Fourier Transforms on Fourier Transforms on Simulated Pulsar Data Simulated Pulsar Data

Gamma-ray Large Gamma-ray Large Area Space Area Space TelescopeTelescope

Basic PulsarBasic Pulsar

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

Period of pulsar P0 = 0.1 s

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

Calculate the number of turns from the pulsar for the photon n

turnsn = F0 * t n

phasen = turnsn – int (turnsn)

h1Entries 495

Mean 0.2999

RMS 0.0001542

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

h1Entries 495

Mean 0.2999

RMS 0.0001542

Phase diagram

Phase

Nu

mb

er

of e

vent

s

Fourier TransformFourier Transform

Calculate the Fourier Transform

power = sqrt (( sin (2 phasen))2 + ( cos (2 phasen))2) /n

Frequency in Hz

Pow

er

Fourier Transform on DifferencesFourier Transform on Differences

Calculate the number between two photons (i,j , i<j )

turnsn = F0 * (tj – ti) where (tj – ti) < max_diff

phasen = turnsn – int (turnsn)

power = sqrt (( sin (2 phasen))2 + ( cos (2 phasen))2) /n

Frequency in Hz

Pow

er

max_diff = 5000s

Re and Im part of the FTRe and Im part of the FT

Power Re Im

Phase offset 0.3 * 2

Re and Im part of the FTRe and Im part of the FT

Power Re Im

Phase offset 0 * 2

Re and Im part of the FTRe and Im part of the FT

Power Re Im

Phase offset 0 * 2

Dependence on max_diffDependence on max_diff

Frequency in Hz

Pow

er

max_diff = 500 000s

0.001

0.01

0.1

100 1000 10000 100000 1000000

max_diff in s

Sig

nal

to

No

ise

Frequency in Hz

Pow

er

max_diff = 500s

Frequency in Hz

Pow

er

max_diff = 10 000s

The influence of max_diff on the Signal to noise ratio

Dependence on max_diffDependence on max_diffThe dependance of the signal with in the power spectrum 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