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
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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
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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
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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
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max_diff = 500s
Frequency in Hz
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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
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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
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max_diff = 100000 s
Pulse width ~ 1/max_diff