A cos (t )
wave polarizations
h  A sin (t )

r  nˆ
 (t )  0  2 
(t  t0 
  E   S ) n1
c
n 0 ( n  1)!
 

r  r (t )  r (t0 )

f (n)
1
A  h0 (1  cos 2  )
2
A  h0 cos
phase evolution
amplitudes

  10kpc      
I
h0  1.05 1027  38 3 2 

  6 
10
kg

m
r
100
Hz


  10 


2
Data characterization
• Stationarity
• Gaussianity
• Impulsive noise
• Holes in the data
• Correct data timing
Detection of periodic signals
• If we had a monochromatic signal, the most natural
strategy would be that of looking for significant peaks in
the spectrum of data
• Due to the frequency modulation and spin-down the signal
power is spread in a large number of frequency bins
• If the signal frequency evolution is known we can
correct for the modulation (targeted search)
• Otherwise we need to perform a ‘rough’ search, select
some candidates and refine the analysis on them (blind
search -> hierarchical methods)
Targeted vs blind searches
• Targeted:
 possibility to apply optimal methods
 computationally not expensive
 upper limits
• Blind:
 oriented to detection
 computationally expensive
 non optimal methods
Targeted search for isolated NS - 1
• Assume sky position, emission frequency and spin-down
are known
• Amplitude, source inclination, polarization angle, initial
phase typically are unknown
• Allow to use optimal DA methods
• Nominal sensitivity hSNR1 
2S n ( f 0 )
Tobs
S n : one sided noise spectral density
Tobs : observatio n time
• For f.a.p=0.01 and f.d.p.=.1 the sensitivity, averaging over
source position and inclination and wave polarization, is
1/ 2
7

Sn ( f0 )
10
s
 26

h  11.4
~ 3 10 
Tobs
 Tobs 
Targeted search for isolated NS - 2
• Time-domain methods
• Re-sampling procedure: use a sampling frequency
proportional to the varying received frequency
Targeted search for isolated NS - 3
• Heterodyning procedure (Abbott et al., PRL94 181103, 2005):
j (t )
multiply the signal by e
• Allow to take into account complex phase evolution, e.g.
using data from radio telescopes, in a straightforward
way
• Frequency domain methods
• F-statistics (Jaranowski et al., PRD58 063001, 1998; Abbott et al. PRD69
082004, 2004) : based on the maximization of the likelihood
function
Targeted search for isolated NS - 4
• Can be used as coherent step in a hierarchical procedure
• Analytical signal (Astone et al., PRD65 022001, 2001) : start
from a set of short FFTs, compute the analytical signal,
correct for the frequency variations, compute the new
spectrum
• Can be used as coherent step in a hierarchical procedure
Blind search - 1
• Assume source position, frequency and spin-down are not
known (or only loosely constrained)
• Cannot be performed with optimal methods due to the
huge number of points in the source parameter space
T
N f  obs  2 1010
2t
Ndb  104 N f  2 106
Tobs  107 s
t  2.5 10  4 s
 min  
f
f
 10 4 yr
max
2
N sky  4Ndb
 5 1013
T
N sd  2 N f  obs
  min
j
 1.3 106 j  1
  
j2
 44
Ntot  N f  N sky   N sd  61031
j
Unreachable
computing power
Blind search - 2
• Hierarchical methods have been developed which strongly
reduce the needed computing power at the price of a small
sensitivity loss (Rome, Potsdam…)
• Typically based on alternation between coherent steps
and incoherent ones.
• Two kinds of incoherent steps are popular:
• stack-slide (Radon transform)
• Hough transform
• Both methods start from a collection of ‘short’ FFTs:
their length is such that a signal would be confined within
a frequency bin
Radon transform
1. Compute periodograms from short FFTs
2. Shift (slide) periodograms according to the frequency
evolution
f
t
3. Sum (stack) the periodograms
Hough transform - 1
• Parameter estimator of patterns in digital images
• Developed in the ’60 by P. Hough to analyze particle tracks
in bubble chamber images
• Example: find parameters of a straight line y=mx+q
y
q
(xi, yi)
q=yi-mxi
x
m
Hough transform - 2
• In our case the HT connects the time-frequency plane to
the source parameter space
• On the periodograms select peaks above a threshold
Hough transform - 3
• For each point in the peak-map we have a circle in the sky

v  nˆ
f (t )  f 0 (t )  f 0 (t )
c
Hough transform - 4
Hough transform - 5
• Slightly less sensitive than Radon (~12% in amplitude)
Hough vs. Radon
Sensitivity ratio
Threshold for peak selection
• More robust against non-stationarities and disturbances
• Computing power reduced by ~10
Hierarchical method outline
h-reconstructed data
SFDB
SFDB
peak map
peak map
hough transf.
hough transf.
candidates
coincidences
coherent
step
events
candidates
Short FFT database - 1
• Time
domain disturbances
• identification of events through an adaptive threshold
(on the CR);
• background estimated from the AR mean of the
absolute value and square of the samples;
• events removal.
(Astone et al., CQG22, S1197)
Short FFT database - 2
• Construction
of the short FFT database
• Maximum length
Tmax
c
1.1105
 TE 

s
2
4 RE f
f
TE : Earth rotation period
R E : Earth radius at the detector latitude
• 4 SFDB for frequency bands [0,31.25Hz],
[31.25Hz,125Hz], [125Hz,500Hz], [500Hz,2kHz]
• Estimation
of the average spectrum
• based on AR estimation;
• used for peaks selection and in the Hough transform;
C6 spectrum and average spectrum
Peak map
• Construction based on the ratio R between the spectrum
and its AR estimation;
• A threshold is set on R and all the local maxima above it
are selected as peaks;
• The threshold is chosen in order to maximize the CR on
the Hough map (see next)
• With thr=2.5 we have that ~1/12 of the frequency bins are
selected
Hough transform
• For each search frequency takes a Doppler band around
it and compute the hough map for all the possible spindown values
• Adaptive hough transform for non-stationarities
(Palomba et al., CQG22, S1255, 2005)
• Computationally heaviest part of the hierarchical analysis
• Efficient implementation needed
• Use of computing grids
Hough statistics
• In the case of pure noise the number count in a Hough map
follows a binomial distribution
N n
P(n |   0)   0 ( )(1  0 ( )) N n
n 
N: total number of spectra
0(q): peak selection probability (depending on the
threshold q)
• In presence of a signal
h02TFFT

2S n
N n
P(n |  )    (1   ) N n
n 
  0 (1   )
(e.g. Krishnan et al., PRD70,
082001)
• The choice of the threshold is done maximizing the
critical ratio CR:
CR 
N (  0 )
N0 (1  0 )
• The optimal choice would be   1.6
•   2.5 is still nearly optimal and reduce the prob. of
peak selection to ~1/12
• We select candidates putting a threshold on the number
count
• The threshold is chosen on the basis of the maximum
number of candidates we can manage (e.g. 109 )
Sensitivity - 1
• Loss factor for nominal sensitivity
• Nominal sensitivity
 Tobs 


 TFFT 
1
4
1
4
H

 10 s   f 
hSNR 1  5 1025  22 1/ 2 



10
Hz
T
500
Hz

 obs  

7
• Number of points in the parameter space
Ntot  N f  N sky   N sd
j
N tot
 T

 2.64 10  FFT 
 4000s 
14
4
3
 Tobs  10 s 

 7 
10
s

t



4
 10 4 yr 





• The number of ‘basic’ hough operations (increasing
by 1 the number count in a pixel) is
 T

Nbasic  Ntot  obs 
 12TFFT 
1
8
Sensitivity - 2
• The needed computing power is
CP 
2 10 9 N basicn fl
Tobs
Gflops
where n fl is the ‘number of equivalent floating point
operations’ needed for pixel increase and the analysis
time is assumed to be half of the observation time
• Computing power of the order of 1Tflops needed for
Tobs  107 s
  10 4 yr
109 candidates selected
• Larger CP available, reduce the spin-down age
Sensitivity - 3
• To compute the ‘effective’ sensitivity loss of the
hierarchical method we have to take into account
candidate selection
• For the optimal method, the loss due to the selection of
10^9 candidates is (exponential statistics)
 109 
  7
   ln 
 N tot 
• The Hough number count distribution is binomial
• Using the gaussian approximation, the threshold as a
function of the number of selected candidates is
N 
thr      erfc1  cand 
 Ntot 
Sensitivity - 4
  Np
hough map mean value
  Np(1  p)
hough map std. dev.
p  1 / 12
peak selection prob. in the peak map
• The 10^9 sensitivity reduction factor is ~2.2-2.8
depending on the frequency band
• The ‘effective’ loss respect to the optimal method is
 Tobs
(0.2  0.4)
 T FFT



1
4
2 – 4 depending on the freq. band
Sensitivity - 5
Tobs  107 s
  10 4 yr
  10 6
109 candidates selected
  10 7
  10 8
  10 9
Pulsars in binary systems - 1
• Orbital parameters must be taken into account (up to 5)
• Orbital Doppler shift may give more stringent limits to
the maximum length of FFTs
(from Dhurandhar & Vecchio, PRD63, 122001)
Pulsars in binary systems - 2
• Optimal methods can be applied only if the system
parameters are known or the uncertainties are small
• Otherwise, hierarchical non-optimal methods are
needed
Accreting NS (e.g. LMXB)
• Same parameters as in the previous case
• The frequency will change randomly due to fluctuations in
the rate of matter accretion
• In LMXB a clustering of frequencies is observed, though
the exact rotation frequency is not known
• Possibility to apply coherent methods over short time
period (few hours) (Vecchio, GWDAW10)
Summary of results
• Coherent analysis
• Explorer: 0.72Hz, 1 sd, all-sky, h90=1e-22
• LIGO S2: 28 isolated NS, h95>1.7e-24
• LIGO S2: Sco-X1, h95=2e-22
• Incoherent analysis:
• LIGO S2: all-sky, 1 sd, 200-400Hz, h95=4.4e-23
SPARE SLIDES
• Two thresholds enter into the game:

nthr
peaks selection
candidates selection
• False dismissal probability
 N  nthr 

  erfc

 N (1   ) 
  0 (1   )
h02TFFT

2S n