Tuning the hierarchical Tuning the hierarchical procedureprocedure

searching for the key where there is lightsearching for the key where there is light

S.Frasca – Potsdam, December 2006S.Frasca – Potsdam, December 2006

The hierarchical procedure to detect The hierarchical procedure to detect periodic sources (rough scheme)periodic sources (rough scheme)

Data set divisionData set division: the data are divided in a small number of sets (2, : the data are divided in a small number of sets (2, 3,...)3,...)

Coherent stepCoherent step: for each set the data are divided in chunks of the : for each set the data are divided in chunks of the length Tlength Tcohcoh and for each chunk a coherent analysis is done and for each chunk a coherent analysis is done

Incoherent stepIncoherent step: from the results of this first step, an incoherent step : from the results of this first step, an incoherent step follows (Hough or Radon transform), that takes all the data of the set. follows (Hough or Radon transform), that takes all the data of the set. This step is normally the most computationally heavy. Candidate This step is normally the most computationally heavy. Candidate source of level 1 are produced.source of level 1 are produced.

CoincidenceCoincidence: the candidates from 2 or more sets are searched for : the candidates from 2 or more sets are searched for coincidence within the parameters (frequency, spin-down, sky coincidence within the parameters (frequency, spin-down, sky position), obtaining candidate sources of level 2position), obtaining candidate sources of level 2

Refining the analysisRefining the analysis: this candidates are “followed up” with a new : this candidates are “followed up” with a new more refined coherent step followed by a new incoherent stepmore refined coherent step followed by a new incoherent step

Coherent step: limit on TCoherent step: limit on Tcohcoh to use just a to use just a

periodogram as the first coherent stepperiodogram as the first coherent step(as those produced by the SFTs) (as those produced by the SFTs)

1/ 24

max 0 2

1001.1 10

4coh EE

c HzT T T s

R

““Input” parameters for a hierarchical Input” parameters for a hierarchical procedureprocedure

Resources:Resources:

• The data: observation time TThe data: observation time Tobsobs (normally fragmented and with varying (normally fragmented and with varying antenna(s) sensitivity)antenna(s) sensitivity)

• Available computing powerAvailable computing power• Maximum number of candidates we can manageMaximum number of candidates we can manage

Target choiceTarget choice

• Sky areaSky area• Frequency rangeFrequency range• Spin-down interval(s)Spin-down interval(s)

Procedure tuningProcedure tuning

• First coherent step time TFirst coherent step time Tcohcoh

• Number of sets of data for separate hierarchical analyses in order to do Number of sets of data for separate hierarchical analyses in order to do candidate coincidencecandidate coincidence

““Output” parameters for a Output” parameters for a hierarchical procedurehierarchical procedure

SensitivitySensitivity

Needed computing power (to do the Needed computing power (to do the analysis in a “reasonable” time)analysis in a “reasonable” time)

Probability of “success”Probability of “success”

Now we discuss only a simple case of hierarchical procedure, neglecting the data set division problem and considering the case of using the Hough transform for the incoherent step.

The coherent time is taken variable (smaller or larger of T0).

Some parameters, like the sensitivity, will be normalized to the case of using Tcoh=T0.

Number of points in parameter Number of points in parameter space for the incoherent stepspace for the incoherent step

2coh

freq

TN

t

410DB freqN N

24sky DBN N

( )

min

2j

j obsSD freq

TN N

( )jtot freq sky SD

j

N N N N

Number of frequency bins

Freq. bins in the Doppler bandat the mid freq. of the band

Sky points

Spin-down points

Total number of points

4 4 4 48 15

min min

0.00025 1010 4.22 10

2000 4coh obs coh obs

tot

T T T T s yearsN

t s months t

Basic equations: needed computing Basic equations: needed computing powerpower

39

2

10op

cohobs

N GflopsCP T

T

2

3obs

op DB fl freq SDcoh

TN N k N N

T

Number of operations

for the incoherent step

Needed computing power

(to do the job in ½ of Tobs)

Basic equations: sensitivityBasic equations: sensitivity

obs

hODCR T

Sh

4)(1

( )41 1

OD obsCR CR

coh

Th h

T

Optimal detection nominalsensitivity

Hierarchical methodnominal sensitivity

1( ) 441 1OD obs

SNR SNR cohcoh

Th h T

T

Optimal detection (whole obs. time) nominal sensitivity:

Hierarchical methodnominal sensitivity

1

2( ) 26

1 22

4: 4.3 10

10 /nOD n

SNRobs obs

SS monthsh

T THz

Nominal sensitivity vs CP

1

121 !!!SNRh CP

To double the “nominal” sensitivity, we need 4048 times more CP

1 billion candidate sensitivity reduction1 billion candidate sensitivity reduction To reduce the number of candidates to a manageable number (e.g. 1 billion) To reduce the number of candidates to a manageable number (e.g. 1 billion)

we must put a threshold on the Hough map. This reduces the sensitivity, we must put a threshold on the Hough map. This reduces the sensitivity, respect to the “nominal” (SNR=1) by a factor given in figure:respect to the “nominal” (SNR=1) by a factor given in figure:

SD

4 months

Band 0 2000 Hz

10000 years

obsT

This farther reduces the dependence of the sensitivity on CP.

1G Sensitivity vs CP1G Sensitivity vs CP

The sensitivity is normalized to the case of using Tcoh=T0

In red there is the cost of only the incoherent step, sperimposed in blue there is the entire cost.

We see that to gain a factor 2 in sensitivity, we need to increase the CP of a factor of more than 10000.

Source distributionSource distributionReasonably the distribution of the amplitude of the sources Reasonably the distribution of the amplitude of the sources ((at at the detectorthe detector)), in the “threshold” range, can be believed as a , in the “threshold” range, can be believed as a power law of exponent m (for example m=2). So the probability to power law of exponent m (for example m=2). So the probability to have a source over a certain threshold ishave a source over a certain threshold is

1( 1) m

AP x

m x

and

1

1 2

2 1

mP x x

P x x

So a gain in sensitivity of the detection algorithm of the order of 2, that we paid 10000 times more in computing power, gives us only a factor 2 in “success” probability.

So what ?So what ?

It is better to enhance the sensitivity with It is better to enhance the sensitivity with longer Tlonger Tcohcoh or enlarge the possible targets or enlarge the possible targets

investigating more spin-down and more sky ?investigating more spin-down and more sky ?