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.