Calibration of TAMA300 in Time Domain Souichi TELADA, Daisuke TATSUMI, Tomomi AKUTSU, Masaki ANDO, Nobuyuki KANDA and the TAMA collaboration
Calibration of TAMA300in Time Domain
Souichi TELADA, Daisuke TATSUMI,Tomomi AKUTSU, Masaki ANDO,
Nobuyuki KANDAand
the TAMA collaboration
Contents
1. Overview of DAQ System of TAMA300.
2. Calibration.
3. Reconstruct data in Time domain.
4. Summary
DAQ System of TAMA300
Optical Configuration
The interferometer is controlled by feed-back servo.
Most important is L- servo which includes GW signals.
The feed-back signal of L- is acquired by DAQ system through a whitening filter.
Power Recycled Fabry-Perot Michelson Interferometer.
The optical signal is detected, through the electric filters and then feed back to the displacement of the mirrors of the both arms anti-symmetrically.
)( fG
)( fW
Coil-Magnet Actuator on Suspended Mirror
Each Mirror is suspended by double pendulum.
Upper Mass is dumped by Eddy-current dumping
4 Magnets on the Mirror and 4 Coils on the cage of the pendulum consist of Actuator.
Coil-Driver
Eddy-current dumping
CoilMagnet
The transfer function from input voltage of the coil driver to the displacement of the mirror is almost a transfer function of 2nd order low-pass filter with the cut off frequency of 1Hz and the Q-value of 3.
Reconstruct Data in Fourier Space
)( fG
)( fA
)( fW
Open Loop Transfer Function of the Servo
Transfer Function of Coil-Magnet Actuator
from Voltage to Displacement
Transfer Function of Whitening Filter
)(~
)(
)(1)(
)(
1
m300
1)(
~fV
fG
fGfA
fWfh
)( fG
)( fW
tV
Calibration Signal
Calibration Signal is injected at just before the Coil-driver with Sum-amp..
The Calibration Signal is sinusodial wave of 625 Hz which is generated by dividing sampling frequency (20kHz) of ADC by 32.
Signals Before and After the Sum-amp are acquired through the Whitening Filters.
)( fG
)( fW
)( fW
625Hz
A/DcalS
20kSPS
Calibration Signal
Extract 625 Hz components from both acquired signals and then divide Before Signal by After Signal. We get G(f =625Hz).
cal
Hz625
Hz625
1S
fG
fGfW
calHz6251
1S
fGfW
Hz625fG
Before Sum-amp.
After Sum-amp.
)( fG
)( fW
)( fW
625Hz
A/DcalS
20kSPS
Calibration Signal
Changeable parameters are two.One is the Optical gain. - Flat characterization for frequency.The other is Cavity pole of arm FP-cavity. - 1st order low pass filter (fc=500Hz).
The Optical gain is obtainedfrom the amplitude of G(f =625Hz).
The Cavity pole is obtainedfrom the phase of G(f =625Hz).
Actually
The Optical gain changed.The Cavity pole didn’t change.
)( fG
)( fW
)( fW
625Hz
A/DcalS
20kSPS
Reconstruct Data in Time domain
10 k
kjkk
kjkj OdIcO
We use Infinite Impulse Response (IIR) filters to reconstruct data in time domain.
tV thIIR
In order to analyze the observational data more generally, we need to produce the strain data h(t) in time domain.
About IIR Filter
Notation of Infinite Inverse Response Filter
jth Output data of IIR Filter.jO
jI jth Input data.
With various set of kk dc , , various filters can be constructed.
1st order Low Pass Filterwith cut off frequency of fc.
20
1
1
22
22
1
kd
d
k
Tf
Tf
c
c
20
1
1
1
1
221
220
kc
c
c
k
Tf
Tf
c
c
Ex.
freqency offCut :
Time Sampling:
cf
T
Reconstruct Data in Time domain
)( fG Open Loop Transfer Function of the Servo
)( fA Transfer Function of Coil-Magnet Actuator from Voltage to Displacement
)( fW Transfer Function of Whitening Filter
)(~
)(
)(1)(
)(
1
m300
1)(
~fV
fG
fGfA
fWfh
TAMA300 in Fourier Space
General Characterization of IIR Filter
Stable or Unstable. Not all factor set { ck, dk } is stable.
IIR filters emulate analog filters not completely - There are differences in higher frequency. -
In practice, there are problems with overflow and precision in calculation of computer.
We made special IIR Filter Set, which is good agreement with analog filter in our observation range. But in higher frequency range ( out of observation range ), it is far from analog filter.We produced Functions for calculation of closed loop.
Specialties for our IIR
Reconstruct Data in Time domain
About fG Open Loop Transfer Function of the Servo.
Openloop of Servo
Frq (Hz)
Am
plitud
e
-1 0 1 2 3 410 10 10 10 10 10
-4
-2
0
2
4
6
8
10
12
10
10
10
10
10
10
10
10
10 The unity gain frequency is about 1 kHz.(It is changeable to depend on the optical gain)
InvFeedbackSig of Servo
Frq (Hz)
Am
plitud
e
10 10 10 10 10 10-1 0 1 2 3 4
10
10
10
10
10
-1
0
1
2
3
Reconstruct Data in Time domain
About
fG
fG1Closed Loop Transfer Function of the Servo at Feed-back point.
Blue line is the transfer function of actual servo (Analog).Pink line is the transfer function of IIR filter.
There are differences between Analog transfer function and Digital transfer function at the higher frequency (near Nyquist frequency).It is characterization of IIR filter and impassible to emulate completely.
Reconstruct Data in Time domain
About fA Transfer Function of Coil-Magnet actuator from input Voltage of coil driver to Mirror displacement.
It is 2nd order low pass filter whose cut off frequency is 1Hz and Q-value is 3.
Openloop of Actuator
Fre (Hz)
Am
plitud
e
10 10 10 10 10 10-1 0 1 2 3 4
-17
-16
-15
-14
-13
-12
-11
-10
-9
-8
-7
1010101010101010101010
Reconstruct Data in Time domain
About fW Transfer Function of the Whitening Filter.
Openloop of Whitening Filter
Frq (Hz)
Am
plitud
e
-1 0 1 2 3 410 10 10 10 10 10-12
-10
-8
-6
-4
-2
0
2
4
10
10
10
10
10
10
10
10
10
InvOpenloop of Whitening Filter
Frq (Hz)
Am
plitud
e
-1 0 1 2 3 410 10 10 10 10 10
-4
-2
0
2
4
6
8
10
12
10
10
10
10
10
10
10
10
10
Reconstruct Data in Time domain
About fW
1Transfer Function of the Inverse Whitening Filter.
Blue line is the transfer function of Analog filter.Pink line is the transfer function of IIR filter.
Inverse Whitening Filter cannot be calculated.Because DC component and lower frequency components are infinite or extremely big. Use additional high pass filter about IIR filter.
Difference
Frq (Hz)
Am
plitud
e re
tio
1 2 3 410 10 10 10
0.940.960.981.001.021.041.06
Difference
Frq (Hz)
Pha
se (de
g.)
1 2 3 410 10 10 10-45
04590
135180225270315
Reconstruct Data in Time domainTotal difference between frequency model and time domain model.
The difference at the lower frequencies caused by additional high pass filter in inverse whitening filter.
Reconstruct Data in Time domain
If without the additional high pass filter. But impossible to calculate !!
Total difference between Frequency model and Time domain model.
Difference
Frq (Hz)
Am
plitud
e re
tio
10 10 10 101 2 3 40.940.960.981.001.021.041.06
Difference
-6-4-20246
Frq (Hz)
Pha
se (de
g.)
10 10 10 101 2 3 4
Reconstructed Data in Fourier Space & Time Domain
Sensitivity of DT9 TAMA300
Hz
1/rt
Hz
FourierTimeDomain
-24
-22
-20
-18
-16
-14
-12
-10
-8
-6
-4
1010101010101010101010
0 1 2 3 410 10 10 10 10
FFT
tV
fV~
fh~
Reconstruct FFT
tV
fh~
Reconstruct
th
Time Domain Signals
We could produce
We can also produce
thtV
tVth
It is useful for some analysis !!
Simulation signals are injected to the observation data in Time domain.Ex.
Show some waves as V(t).
Time Domain Signals
Chirp Signal of 1.4-1.4 Solar-Mass
Chirp 1.4-1.4 Solar-Mass
-3.E-17-2.E-17-1.E-170.E+001.E-172.E-173.E-17
0.000 0.020 0.040 0.060 0.080 0.100 0.120Time (sec.)
h(t)
Chirp 1.4-1.4 Solar-Mass
-2.E+00
-1.E+00
0.E+00
1.E+00
2.E+00
0.000 0.020 0.040 0.060 0.080 0.100 0.120Time (sec.)
V(t
)
Time Domain Signals
Dimmelmeier’s Burst catalogue
signal_A1B1G1_N
-1.5-1.0
-0.50.0
0.51.0
1.5
0.06 0.07 0.08 0.09 0.10 0.11 0.12Time (sec.)
h(t)
signal_A1B1G1_N
-3.0-2.0
-1.00.0
1.02.0
3.0
0.06 0.07 0.08 0.09 0.10 0.11 0.12Time (sec.)
V(t
)
x1E+16
Time Domain Signals
Dimmelmeier’s Burst catalogue
signal_A3B2G1_R
-3.0-2.0-1.00.01.02.03.0
0.08 0.09 0.10 0.11 0.12Time (sec.)
h(t)
signal_A3B2G1_R
-1.5-1.0-0.50.00.51.01.5
0.08 0.09 0.10 0.11 0.12Time (sec.)
V(t
)
Extract Hardware Signal Injection
In the Data Taking Run 8 (DT8), Some simulated GW signals were injected at just before the Coil driver with Sum-amp..
)( fG
)( fW
A/DGW Sig.
Extract Hardware Signal InjectionUpper graph is injected signal.Lower graph is reconstructed data.
Band Pass Filter
Zoom in
Curve Fit&
Subtract the Curve
Reconstruct h(t)
Injected Sig
-4.E-16
-2.E-16
0.E+00
2.E-16
4.E-16
8.30 8.35 8.40 8.45 8.50 8.55 8.60 8.65 8.70
Time (sec.)
h(t)
Extracted Sig
-1.E-10
-5.E-11
0.E+00
5.E-11
1.E-10
8.30 8.35 8.40 8.45 8.50 8.55 8.60 8.65 8.70
Time (sec.)
h(t)
Extract Hardware Signal InjectionUpper graph is injected signal with band pass filter.Lower graph is reconstructed data with band pass filter.
Band Pass Filter
Zoom in
Curve Fit&
Subtract the Curve
Reconstruct h(t)
Injected Sig
-4.E-16
-2.E-16
0.E+00
2.E-16
4.E-16
8.30 8.35 8.40 8.45 8.50 8.55 8.60 8.65 8.70
Time (sec.)
h(t)
Extracted Sig
-2.E-14-1.E-14-5.E-150.E+005.E-151.E-14
8.30 8.35 8.40 8.45 8.50 8.55 8.60 8.65 8.70
Time (sec.)
h(t)
Extract Hardware Signal InjectionZoom in time scale at the point of injection.You can see small structure on the reconstructed curve.
Band Pass Filter
Zoom in
Curve Fit&
Subtract the Curve
Reconstruct h(t)
Injected Sig
-4.E-16
-2.E-16
0.E+00
2.E-16
4.E-16
8.460 8.465 8.470 8.475 8.480
Time (sec.)
h(t)
Extracted Sig
-1.E-14
-5.E-15
0.E+00
5.E-15
1.E-14
8.460 8.465 8.470 8.475 8.480
Time (sec.)
h(t)
Extract Hardware Signal Injection
Remaining signal after subtracting the curve.
Band Pass Filter
Zoom in
Curve Fit&
Subtract the Curve
Reconstruct h(t)
Injected Sig
-4.E-16
-2.E-16
0.E+00
2.E-16
4.E-16
8.460 8.465 8.470 8.475 8.480
Time (sec.)
h(t)
Extracted Sig
-4.E-16
-2.E-16
0.E+00
2.E-16
4.E-16
8.460 8.465 8.470 8.475 8.480
Time (sec.)
h(t)
Similar or Not Similar !?
Summary
We could reconstruct from V(t) to h(t) by using IIR filter.
We could also produce from h(t) to V(t) by using IIR filter.
We could extract the hard ware injection signals.
Future
Do some analysis in Time Domain.