TR2: Design of whitening filters for TR2
National Astronomical Observatory, TAMA project
Koji AraiApril 14, 2002

1Introduction

For gravitational wave detection by an interferometric detector, diﬀerential length changeof the two orthogonal arms is detected as an analog signal and stored by digital devices.
Photodetectors transform the intensity of light from the interferometer into photocurrent.
The output signal of the photodetector is stored in a storage system by converting theanalog signals to sequence of the digital numbers.
The whitening ﬁlters are important circuits to prevent from contaminating the acquired
signals by noise and overﬂow of the Analog-To-Digital converter. The previous whitening
ﬁlters for TAMA300 δL− signal have been designed and tested on the ﬁfth Data Taking(DT5, 2001/3/2∼8). These are called “WTF#1” and “WTF#2”. The design priciples andthe performances of the whitening ﬁlters have been described in the earlier documents [1, 2].
In the ongoing power-recycling experiment, the noise level of the interferometer wastemporarily degraded. Therefore the whitening ﬁlters for Test Run 2 (TR2, 2002/5/9∼10)
had to be modiﬁed in order not to hit the range of the HDAQ input. In this document, thecircuit of the modiﬁed ﬁlters, measured transfer functions, and the result of the transfer
function indentiﬁcation are described.

2Circuit

The circuit diagram of the whitening ﬁlter is shown in Figure. 1. With the current voltage
level of the detector output, many range-over of the HDAQ Channels were observed. areinput (±5V at the input of the HDAQ diﬀerential driver). Since there was no saturation
observed in the whitening ﬁlters themselves, only the gain of the output stage has been
reduced from ×51 to ×5.1. Since the input noise level of WTF#1 and WTF#2 are very√
good (2 ∼ 3nV Hz) and determined by the ﬁrst two stages, this modiﬁcation did notchanged the noise levels of these ﬁlters.

3Transfer function

After the modiﬁcation of the ﬁlters, the gain matching at 625Hz were perfomed again. Also,
the transfer funcitons of the ﬁlters were measured again.
In order to simplify the calibration process, the gains of the two whitening ﬁlters must
be adjusted to be same as far as possible. The diﬀerence of the gains measured by an FFT1

.

Figure 1: Circuit of the whitening ﬁlter actually used in TR22

spectrum analyzer are given by|HWTF#2 (625Hz)/HWTF#1 (625Hz)| = −0.0078 [dB](1)

where HWTF#j (f ) is a transfer function of WTF#j. This value corresponds to diﬀerenceof 0.09%.
In order to characterize the WTFs, the measured transfer functions were ﬁtted using
LISO [3]. The source code for the ﬁtting are shown below.

a0 0a1 0a2 0a3 0a4 0a5 1zero 4.3539468
pole 102.89671
pole 37.835922
pole 351.72098 692.57935m
pole 101.75027 1.3011752
pole 86.285517 534.77896m
pole 102.51971k
factor -6.4453806n#param a0: 1 10
#param a1: 1 10
#param a2: 1 10
#param a3: 1 10
#param a4: 1 10
#param a5: 1 10
param zero0:f 1 10
param pole0:f 20 1500
param pole1:f 20 1500
param pole2:f 200 1000
param pole2:q 0.1 10
param pole3:f 10 200
param pole3:q 0.1 10
param pole4:f 10 200
param pole4:q 0.1 10
param pole5:f 1e3 1e6param factor 1e-15 1e-6
fit llm_WTF1_tf.bod absdeg semi
rewrite samebetter#gnuterm cps # uncomment for making the eps filetfoutput abs:deg
freq log 10 9.891k 400 ### from data file3

A
f0 (Hz)f1 (Hz)f2 (Hz)f3 (Hz)q3
f4 (Hz)q4
f5 (Hz)q5
f6 (Hz)

WTF#1

WTF#2

−6.4453806n
4.3539468
37.835922
102.89671
351.72098
0.69257935
101.75027
1.3011752
86.285517
0.53477896
102.51971k−12.643198n
4.0060725
37.128395
35.607634
349.69349
0.69233685
101.07655
1.3119331
102.25272
0.53696242
221.72274k

Table 1: Identiﬁed poles and zeros from the measurements of the transfer functions.

Overall transfer functions were parametrized as the following:H(f ) = A1
(if )2(if )2
1
h(f, f0 ) if,h(f, f1 ) h(f, f2 ) h(f, f3 , q3 ) h(f, f4 , q4 ) h(f, f5 , q5 ) h(f, f6 )(2)

where a single pole (or zero) on the imaginary axis and a double pole pair are respectively
expressed byh(f, f0 ) = 1 + i

ff0f−h(f, f0 , q0 ) = 1 + i
f0 q0(3)
ff0

2.

(4)

Note that 1/h(f, f6 ) was added so as to represent phase delay caused by the opamps.
The result of the ﬁtting are shown in Table. 1. Figures. 2 and 3 show the calculated
and measured transfer functions of the whitening ﬁlters. Figure. 4 shows the errors in theﬁtting. The ﬁtting has errors smaller than ±1% and ±1 degree in magnitude and phasefrom 20Hz to 2kHz.

4Summary

The whitening ﬁlters for TR2 were built and tested. Two circuits has been adjusted to havea machted gain at 625Hz with precision of 0.09%. The transfer functions of the ﬁlters weremeasured and characterized by pole-zero models with accuracy of 1% in amplitude and 1
Degree in phase between 20Hz to 2kHz.

4410Magnitude [V/V]

210010101010-2-4-6Phase [Degree]

10010

1102103104510180120600-60-120-180
100101102103104510Figure 2: Measured (solid) and ﬁtted (dashed) transfer functions of WTF#1. Values from
10Hz to 10kHz were used for the ﬁtting.

410Magnitude [V/V]

2100101010

Phase [Degree]

10-2-4-610180120
600-60-120-180
100

01010

11101022101033101044510

510Frequency [Hz]Figure 3: Measured (solid) and ﬁtted (dashed) transfer functions of WTF#2. Values from
10Hz to 10kHz were used for the ﬁtting.

5

Phase difference (Meas.-Fit.) Ratio (Meas./Fit.)[Degree]

1.02
1.01
1.00
WTF#1
WTF#2

0.99
0.98110231010104210-1-2110231010104

Frequency [Hz]Figure 4: Comparison between the measured transfer functions and the ﬁtted ones. Upperplots shows the magnitude ratio of the measured one and the calculated one. Lower plotsshow the diﬀerence of their phases.

References[1] K. Arai, “DT5: Design of whitening ﬁlters for DT5”,
http://tamago.mtk.nao.ac.jp/
tama/recom/general lib/circuits/010221 DT5 WTF/whitening.pdf[2] K. Arai, “DT5: Design of whitening ﬁlters for DT5 part 2”,
http://tamago.mtk.nao.ac.jp/
tama/recom/general lib/circuits/010311 DT5 WTF/whitening2.pdf[3] G. Heinzel, “LISO - Program for Linear Simulation and Optimization of analog electronic citcuits”.

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