Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 1

Fourier Transform with differences on Fourier Transform with differences on Simulated and Real Pulsar Data Simulated and Real Pulsar Data Gamma-ray Large Gamma-ray Large

Area Space Area Space TelescopeTelescope

Marcus Ziegler

Bill Atwood

Brian Baughman

Robert Johnson

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 2

Fourier Transform on DifferencesFourier Transform on Differences

Calculate the time between two photons

tdiff = tj – ti where (tj – ti) < max_diff

phasen = F0 * tdiff

power = sqrt (( sin (2 phasen))2 +

( cos (2 phasen))2) /n

Frequency in Hz

Pow

er

max_diff = 5000s

Create series of evens distributed over 500 000s (5.7 days)

Period of pulsar P0 = 0.1 s (10 Hz)

Number of events ~ 500 (= bright Geminga like pulsar)

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 3

Dependence on max_diffDependence on max_diff

The dependence of the signal width on max_diff

-0 .001 -0.0005 0 0.0005 0.0010

0.2

0.4

0.6

0.8

1

FT+differnces

F + f in Hz0

Po

we

r

max_diff = 2000 s

-0. 001 -0.0005 0 0.0005 0.0010

0.2

0.4

0.6

0.8

1

FT+differnces

F + f in Hz0

Pow

er

max_diff = 10000 s

-0. 001 -0.0005 0 0.0005 0.0010

0.2

0.4

0.6

0.8

1

FT+differnces

F + f in Hz0

Pow

er

max_diff = 100000 s

Pulse width ~ 1/max_diff

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 4

Dependence on max_diffDependence on max_diff

Frequency in Hz

Pow

er

max_diff = 500 000s

Frequency in Hz

Pow

er

max_diff = 500s

Frequency in HzP

ower

max_diff = 10 000s

The influence of max_diff on the Signal to noise ratio

0

0.005

0.01

0.015

0.02

0.025

0.03

100 1000 10000 100000 1000000

max_diff in s

No

ise

RM

S

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 5

Adding a Gaussian distributionAdding a Gaussian distribution

Phase diagram

Frequency in Hz

Frequency in Hz

Frequency in Hz

Frequency in Hz

Frequency in Hz

Frequency in Hz

Normal FT

FT on differences

Sigma = P/5Sigma = P/20Sigma = P/100

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 6

Adding a Second PeakAdding a Second Peak

Frequency in Hz Frequency in Hz

Two peaks with 180 deg phase difference cancel

the contribution to the power spectrum.

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 7

Adding a period derivativeAdding a period derivative

max_diff = 100 000 s

F - F0

F - F0

Pd 1e-14

F - F0

F - F0

Pd 1e-13

F - F0

F - F0

Pd 3e-13

F - F0

F - F0

Pd 1e-12

F - F0

F - F0

Pd 3e-12

Phase diagram

Normal FT

FT on differences

Pd = 1E-14 1E-13 3E-13 1E-12 3E-12

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 8

Changing Max_diff with a PdChanging Max_diff with a Pd

F - F0

F - F0

max_diff 100 000s

Pd 3e-12

F - F0

F - F0

max_diff 20 000s

Pd 3e-12

F - F0

F - F0

max_diff 5 000s

Pd 3e-12

F - F0

F - F0

max_diff 1 000s

Pd 3e-12

F - F0

F - F0

max_diff 500s

Pd 3e-12

Phase diagram

Normal FT

FT on differences

Pd = 3E-12

max_diff = 100 000s 20 000s 5 000s 1 000s 500s

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 9

S/N vs. max_diffS/N vs. max_diffThe dependence of Signal/Noise on max_diff

0

10

20

30

40

50

60

70

80

100 1000 10000 100000 1000000

Sig

nal/N

oise

max_diff in s

P1 = 3E-12P1 = 3e-12 F1 = 3E-10 1/s2

max_diff = 1/ sqr(F1)max_diff = 57 000s

phas

e sh

ift

time

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 10

Looking at real pulsar dataLooking at real pulsar data

Brian Baughman wrote a Program to- read in the EGRET low level data- calculate the solar system baric center corrected photon arrival times- create new output file with corrected times

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 11

Looking at real pulsar dataLooking at real pulsar data

Select photons within a 2 deg radius around the source.

(no cut on the photon energy was made yet)

- 1200 photons were collected for this viewing period (about 14 days)

Ephemeris for GEMINGAEpoch 43946

F0 4.21775012277

F1 -0.1952379E-12

F2 0.28E-24

h3Entries 1200

Mean 0.468

RMS 0.3001

0 0.2 0.4 0.6 0.8 1

10

20

30

40

50

60

70

80

90h3

Entries 1200

Mean 0.468

RMS 0.3001

Phase

coun

ts

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 12

Ft on differencesFt on differences

Max_diff 10 000s

Po

we

r

4.2172 4.2178 4.21840

0.005

0.01

0.015

0.02

0.025

0.03

0.035 Power (f)

Po

we

r

Max_diff 100 000s

Frequency scan around GEMINGAs F0 with correction for F1

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 13

Search for GEMINGASearch for GEMINGA

Search for GEMINGA

no correction for F1, max_diff = 10 000s

Marcus Ziegler SCIPP/UCSC Pulsar Meeting 1/17/2006 14

Conclusion and OutlookConclusion and Outlook

The Fourier Transform on the differences is very simple and powerful

This method tolerates uncertainties in the ephemeris data

A very good “noise” separation can be achieved by tuning max_diff

Further optimize the algorithm

Look for pulsars in the EGRET data